![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Is there a wikipedia policy for where pronunciations should appear? I dislike the redundancy of placing the pronunciation of Riemann both at this article and also at Bernhard Riemann. If a user wants to know the pronunciation they can simply follow the name link to find it. - Gauge 04:44, 2 September 2005 (UTC)
I'd like to hold a survey regarding the article
Riemann zeta function, to help determine its general comprehensibility and identify areas where it may be incomplete. Please indicate your perceptions of the article below, and feel free to expand the survey or article as you see fit.
‣ᓛᖁ
ᑐ
21:07, 9 September 2005 (UTC)
Do you currently understand this article?
Yes
No
Comment
If not, do you feel you could understand it after following its internal links?
Yes
No
The article's lead section states the Riemann zeta function is "of paramount importance in number theory". From reading the article, do you understand why this function is important?
Yes
No
I removed the following text:
I couldn't understand what it was trying to say. Its trying to describe some dirichlet series maybe??. linas 00:46, 21 December 2005 (UTC)
I found this section very helpful for understanding the connection between the Zeta function and prime numbers. BringCocaColaBack 11:29, 13 January 2006 (UTC)
Following question from an anon contributor moved from the Globallly convergent series section on the article page. Gandalf61 08:35, 20 April 2006 (UTC)
That question applies more properly to the formula in the section above Globally convergent series, called Series expansions, which contains the formula:
What's that 'x' doing there? - GTBacchus( talk) 13:38, 20 April 2006 (UTC)
In the series expansion section, it's written that "Another series development valid for the entire complex plane is.." I can't figure out what the variable 'x' is supposed to be in the expansion that follows. x=s? —The preceding unsigned comment was added by 12.208.117.177 ( talk • contribs) .
Is this function called the zeta-function or the zeta function? The article uses both. toad ( t) 12:10, 10 February 2006 (UTC)
Zeta-function refers to all zeta function is general. But in this case its just Riemann zeta function. -- He Who Is[ Talk ] 01:59, 29 June 2006 (UTC)
No graphic of the graph in the complex plane? Surely the article should include one or perhaps three; real part, imaginary part and absolute value, as is the standard on Mathworld. Soo 14:17, 16 July 2006 (UTC)
The article titled 42 (number) says:
I don't know what this means. Here's a guess:
I'm accustomed to the definition of momnets of probability measures; if ζ were a probability density function then the integral above would be the third moment of the corresponding probability distribution. But ζ is negative in some places, and from the way ζ(s) blows up at s = 1 it seems we'd have to be thinking of a Cauchy principal value or something like that.
Can someone make the article's statement clearer? Michael Hardy 17:52, 5 July 2006 (UTC)
When k=3 you get the "sixth moment". The constant 42 comes up as a scaling factor in a conjecture by Conrey & Ghosh for the leading order term of this integral. .—Preceding unsigned comment added by 74.74.128.91 ( talk • contribs) *****
“ | So, feel free to post this comment for me:
I don't know anything about the "3rd moment of the Riemann zeta function", but perhaps what's meant is the 3rd moment of the distribution of spacings between zeroes of the Riemann zeta function. There's a lot of evidence relating the distribution of these spacings to the distribution of spaces between eigenvalues of a large random self-adjoint matrix. For lots more, try this: http://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htm and for general connections between the Riemann zeta function and quantum mechanics, try: http://www.maths.ex.ac.uk/~mwatkins/zeta/physics1.htm Best, jb |
” |
I stored in commons a graph of zeta (x) with -20 < x < 10.
see
Perhaps it can go into the article
-- Brf 10:05, 31 August 2006 (UTC)
Can someone add an explanation about the critical strip? This term's definition is nowhere to be found in Wikipedia. Thanks! By the way, the whole article is fine and readable; everybody with a college degree will understand at least the basics. Hugo Dufort 08:28, 14 November 2006 (UTC)
I added an expanded form of the zeta function I got from Marcus de Sautoy's "Music of the Primes", because I think it helps to visualise exactly what the product is about. Comments? DavidHouse 21:21, 26 December 2006 (UTC)
In a technical report entitled "[ http://cswww.essex.ac.uk/technical-reports/2005/csm-442.pdf An elementary formulation of Riemann’s Zeta function]", myself (Riccardo Poli) and Bill Langdon provided a very simple proof that, for , Riemann's Zeta function can be written as where .
We are not experts in number theory, but we have searched widely and also asked several mathematicians: it appears that our rewrite is new. These people tell us that this is useful formulation. So, we were wondering whether it would make sense to include it in the article.
The paper mentioned above has now been published in arXiv.org in the Mathematics History and Overview section (math.HO/0701160). Perhaps the result could now be included in the article?
I want to add a personal support for the writers of the article. Even though the article necessarily is largely technical, the lead-in paragraph adequately establishes the backround of the function for laymen. -- Cimon Avaro; on a pogostick. 08:51, 12 February 2007 (UTC)
209.226.117.54 16:08, 17 February 2007 (UTC) Jacques Gélinas
Browsing through the talk page, it seemed to me that there had been quite a few complaints about the definition of from people not comfortable with analytic functions, or perhaps, with mathematics in general. I can definitely confirm that the so-called "introduction" to this article is too terse to be of any use. Other articles, such as Riemann hypothesis are much better in this regard. So this is certainly something that needs to be dealt with. For now, I have expanded the definition a bit, it remains a rigorous mathematical definition, so it's unclear to me how much happier would non-mathematicians be with it. Hopefully, it is somewhat gentler to those who are unsure about all the symbols and unfamiliar terminology, although to experts on Riemann zeta function it may appear to be perhaps a little too easy. I do want to point out that Enrico Bombieri, in the description of the Riemann hypothesis in the Millenium Prize book starts by mentioning that the Dirichlet series for is defined only for large , and then explains that it is analytically continued. I definitely feel that it's not something to be taken lightly, especially since analytic continuation of general Artin L-functions is still unknown, and of course, by no means obvious! Perhaps, it would make sense to expand the definition even more, it is a judgement call (or an editorial decision), so I would wait to hear the reaction.
Incidentally, I think that in line mathematical formulas do not look very good in this case, but since it's a highly emotional issue for at least some users, I tried to preserve them. Arcfrk 06:04, 10 March 2007 (UTC)
For those who get lost in this Wikipedia article, I have found the following link to be the clearest description in layman's terms. Those who authored and are maintaining this Wiki article may want to read this to understand how a complicated math concept can be described in normal conversational English:
http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php Overlook1977
I glanced at if for a second or two. Is there anything in this link that actually says what the zeta function is?? If not, I certainly wouldn't say it's clear to either lay persons or anyone else. Michael Hardy 20:45, 16 April 2007 (UTC)
This article needs more information on how the zeta function is connected to prime numbers. My understanding of this function and the Riemann hypothesis is that this function and prime numbers are very deeply connected, but I can not find on Wikipedia or anywhere else an explanation as to why. Sloverlord 12:36, 17 May 2007 (UTC)
The Mellin Tranform section has . Could "where is the Gamma function" be added?
Is this vandalism? At the bottom of the page there's a link to a .gz file. Even so, it should be reworded. '''T'''''o''__m__ 17:34, 7 November 2007 (UTC)
I thought the Riemann zeta function referred to complex numbers - what is the justification for including the harmonic series, the Basel equality, and other zeta functions that don't involve complex numbers here? 24.61.112.3 15:12, 2 December 2007 (UTC)
The Specific Values section contains examples of series for natural numbers only. Don't these belong in an article about Zeta functions "in general"? The sudden transition from discussing natural number constants to the "Zeta zeros" - and hence complex numbers of the Riemann Zeta function - is bizarre and misleading to say the least. Michaelmross 14:53, 21 January 2007 (UTC)
I find it sad that a well-intended comment is misunderstood and then labeled as misleading. I didn't say the topic was about "zeta functions in general" - I clearly suggested that natural number series might belong in a *topic about zeta functions in general*. Because what I'm saying about this section of the article is that it goes from something general about natural numbers that a layperson like myself can understand to something very specific concerning complex numbers and the zeta zeros. And to me - a non-mathematician - this is very confusing. I would like to see a layperson's distinction between a generic zeta function and a Riemann zeta function. This will be my last comment on the matter, so there's no need to flame me any further. Michaelmross 20:12, 22 January 2007 (UTC)
The approximation by Gergő Nemes in "The functional equation" does not work when
In other words, it mostly makes sense as an approximation of zeta for large negative values. For these values, the derivation is simply plugging in Stirling's formula into the functional equation, which only serves to complicate the expression. —Preceding unsigned comment added by 64.3.169.42 ( talk) 21:15, 3 January 2008 (UTC)
I don't know how I missed them before, but the sections purporting to evaluate zeta(2) and zeta(4) can, perhaps, be best characterized as unencyclopaedic garbage, the word that they seem to be using quite a bit. "Zeta constant" already offers a thorough coverage of special values of zeta. The "proofs" themselves are about as tortured as one can imagine, and blatantly fail WP:NOT. I got rid of them. Arcfrk ( talk) 03:45, 3 March 2008 (UTC)
What is the point of this graph in the middle of the article, when the article does not mention it even once? T.Stokke ( talk) 21:05, 1 March 2008 (UTC)
Years ago I did a course in complex analysis. I got a bare pass, which doesn't seem sufficient to understand the section on trivial zeros. Could someone explain explicitly why the negative evens are zeros? RayJohnstone ( talk) 15:50, 28 March 2008 (UTC) —Preceding unsigned comment added by RayJohnstone ( talk • contribs) 15:47, 28 March 2008 (UTC)
As I understand it: zeta(0) = 1/1^0 + 1/2^0 + 1/3^0... = 1 + 1 + 1... = infinity != -1/2
Where do I wrong? 84.108.198.129 ( talk) 23:35, 2 September 2008 (UTC)
Shouldn't ζ(0)=infinity?
05:47, 19 April 2008 (UTC)
How does ζ(0)=-1/2? —Preced PMajer ( talk) 16:21, 8 September 2008 (UTC)ing unsigned comment added by Anon126 ( talk • contribs) 05:59, 15 May 2008 (UTC)
This article shows a kind of a common problem of mathematical articles. It seems that a lot of people are interested in the Riemann zeta function, without beeing so interested in understanding the elementary underlying facts, such as, what is a complex number, or what is the sum of a series, etc. This makes sense, of course, for the sake of a general information (in the same way, I am interested in music even though I cannot play). But in this case one should not complain if an article looks too technical, or if things are not made easier than possible, and one should not be surprised if he or she can't go beyond a row picture of the facts, because it is natural. The point is simply that a complete explanation with all details, e.g. about the zeta function, would require a good part of an elementary course of complex variables, what is beyond the scope of thes articles. On the other hand, there is a minority, made of users with a mathematical culture, that go to a math article looking for a more technical, precise and usually very short information (tipically, in areas of mathematics different from their own). For them wikipedia has become quite a useful tool, so I think it should be a mistake to keep an article to the level of the mean users. Maybe a good thing should be to keep disjoint the general and the specialistic part... PMajer ( talk) 17:48, 8 September 2008 (UTC)
zeta(-2) = 1/1^-2 + 1/2^-2 + 1/3^-2 + ... = 1 + 2^2 + 3^2 + ... = infinity !=0
Someone is wrong - me, or the rest of the world. Who? (and why?) 84.108.198.129 ( talk) 23:43, 2 September 2008 (UTC)
How can a subset of a straight line have a fractal dimension greater than 1? Or does it mean that the Riemann hypothesis is false?-- Yecril ( talk) 23:08, 6 October 2008 (UTC)
Does anyone agree that the "Applications" section is a little silly? The connection to physics is sparse, and no connection to the use of the Riemann-Zeta function is actually motivated by it (would you actually need to use it to show that sum diverges? Surely you want to show something else). Njcnjc ( talk) 03:03, 21 October 2008 (UTC)
Can anyone elaborate on the "prime numbers" section? It says "this is a consequence..." without giving even a hint or flavour of why. -- Doradus 21:25, Sep 17, 2004 (UTC)
I've added what I believe to be a standard treatment of why the two expansions are equivalent. Do any real mathematicians want to do a better job? -- The Anome 23:27, 17 Sep 2004 (UTC)
I've turned the "vigorous handwaving" into an actual proof and have made a number of further changes and additions. Gene Ward Smith 08:42, 29 Jan 2005 (UTC)
There will always be a long standing question on the readability of this article; I feel the need to put my two cents in. I came to the page after reading probably 20-30 pages on abstract math, no formal training in mathematics beyond Differential Equations and Vector Calculus in college. I found it to be sufficiently readable that I was able to "get the gist" of what the Riemann Zeta function is and why it is important. No, I could not do a single calculation regarding it, but from what I gather, the masters of the topic have trouble doing more than the most basic proofs regarding the topic - so is it such a surprise that I have trouble?
Fundamentally, Riemann Zeta is not simple. Not by a long shot. But that doesn't mean its unimportant; it doesn't mean that mathematicians who can leverage the information should be denied such a tool. There's still truly trivial content on wikipedia; lets leave the profound content (like Riemann Zeta) in place. 199.46.245.230 ( talk) —Preceding undated comment was added at 17:04, 26 November 2008 (UTC).
I don't feel there is adequate discourse of using the reciprocal form. Specifically, readers will wonder why it is used when the sum of positive integers from 1 thru N is well established as N*(N+1)/2. Is the manipulation to be able to apply specialized analysis forms or ??-- Billymac00 ( talk) 03:17, 27 January 2009 (UTC)
The trivial zeros do not seem to yield zeros:
For example: ;
which yields infinity.
I will appreciate if someone lets me know where I am mistaken.
128.97.68.15 ( talk) 17:25, 1 February 2008 (UTC)
Based on a false premise, don't you think? Fergananim ( talk) 20:51, 5 February 2009 (UTC)
The following snippet appears to be meaningless. What is s? What is the significance of the variable x? A broader question is: is this formula significant enough to be in the article? If so, could somebody please correct it and give it enough context that it becomes meaningful and manifestly notable for inclusion in the article. Sławomir Biały ( talk) 03:05, 9 July 2009 (UTC)
here dN(x)/dx is just the derivative (as a distribution) of the number of zeros on the critical strip 0 < Re(s) < 1. A proof of this can be found on a work by Guo (see references).
- the derivatives are all respect to 'x'
- The formula is an expansion of the logarithm of on the critical strip, as you can see it reproduces all the poles of
- is just the expression , whith 'gamma' being a sum over the imaginary part of the zeros -- Karl-H ( talk) 09:45, 9 July 2009 (UTC)
this formula is just an expansion of the logarithmic derivative of Zeta function on critical strip, it is interesting (just my opinion) since it involves a sum over the poles of Riemann zeta, i think i found in the paper refereed before or in another context in papers talking about
Gutzwiller Trace formula and Riemann Hypothesis, perhaps this formula would suit better into an article about Riemann Hypothesis or Gutzwiller trace, as you wishes --
Karl-H (
talk)
20:19, 10 July 2009 (UTC)
If so, it wouldn't have had any zero on critical strip Re(s)=1/2, neither any trivial zero.
Clearly, '>' should be replaced by '<'.
See paragraph 1. —Preceding unsigned comment added by 151.53.136.190 ( talk) 21:02, 12 August 2009 (UTC)
The article says that all negative even integers are trivial zeroes, but the value of the zeta function is infinite for them, not 0.
1+4+9+16+..., 1+16+81+256+..., etc. all diverge to infinity. —Preceding unsigned comment added by 75.28.53.84 ( talk) 14:51, 15 August 2009 (UTC)
Given the above endless confusion about the definition of the series, the definition of the function defined by the series and the definition of the zeta function itself, I've changed the wording of the definition to something that I hope is slightly clearer. I attempted to make clear what people on the talk page were saying: that the zeta function is the analytic continuation of the function defined by the series given. Please take note that I have NO UNDERSTANDING AT ALL of analytic continuations. I merely changed the definition to emphasize what has been repeatedly said on this talk page. If I got it wrong, please feel free to correct it. mkehrt ( talk) 09:03, 7 October 2009 (UTC)
The Euler product formula is from about a hundred years before Riemann was born. Is it known who was the first to consider this function? It seems Chebyshev's work connecting prime numbers to this function was before Riemann. Why is it called the Riemann zeta function? Was Riemann the first to define it for complex numbers? A section on history would be nice. Regards, Shreevatsa ( talk) 14:57, 18 March 2009 (UTC)
The article states that the function was named after Riemann because "he introduced it"... Is that the case, or is it as Shreevatsa says, that he was the first to define it for complex numbers? —Preceding unsigned comment added by 24.12.13.8 ( talk) 19:07, 15 October 2009 (UTC)
The argument: (harmonic diverges) -> (euler's formula predicts infinitely many primes) is flawed since Euler's formula depends on there being infinitely many primes to sieve the infinite sum of the riemman-zeta function to '1' —Preceding unsigned comment added by 99.73.17.9 ( talk) 06:17, 7 April 2010 (UTC)
Only the original formula of Riemann zeta function is in the article. I believe it's very important to also include its analytic continuation formula too. It is a crucial piece of information. Also it would be nice to add in the article how this analytic continuation was found. I'm searching for this information myself so if somebody could do it it would be very appreciated. SmashManiac 20:15, 19 April 2007 (UTC)
Thanks for the history of derivation. Can you also provide the actual analyiticall continued formula for complex plane.
-Subhash —Preceding unsigned comment added by 12.144.36.2 ( talk) 19:09, 24 May 2010 (UTC)
Can some insert the actual RZF for complex plane. I found some formula at [1]. But I do not think I am competetive enought to validate the formula. -Subhahsh —Preceding unsigned comment added by 12.144.36.2 ( talk) 19:59, 24 May 2010 (UTC)
For the given functional equation
it follows that
but
Why are these values different? 67.185.99.246 02:40, 11 February 2007 (UTC)
Speaking of that formula, it actually doesn't help find riemann zeta function since it uses circular definition to define it (if you don't know riemann zeta of n or 1-n you can't use it). It can be used to define factorial if you isolate (-n)! and let n=-x. Note: letting x=0 or -1 will result in zero times infinity in the formula, so here is the heads up on those: 0!=1 -1!=+-infinity srn347 —Preceding unsigned comment added by 68.7.25.121 ( talk) 06:33, 19 November 2008 (UTC)
I was likewise confused about the meaning of the functional equation. It says it's valid for all s (except 0 and 1)... but since the gamma function has poles for negative integers, the functional equation is actually meaningless for s = 2, 3, ..., _unless_ it is understood to involve limits in that case. Since the average reader might not be aware of this shorthand, I mentioned it in the article. If you can find a better way to say what I said, go for it. Kier07 ( talk) 03:20, 24 August 2010 (UTC)
Disclosure: I am the son of a world-famous mathematician -- indeed, in her later life a number theorist. However, she mated with a whiskey-soaked advertising copywriter and I was raised on a farm by psychopathic semi-deaf-mutes shunned by their neighbors for the worldly sin of using electricity and automobiles. Mother did not rescue me from this Appalachian pastoral idyll for many years, whereupon she made my bedtime tales out of graph theory.
So, I have only half the genes and half the nurture. I have managed to stagger through a computer engineering career with nothing higher than the calculus -- which only served me once, and that indirectly. On the brighter side, I'm generally able to comprehend intelligent explanations, often in polynomial time.
As it stands, I find this article to be incomprehensible and without merit here. Wikipedia is a general reference work. For content to be included here, it has to pass the Joe test: If you had unlimited time in which to explain it (in far more detail than here) to Joe, the Everyman; if Joe was completely cooperative, intelligent, and patient; if eventually he understood all you intend, then could he imagine any possible way in which this subject might be of interest? This topic fails the Joe test.
This article bears a disturbing resemblance to Graveler. It is an isolated swatch of factoplasm lifted out of a highly specialized context, meaningless outside that context. Nothing has been done to show that it has any application to the Real World. While the subject may be a vital part of its own bubble universe, nothing in the article connects it to anything outside.
To be sure, this is longer than Graveler, and arguably factual; but I think that by the time you get past definitions of terms, it is equally valid to say that it is very like a playing card in a mathematical game. If there is a connection to anything concrete, that needs to be shown.
So, I suggest the article be downsized to a bare description of the function and merged into its parent article. (It does at least have some relationship to a larger mathematical topic, does it not?)
Another possibility is to open a new WikiBook entitled "Higher Mathematics" and expand this article into a whole chapter -- there.
I hold out one other route for improvement. I almost began to see a glimmer of light in Applications. Perhaps a diligent effort could actually find some application of this function to something tangible. If so, rewrite this section and move it to the introduction. MBAs and fools like me can read the 4 or 5 sentences that place the function in context, and then we can graze on.
— Xiong 熊 talk * 05:11, 2005 September 12 (UTC)
Aside from psychoanalysis of this person (is he (or she) really Chinese), he has some point. But he is questioning how wikipedia should be written, and this is probably not a place to do. There are many, too many, hopelessly technical pages already. But the consensus is we should try to revise it so that laymen can understand it, not that eliminate or downsize it from wikipedia. -- Taku 03:31, 14 September 2005 (UTC)
And no matter how uninteresting "Joe" might find this, im sure he's interested by the fact that the Riemann Hypothesis one of the Millenium Problems. As for being useless, this problem is closely related to the primes. Both the distribution of primes and primality testing are two of the biggest problems in number theory, both of which are closely related to cyptanalysis, which is certainly useful. -- He Who Is[ Talk ] 01:58, 29 June 2006 (UTC)
a) This article does not fail the "Joe test" as characterized above. b) Even if it did, the "Joe test" does not reflect Wikipedia policy or intent. -- 98.108.202.144 ( talk) 01:56, 31 August 2010 (UTC)
I have read, reread, and then reread this article. I am not a mathematician, but I am also not ignorant of higher mathematical concepts. This article is so technically oriented that it is virtually impossible to comprehend for a layman. In fact, I would wager that unless one already understood the Riemann zeta function that this article would be so complex that it would be useless. Please don't attack me for saying this, I only wish to improve an article that others have obviously worked very hard on. I hope someone will take up the challenge. —Preceding unsigned comment added by 70.121.7.89 ( talk • contribs)
You may be right that it's impossible for non-mathematicians to understand most of it. But that may perhaps be true of nearly any article that could be written on this sort of topic.
But I think you're wrong to say that only those who already understand the zeta function can understand this article. I think most mathematicians not familiar with the zeta function would understand it.
Michael Hardy
01:30, 28 January 2007 (UTC)
Looking at it now, I'd say the "Definition" section, the "Relationship to primes numbers" section, and the "Specific values" subsection would be readily understood by non-mathematicians (even if not by those who simply dislike math and never study math at any level). So I think you're being a bit alarmist. Certainly there are some things here that few besides mathematicians will understand, but far from everything. Michael Hardy 23:20, 4 February 2007 (UTC)
This paragraph is not formal at all, and it surely quite confusing. Who says that being coprime with a prime and with another one are mutually indipendent? This kind of trick doesn't always work and often one has to introduce a correction factor. Also, what does "s randomly selected integers" mean? What kind of probabilty distribution are we using on the set of natuaral numbers?-- Sandrobt ( talk) 16:17, 7 December 2010 (UTC)
please insert this astonishing new informations from Marcus du Sautoy (" The Music of the Primes"): "the moments of the Riemann zeta function...We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42 ... Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence"
188.46.217.137 ( talk) 21:13, 12 December 2010 (UTC)
Stumbled upon this article from 97, http://arxiv.org/abs/physics/9705021 claiming a new formula for \zeta(2n+1). Money is on it being a crank, however, if its actually right then it seemed like something to throw on the page. -- Differentiablef ( talk) 16:14, 13 April 2011 (UTC)
That formula for zeta(2*n+1) seems interesting since i don't think anyone was able to 'translate' it to any of the other formulas we know... and i don't mean his definition of B(s) := -s*zeta(1-s) and B'(s)=-zeta(1-s)+s*zeta'(1-s) and then zeta(2*n+1) in terms of B'(s) - since that is just a reformatting of the functional equation of zeta and the derivative (of the functional equation). I mean the formula for B(s) in terms of how Woon calls it 'analytic continuation of an operator'. The closest _looking_ formla to Woons formula for B(s) that we have is i think this http://en.wikipedia.org/wiki/Hurwitz_zeta_function#Series_representation and http://en.wikipedia.org/wiki/Bernoulli_polynomials#Another_explicit_formula. They look very similar to Woons - but still slightly different. But the common wisdom is that there are no _easy_ formulas for zeta(2*n+1) - there are definitly ones we can discount because they are not easy - so maybe that can also be applied to Woons formula. But for example what about zeta(n) = C_n*int(B_n(x)*cot(pi*x),x,0,1) for odd integer n? Here B_n(x) is the nth bernoulli polynomial, cot(x) = cos(x)/sin(x), int(f(x),x,0,1) is the integral for x over [0,1] and C_n is expressible in powers of pi and factorial(n) (see further 'Abramowitz and Stegun: Handbook of Mathematical Functions' http://people.math.sfu.ca/~cbm/aands/page_807.htm). Is that worth mentioning? — Preceding unsigned comment added by Petersheldrick ( talk • contribs) 00:09, 8 July 2011 (UTC)
Without wishing to enter into a debate over the modern convention for defining the function, I would like to point out that Riemann himself defined it as a contour integral, not by analytical continuation. It is evident from Riemann's paper that this is so. There is quite a good discussion of this point also in section 1.4 of Edwards' book, which you include in the References section. — Preceding unsigned comment added by 58.168.69.120 ( talk) 00:22, 13 October 2011 (UTC)
The first diagram presents a color-based encoding of the values of ζ(s) in the complex plane. The last sentence of the caption to that diagram says: "Positive real values are presented in red." I'm wondering if that should more precisely read: "Positive real parts are presented in red", especially as the word "value" is used previously to refer to the value of ζ(s), not to the [real] value of its real or imaginary part. — Preceding unsigned comment added by 210.9.140.245 ( talk) 09:53, 18 March 2012 (UTC)
So all values of the zeta function in the entire half-plane Re(s) > 1 are real and positive? Equivalently, the Dirichlet series of which the zeta function is the analytic continuation converges to a positive real number for any s satisfying Re(s) > 1? — Preceding unsigned comment added by 122.110.54.252 ( talk) 03:43, 19 March 2012 (UTC)
Is there some mathematician who can clear up my confusion? If I'm reading the caption correctly, the red expanse in the first diagram in the article indicates that ζ(s) is everywhere real and positive in the half-plane Re(s) > 1. The function obviously takes real values for real s > 1 but is it actually real and positive elsewhere in that half-plane? Or am I misunderstanding the diagram? — Preceding unsigned comment added by 210.9.140.245 ( talk) 20:21, 21 March 2012 (UTC)
Still no answer to the previous question but in the meantime I've made a slight correction to the series definition of ζ(s) in the Definition section, where it had a set membership sign (read as "σ belongs to the real part of s > 1") that was obviously meant to be an equals sign. — Preceding unsigned comment added by 210.9.140.245 ( talk) 07:45, 6 April 2012 (UTC)
Thanks for your response but I'm still confused. I think the caption needs to be rewritten for clarity. The relationship between color and hue is quite unclear, and I can't tell whether they're meant to be independent attributes of each pixel. There seems to be a "parent" attribute (color) with two child attributes, "dark color" and "hue", which makes no logical sense to me.
If there are two independent attributes of each pixel that constitute the encoding, then the significance of each needs to be clearly defined, and with no interaction between them. If I knew enough about the RZF I could probably work out what the caption is trying to say and could redraft it myself. But I don't, so I can't. As someone who understands the rudiments of complex-valued functions of a complex variable (which is all the knowledge that the diagram presupposes), I find the caption in its current form very confusing. I appreciate that my difficulty may not be comprehensible to someone with a good understanding of the RZF but I presume that the article, and certainly that diagram, is written for the likes of myself, not for RZF specialists. — Preceding unsigned comment added by 210.9.140.245 ( talk) 10:18, 9 April 2012 (UTC)
I wasn't and thank you. If I'm understanding things correctly, the diagram uses two of the three HSL attributes, viz., hue (to represent the argument) and lightness (to represent distance from zero, or equivalently the modulus).
Actually I don't think it's either necessary or desirable to require or assume familiarity of the reader with the HSL model, or to have to point the reader to any explanation of it. In any case, since hue is referred to explicitly (with a link to its Wikipedia entry) I would expect the same for lightness, which as far as I can tell is referred to only implicitly (in the phrase "dark colors"). The fact is, the information that needs to be conveyed in the caption is simple enough that informal English can be used for greater clarity and without any loss of precision. Thus, I would suggest use "color" and avoid "hue" entirely, and perhaps use "intensity" to capture the lightness attribute. In that way, the caption would be immediately intelligible to any reader regardless of their familiarity with HSL and without the reader having to wonder about the difference between "color" and "hue", or indeed whether there is any (which was another source of my confusion).
I'm still a little confused by the statement "Positive real values are presented in red", which doesn't seem to sit well with either the previous use of the term "values" ("dark colors denote values [of the function] close to zero") or the HSL model. If those values are of the argument (as previously explained by sandrobt) it seems a weird thing to say, as arguments are always real anyway. This was the source of my original confusion. I would therefore suggest rewriting the last sentence of the caption to avoid the word "real" and to make it clear that red represents values of the function that have a positive argument. I find the use of the unqualified word "values" in the last sentence ambiguous and confusing.
Finally, a question: At the scale at which the diagram is drawn, would no significant variation in the lightness of the red expanse be visible? In other words, over the red-colored half-plane does the function take only values whose (close) distance from zero can't be visually distinguished on the scale of the diagram? I'm assuming that the red is meant to be a "dark" version of that color (= hue) and thereby denotes proximity to zero as per the caption ("dark colors denote values close to zero").
This may all sound somewhat pedantic but I'm speaking from the point of view of the audience to whom I believe the article is addressed, viz., someone literate in English and partially literate in complex functions. It's a nice diagram, which uses a powerful visual technique to capture the reader's attention and illustrate some salient aspects of the behavior of the function. I think it's therefore reasonable to expect a comparable level of clarity in the wording of the caption as in the mathematical discussion that it accompanies.
Thanks for the further clarification. So it's true that positive real values are encoded in red — as well as a lot of other values! I was mistakenly (though I think understandably) reading the sentence as a definition of "red" and thus inferring the unintended converse of your statement. Your new wording is much better. I would suggest one more improvement for maximum clarity: "Values with arguments close to zero (in particular, positive real values on the real half-line) are encoded in red". This just makes it absolutely clear that the red encoding applies to positive reals by virtue of a more inclusive encoding criterion, and also tells the reader where those positive real values actually occur among that vast expanse of red. — Preceding unsigned comment added by 210.9.140.245 ( talk) 21:06, 11 April 2012 (UTC)
It's clear now, thank you. — Preceding unsigned comment added by 210.9.140.245 ( talk) 09:49, 14 April 2012 (UTC)
That diagram is not explained very well ... where is the "white spot" at s1. First of all, why isn't "S" labelled on the axis. Which Axis? If you label an axis "S" then we could find S1 but you didn't and then I don't see any "white spot" anyways. So, if you could label which axis is S and then maybe make the "white spot" maybe bigger so we can actually see it because I don't see a white spot do you?
Ty
173.238.43.211 ( talk) 18:53, 6 May 2012 (UTC)
Ya well it shouldn't take a lot of ingenuity to figure out THERES NO WHITE SPOT ON THAT GRAPH. Its bad enough you think a reader can just assume which axis you are talking about but its even worse that you claim theres a "white spot" and then not even label which axis its on. I can only assume that kinda black/green/white "bubble" area in the middle of the chart is supposed to be a "white spot" but it happens to have black and green in it also so no idea where this "white spot" is or which is s1 so does anyone know who made this graph so he can explain it so we get it? Do you mean the black egg shaped spot in the centre?? Is that black spot the "white spot"?
173.238.43.211 ( talk) 08:24, 7 May 2012 (UTC)
Ok, thank-you. Perhaps someone else that doesn't see the "obvious white spot on the unmarked axis consisting of blue/green/ and yellow colors" will hopefully read through this talk page to find out that the blue/green/yellow spot is the "white spot". Do you see how small that "white spot" is? Its a pin prick. If only the supposed "white spot" area was circled then noone would ever question where this "white spot" that has blue, green, and yellow in it actually is. That was only the point of me asking where it was. It was where I kind of guessed it is but wasn't sure since the diagram in no way makes it easy to know for sure was all I wanted to get across. Sorry if my question bothered you. We all win in the end.
173.238.43.211 ( talk) 18:19, 9 May 2012 (UTC)
Oh and this article and graph about the zeta function is less problematic to understand in my opinion;
http://simple.wikipedia.org/wiki/Riemann_hypothesis
173.238.43.211 ( talk) 12:31, 10 May 2012 (UTC)
Maybe calling it a "singularity" instead of "white spot" at s1 would be better since its really blue/green/ and yellow and the axis isnt labelled so we could know for sure - just my opinion
173.238.43.211 ( talk) 22:39, 10 May 2012 (UTC)
The link to Möbius function leads to a function defined only for integers. I suspect that this is not the function intended. Klausok ( talk) 08:13, 23 October 2012 (UTC)
Hi, I think this should be mentioned at the original page
Log[2/3/E^(5 (Pi/2)) + E^(7 Pi)/(7/2/E^(7 (Pi/2)) + 5/2/E^(5 (Pi/2)) + 3/2/E^(3 (Pi/2)) + E^(5 (Pi/2)) + 2 Pi)] = 14.134725141734373744769652011837891376454997031685134...
But the first zero on the critical line is Zeta(1)=
14.134725141734693790457251983562470270784257115699...
Have a nice day, Dietmar — Preceding unsigned comment added by 217.81.127.212 ( talk) 16:08, 5 November 2012 (UTC)
How is the zeta function of a half calculated? In fact, how is the zeta function of any number between zero and one calculated?! — Preceding unsigned comment added by 86.136.129.187 ( talk) 19:12, 3 December 2012 (UTC)
Would it be possible for me to re-write the article from scratch, and then to go from there?
The article is in a deplorable state. It is too technical and in addition the technical parts are not central to the current (or past) research on $\zeta(s)$. The emphasis on "Selberg's conjecture" (not a conjecture in the first place but more of a remark in Selberg's paper) is badly misleading.
It seems that somebody very fond of Karatsuba keeps putting these here. It is not normal to have 8 references to Karatsuba and 1 to Shanker (never heard of him) and no references to Hardy-Littlewood, Ingham, Bohr-Landau, Titchmarsh and more recently to for example Conrey, Soundararajan, Iwaniec, etc.
Anyway, if this is not possible then I will go away and the article will remain in its deplorable state... — Preceding unsigned comment added by Karatsuba ( talk • contribs) 07:50, 9 June 2013 (UTC)
Basically everything after and including the section "Zeros, the critical line, ..." needs to be purged and re-written. There is too much misleading emphasis (i.e Karatsuba, identities which nobody uses) there and too much original research (i.e Shanker).
The Section "Estimates of the maximum of the modulus of the zeta function" contains a typo: "The case was studied by Ramachandra" should read "The case was studied by Ramanujan". I'd not go as far as editing the article as I would not dare to touch an article on mighty maths, just a suggestion. — Preceding unsigned comment added by 152.66.244.111 ( talk • contribs) 15:13, 13 May 2011 (UTC)
No the result is due to Ramachandra and not Ramanujan. — Preceding unsigned comment added by Karatsuba ( talk • contribs) 07:56, 9 June 2013 (UTC)
Recently text on the so-called Selberg conjecture was removed by an anonymous IP editor. This removal was reverted (on somewhat superficial grounds). It seems to me that the IP has a legitimate point. The material under discussion at best seems like undue weight, and at worst original research. (See the comments also in the preceding section, which pertain to the same material). May I suggest that we leave this material out unless the case for its inclusion is made more clear. Sławomir Biały ( talk) 01:55, 11 June 2013 (UTC)
There should be more on zero density theorems here or in the article on the Riemann hypothesis. — Preceding unsigned comment added by 86.181.154.200 ( talk) 12:55, 28 December 2013 (UTC)
The recent addition of an external link to Brady Haran's video on the zeta function was reverted as Added link does not add any new information about the subject (just mentions it). The subject of the video was "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12". This brings up three points.
I'm not interested in an edit war, so I'll neither put the link back nor remove the deminishment of Euler. But somebody should. user:JMOprof ©¿©¬ 18:22, 10 January 2014 (UTC)
There seems to be an edit war going on at the moment, and we should resolve it here on the talk page rather than by a bunch of different editors reverting and re-reverting.
Is Zeta(-1) = 1 + 2 + 3 + 4 .... ? Until we can come to consensus and reliably source this as settled, we probably shouldn't put it in the article, or if we do we need to be careful to provide context so that it is not misunderstood by the lay reader. My understanding of wiki policy is to leave it out until a consensus is reached.
A couple of observations:
There's a quite misleading video making the rounds, and many users are coming here to "verify" whether it's true or not. We need to provide a solid article. Looking forward to comments. Mr. Swordfish ( talk) 19:03, 21 January 2014 (UTC)
As of today, the formula
is back in the article. Are we going to just keep removing and restoring it on a weekly basis, or are we going to try to come to some consensus here on the talk page? Let's hear the reasons for including it. Mr. Swordfish ( talk) 16:33, 17 February 2014 (UTC)
Such a statement obviously presents philosophical difficulties. Namely, one is forced to ask how the “sum” of a divergent series of entirely positive terms can be negative. Yet the manipulations involved in our determination of s are no more outlandish than those used in determining 1 − 1 + 1 − 1 + ··· = 1/2. We will see later that in a very precise sense, −1/12 is the correct value of 1 + 2 + 3 + 4 +···.
With 64 threads, some of which are approaching their second decade, how about implementing auto archive? I will volunteer to do it if there is consensus. Mr. Swordfish ( talk) 16:55, 18 February 2014 (UTC)
You say that the Riemann zeta function can also be defined by the integral:
But you never stated whether this definition is valid everywhere on the complex plane (wherever the function exists). People just learning about Zeta function regularization of what I like to call the Ramanujan (1+2+3+...) are interested in the analytic continuation of the Riemann zeta function over the entire complex plane. It seems trivial to me that the aforementioned definition is valid over the entire plane, but I might be wrong. You need to either state that this equation is valid for the entire complex plane, or you need to help people like me who will look at the equation, do a bit of sloppy mental math and conclude (falsely?) that the expression does define the function everywhere that it exists.-- guyvan52 ( talk) 15:18, 21 December 2014 (UTC)
The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function, following Riemann.)
The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:
The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Eqivalentlyreference?, it can also be defined everywhere that it exists on the complex plane by the integral
-- guyvan52 ( talk) 15:18, 21 December 2014 (UTC)
Under Various properties and Reciprocal is the statement "the claim that this expression is valid". This expression? Which one? John W. Nicholson ( talk) 04:54, 15 March 2015 (UTC)
Someone changed the equation into: (i.e.: minus 1 changed into plus 1 in the denominator)
This seems suspicious, since there is no explanation, and no other edits from that IP. Can someone looks into it to verify if it is legitimate? — Preceding unsigned comment added by Dhrm77 ( talk • contribs) 05:20, 25 March 2016 (UTC)
This paragraph first states "it was first proved by blah in 19xx" but then says "Nevertheless, none of the proofs above are definitive or complete, since a definitive proof would imply that Riemann's Hypothesis is true."
Is it proved or not proved? Whats missing? This is confusing to me. — Preceding unsigned comment added by 32.215.35.33 ( talk) 18:54, 21 June 2016 (UTC)
Well yes, it is not written exactly clearly. But idea is, that someone proved something while considering that Riemann Hypothesis is true, so basically if Riemann Hypothesis is true then whatever that someone proved will be also true, but if it is not, then it might not be. So basically they proved something, but took an unproven hypothesis as basis for their proof. Trimutius ( talk) 15:40, 22 June 2016 (UTC)
That whole section from that point down starts assuming that the reader knows things that have not yet been explained or defined. I would say the entire page needs to be reworked, starting at the top with definitions. 71.48.255.210 ( talk) 20:56, 14 July 2016 (UTC)
i understand nothing to this notion of «ζ(x) zero», so if true, i think we should ad it to explain — Preceding unsigned comment added by BeKowz ( talk • contribs) 12:59, 3 December 2016 (UTC)
The article mentions the following relationship between and under the heading "Mellin transform".
I was hoping to explore the relationship above using the Fourier series representation for which is an infinite series of Fourier series where each Fourier series consists of an infinite series of Sine terms (see Fourier Series Representations of Prime Counting Functions). Unfortunately a term of the form resists integration. However, I have been able to explore the relationship above using the relationship below was derived from the original relationship above via integration by parts.
I haven't been able to explore the second relationship above using the Fourier series representation for because a term of the form also resists integration, but the second relationship above can be evaluated by writing the integral as a sum as follows.
Using the sum formula for the second relationship above, I confirmed the the real part of converges to some degree for the two relationships above for , but the imaginary part of diverges significantly using the sum formula above which makes me suspect the validity of the two relationships above.
StvC ( talk) 01:04, 5 November 2016 (UTC) StvC ( talk) 00:49, 15 December 2016 (UTC) StvC ( talk) 05:15, 16 December 2016 (UTC) StvC ( talk) 16:46, 16 December 2016 (UTC)
In the Riemann zeta function page it is written that:
and that:
.
However looking at the Gamma function definition in Wikipedia one can see that:
.
Correspondingly .
More looking at Riemann's paper he has the s-1 also.
However Riemann instead of Gamma used factorial .
I tried to fix the page but it was rejected as vandalism.
Please correct it or explain to me what am I wrong about.
Adikatz ( talk) 06:52, 26 July 2017 (UTC)
In the definition of the zeta function the definition of the Gamma function is wrong. It should be s-1 instead of s. The s-1 should also be in the integral representing the gamma function. I have tried to fix but it was revoked as vandalism.
. Or . But not . Sapphorain ( talk) 07:57, 26 July 2017 (UTC)
Sorry. I was mistaken since I did not pay attention to the fact that the dx was divided by x. Adikatz ( talk) 14:21, 26 July 2017 (UTC)
Here are some more representations, but I don't know what subsection to put them in, or what commentary to give on them:
for positive integer n
for 0 < Re(s) < 1, where frac is the fractional part
for Re(s) > 1
(Reference: http://mathworld.wolfram.com/RiemannZetaFunction.html)
-- AndreRD ( talk) 16:29, 27 July 2017 (UTC)
If the complex number s = 1/2 + 0j is put into the Riemann functional equation after the equation is rearranged so the left hand side reads E{s}/E{1-s} then the left hand side with a value of 1 does not equal the right hand side and this means that the complex part is not zero and so there can be no solution on the 1/2,0 coordinate of the complex plane as Riemann says there is. Have I got this right?
Soopdish ( talk) 12:48, 8 August 2017 (UTC)Soopdish
The Integral sections shows a formula for the zeta function that is super wrong. The correct formula can be found here
Abel–Plana formula
It should look like this:
The provided link #24 is also wrong. It links to an arc length calculus page.
--
EmpCarnivore (
talk)
21:19, 7 September 2018 (UTC)
Is there a formula out there that relates the zeta function to the perfect powers (1, 4, 8, 9, 16, 25, 27, 32, etc) in a similar way that Euler's product formula does for primes? The reason I ask is because I discovered a very simple one a while back, but I can't find information about any other such formulas. Thanks. -- Vagodin 14:47, August 21, 2005 (UTC)
The 'globally convergent series' found by Hasse appears to be essentially just the Euler transform applied to the Dirichlet eta function. — Preceding unsigned comment added by Carifio24 ( talk • contribs) 23:22, 4 July 2007 (UTC)
I programmed the second of the two Hasse series, and the second one seems to be wrong. It returns large incorrect values. If there's a bug, I must be blind:
The first series (which works correct) uses the following code:
double hasse0(double s, uint64_t limit) {
double u = 0; for (uint64_t n=0; n<limit; ++n) { double r = 0; for (uint64_t k=0; k<=n; ++k) { double t = choose(n,k) / pow(k+1,s); if (k%2) r -= t; else r += t; } u += r/pow(2,n+1); } return u/(1-pow(2,1-s));
}
The second series (which does not work) uses the exact same template:
double hasse1(double s, uint64_t limit) {
double u = 0; for (uint64_t n=0; n<limit; ++n) { double r = 0; for (uint64_t k=0; k<=n; ++k) { double t = choose(n,k) / pow(k+1,s-1); if (k%2) r -= t; else r += t; } u += r/(n+1); } return u/(s-1);
} — Preceding unsigned comment added by Pcp071098 ( talk • contribs) 23:07, 27 August 2013 (UTC)
Under "Specific values", the graph seems to be of three functions, only one of which is the Zeta function. The other two seem to be based on a finite number of terms of the infinite series in the definition of the Zeta function. — Preceding unsigned comment added by 79.79.29.112 ( talk) 08:15, 28 July 2017 (UTC)
Also, for we have:
where, in both expressions, refers to the
Euler-Mascheroni constant. — Craciun Lucian.
In the section on the proof of the functional equation, the text keeps mentioning criteria on sigma, like sigma > 0, sigma > 1. But then the actual computations have no sigma. They're all in terms of s, is sigma meant to be s here? - lethe talk + 19:04, 29 January 2020 (UTC)
Why use , not , as ? I think is more “natural”. Three similar functions are , and . — Preceding unsigned comment added by Xayahrainie43 ( talk • contribs) 13:54, 24 September 2018 (UTC)
The following anonymous edit comes from an IP with a very checkered history. It needs vetting:
Thanks. -- Wetman 13:02, 18 Apr 2005 (UTC)
This section seems suspect since 1) the reference [21] is just a reference to the cauchy integral theorem, and 2) the series presented does not appear to be a dirichlet series. Can someone confirm if this is even true? — Preceding unsigned comment added by 208.38.59.163 ( talk) 20:19, 23 September 2020 (UTC)
Note: it does give the correct value of zeta at 2, but i am still not seeing where it comes from? — Preceding unsigned comment added by 208.38.59.163 ( talk) 20:29, 23 September 2020 (UTC)
Under "Globally convergent series", I think that Knopp's conjecture was made in 1926. — Preceding unsigned comment added by 213.48.238.18 ( talk) 13:42, 3 November 2020 (UTC)
Under "Generalizations", the Clausen function is mentioned, ungrammatically. — Preceding unsigned comment added by 79.77.163.188 ( talk) 13:01, 26 April 2021 (UTC)
This article uses the picture /info/en/?search=Riemann_zeta_function#/media/File:Zeta_polar.svg
Note that it is not a polar graph in any sense, it is a regular Cartesian graph, with and , . It should be corrected and renamed; the current version is misleading. A1E6 ( talk) 18:17, 21 July 2021 (UTC)
There is of course -s function equation and alternating series form but what about actual closed form? Valery Zapolodov ( talk) 10:13, 4 January 2022 (UTC)
On Riemann sphere that is? Valery Zapolodov ( talk) 13:50, 4 January 2022 (UTC)
@ Jomu5221, Tanfer Tanriverdi, and JayBeeEll: I just removed a recently added result from the article because it is not notable:
The recent addition of a claim about "the first work that led to the derivation of the formula" is even less suitable for the article:
Cheers, -- Macrakis ( talk) 00:28, 17 January 2022 (UTC)
I am fairly certain that this article uses log() to denote logarithms in base e rather than base 10. If so, please confirm this and I will add the following template: Toadspike ( talk) 16:32, 12 May 2022 (UTC)
The first series under the "Representation" section does not seem to converge correctly (e.g. near zero, it converges to zero when it should converge to -1/2). Other representations seem to work fine; indeed the very next series converges to -1/2 as expected. Moreover, I chased the given reference, and did not see (either) series listed in the given location. It is possible, perhaps likely, that I have made an error. — Preceding unsigned comment added by Geometrian ( talk • contribs) 03:26, 2 June 2022 (UTC)
https://en.wikipedia.org/?title=Riemann_zeta_function&oldid=prev&diff=1102657060
Oh I guess I should've linked to this
/info/en/?search=Convergence_tests#p-series_test
instead of that
but still?
Thewriter006 ( talk) 07:27, 6 August 2022 (UTC)
A formula for involving the Thue-Morse sequence was recently added. While this result is interesting, I have my doubts that it is sufficiently notable for an encyclopedia article on the Riemann zeta function. I note that the same editor added similar content to the article on Apery's constant. 74.111.98.156 ( talk) 16:57, 18 December 2022 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Is there a wikipedia policy for where pronunciations should appear? I dislike the redundancy of placing the pronunciation of Riemann both at this article and also at Bernhard Riemann. If a user wants to know the pronunciation they can simply follow the name link to find it. - Gauge 04:44, 2 September 2005 (UTC)
I'd like to hold a survey regarding the article
Riemann zeta function, to help determine its general comprehensibility and identify areas where it may be incomplete. Please indicate your perceptions of the article below, and feel free to expand the survey or article as you see fit.
‣ᓛᖁ
ᑐ
21:07, 9 September 2005 (UTC)
Do you currently understand this article?
Yes
No
Comment
If not, do you feel you could understand it after following its internal links?
Yes
No
The article's lead section states the Riemann zeta function is "of paramount importance in number theory". From reading the article, do you understand why this function is important?
Yes
No
I removed the following text:
I couldn't understand what it was trying to say. Its trying to describe some dirichlet series maybe??. linas 00:46, 21 December 2005 (UTC)
I found this section very helpful for understanding the connection between the Zeta function and prime numbers. BringCocaColaBack 11:29, 13 January 2006 (UTC)
Following question from an anon contributor moved from the Globallly convergent series section on the article page. Gandalf61 08:35, 20 April 2006 (UTC)
That question applies more properly to the formula in the section above Globally convergent series, called Series expansions, which contains the formula:
What's that 'x' doing there? - GTBacchus( talk) 13:38, 20 April 2006 (UTC)
In the series expansion section, it's written that "Another series development valid for the entire complex plane is.." I can't figure out what the variable 'x' is supposed to be in the expansion that follows. x=s? —The preceding unsigned comment was added by 12.208.117.177 ( talk • contribs) .
Is this function called the zeta-function or the zeta function? The article uses both. toad ( t) 12:10, 10 February 2006 (UTC)
Zeta-function refers to all zeta function is general. But in this case its just Riemann zeta function. -- He Who Is[ Talk ] 01:59, 29 June 2006 (UTC)
No graphic of the graph in the complex plane? Surely the article should include one or perhaps three; real part, imaginary part and absolute value, as is the standard on Mathworld. Soo 14:17, 16 July 2006 (UTC)
The article titled 42 (number) says:
I don't know what this means. Here's a guess:
I'm accustomed to the definition of momnets of probability measures; if ζ were a probability density function then the integral above would be the third moment of the corresponding probability distribution. But ζ is negative in some places, and from the way ζ(s) blows up at s = 1 it seems we'd have to be thinking of a Cauchy principal value or something like that.
Can someone make the article's statement clearer? Michael Hardy 17:52, 5 July 2006 (UTC)
When k=3 you get the "sixth moment". The constant 42 comes up as a scaling factor in a conjecture by Conrey & Ghosh for the leading order term of this integral. .—Preceding unsigned comment added by 74.74.128.91 ( talk • contribs) *****
“ | So, feel free to post this comment for me:
I don't know anything about the "3rd moment of the Riemann zeta function", but perhaps what's meant is the 3rd moment of the distribution of spacings between zeroes of the Riemann zeta function. There's a lot of evidence relating the distribution of these spacings to the distribution of spaces between eigenvalues of a large random self-adjoint matrix. For lots more, try this: http://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htm and for general connections between the Riemann zeta function and quantum mechanics, try: http://www.maths.ex.ac.uk/~mwatkins/zeta/physics1.htm Best, jb |
” |
I stored in commons a graph of zeta (x) with -20 < x < 10.
see
Perhaps it can go into the article
-- Brf 10:05, 31 August 2006 (UTC)
Can someone add an explanation about the critical strip? This term's definition is nowhere to be found in Wikipedia. Thanks! By the way, the whole article is fine and readable; everybody with a college degree will understand at least the basics. Hugo Dufort 08:28, 14 November 2006 (UTC)
I added an expanded form of the zeta function I got from Marcus de Sautoy's "Music of the Primes", because I think it helps to visualise exactly what the product is about. Comments? DavidHouse 21:21, 26 December 2006 (UTC)
In a technical report entitled "[ http://cswww.essex.ac.uk/technical-reports/2005/csm-442.pdf An elementary formulation of Riemann’s Zeta function]", myself (Riccardo Poli) and Bill Langdon provided a very simple proof that, for , Riemann's Zeta function can be written as where .
We are not experts in number theory, but we have searched widely and also asked several mathematicians: it appears that our rewrite is new. These people tell us that this is useful formulation. So, we were wondering whether it would make sense to include it in the article.
The paper mentioned above has now been published in arXiv.org in the Mathematics History and Overview section (math.HO/0701160). Perhaps the result could now be included in the article?
I want to add a personal support for the writers of the article. Even though the article necessarily is largely technical, the lead-in paragraph adequately establishes the backround of the function for laymen. -- Cimon Avaro; on a pogostick. 08:51, 12 February 2007 (UTC)
209.226.117.54 16:08, 17 February 2007 (UTC) Jacques Gélinas
Browsing through the talk page, it seemed to me that there had been quite a few complaints about the definition of from people not comfortable with analytic functions, or perhaps, with mathematics in general. I can definitely confirm that the so-called "introduction" to this article is too terse to be of any use. Other articles, such as Riemann hypothesis are much better in this regard. So this is certainly something that needs to be dealt with. For now, I have expanded the definition a bit, it remains a rigorous mathematical definition, so it's unclear to me how much happier would non-mathematicians be with it. Hopefully, it is somewhat gentler to those who are unsure about all the symbols and unfamiliar terminology, although to experts on Riemann zeta function it may appear to be perhaps a little too easy. I do want to point out that Enrico Bombieri, in the description of the Riemann hypothesis in the Millenium Prize book starts by mentioning that the Dirichlet series for is defined only for large , and then explains that it is analytically continued. I definitely feel that it's not something to be taken lightly, especially since analytic continuation of general Artin L-functions is still unknown, and of course, by no means obvious! Perhaps, it would make sense to expand the definition even more, it is a judgement call (or an editorial decision), so I would wait to hear the reaction.
Incidentally, I think that in line mathematical formulas do not look very good in this case, but since it's a highly emotional issue for at least some users, I tried to preserve them. Arcfrk 06:04, 10 March 2007 (UTC)
For those who get lost in this Wikipedia article, I have found the following link to be the clearest description in layman's terms. Those who authored and are maintaining this Wiki article may want to read this to understand how a complicated math concept can be described in normal conversational English:
http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php Overlook1977
I glanced at if for a second or two. Is there anything in this link that actually says what the zeta function is?? If not, I certainly wouldn't say it's clear to either lay persons or anyone else. Michael Hardy 20:45, 16 April 2007 (UTC)
This article needs more information on how the zeta function is connected to prime numbers. My understanding of this function and the Riemann hypothesis is that this function and prime numbers are very deeply connected, but I can not find on Wikipedia or anywhere else an explanation as to why. Sloverlord 12:36, 17 May 2007 (UTC)
The Mellin Tranform section has . Could "where is the Gamma function" be added?
Is this vandalism? At the bottom of the page there's a link to a .gz file. Even so, it should be reworded. '''T'''''o''__m__ 17:34, 7 November 2007 (UTC)
I thought the Riemann zeta function referred to complex numbers - what is the justification for including the harmonic series, the Basel equality, and other zeta functions that don't involve complex numbers here? 24.61.112.3 15:12, 2 December 2007 (UTC)
The Specific Values section contains examples of series for natural numbers only. Don't these belong in an article about Zeta functions "in general"? The sudden transition from discussing natural number constants to the "Zeta zeros" - and hence complex numbers of the Riemann Zeta function - is bizarre and misleading to say the least. Michaelmross 14:53, 21 January 2007 (UTC)
I find it sad that a well-intended comment is misunderstood and then labeled as misleading. I didn't say the topic was about "zeta functions in general" - I clearly suggested that natural number series might belong in a *topic about zeta functions in general*. Because what I'm saying about this section of the article is that it goes from something general about natural numbers that a layperson like myself can understand to something very specific concerning complex numbers and the zeta zeros. And to me - a non-mathematician - this is very confusing. I would like to see a layperson's distinction between a generic zeta function and a Riemann zeta function. This will be my last comment on the matter, so there's no need to flame me any further. Michaelmross 20:12, 22 January 2007 (UTC)
The approximation by Gergő Nemes in "The functional equation" does not work when
In other words, it mostly makes sense as an approximation of zeta for large negative values. For these values, the derivation is simply plugging in Stirling's formula into the functional equation, which only serves to complicate the expression. —Preceding unsigned comment added by 64.3.169.42 ( talk) 21:15, 3 January 2008 (UTC)
I don't know how I missed them before, but the sections purporting to evaluate zeta(2) and zeta(4) can, perhaps, be best characterized as unencyclopaedic garbage, the word that they seem to be using quite a bit. "Zeta constant" already offers a thorough coverage of special values of zeta. The "proofs" themselves are about as tortured as one can imagine, and blatantly fail WP:NOT. I got rid of them. Arcfrk ( talk) 03:45, 3 March 2008 (UTC)
What is the point of this graph in the middle of the article, when the article does not mention it even once? T.Stokke ( talk) 21:05, 1 March 2008 (UTC)
Years ago I did a course in complex analysis. I got a bare pass, which doesn't seem sufficient to understand the section on trivial zeros. Could someone explain explicitly why the negative evens are zeros? RayJohnstone ( talk) 15:50, 28 March 2008 (UTC) —Preceding unsigned comment added by RayJohnstone ( talk • contribs) 15:47, 28 March 2008 (UTC)
As I understand it: zeta(0) = 1/1^0 + 1/2^0 + 1/3^0... = 1 + 1 + 1... = infinity != -1/2
Where do I wrong? 84.108.198.129 ( talk) 23:35, 2 September 2008 (UTC)
Shouldn't ζ(0)=infinity?
05:47, 19 April 2008 (UTC)
How does ζ(0)=-1/2? —Preced PMajer ( talk) 16:21, 8 September 2008 (UTC)ing unsigned comment added by Anon126 ( talk • contribs) 05:59, 15 May 2008 (UTC)
This article shows a kind of a common problem of mathematical articles. It seems that a lot of people are interested in the Riemann zeta function, without beeing so interested in understanding the elementary underlying facts, such as, what is a complex number, or what is the sum of a series, etc. This makes sense, of course, for the sake of a general information (in the same way, I am interested in music even though I cannot play). But in this case one should not complain if an article looks too technical, or if things are not made easier than possible, and one should not be surprised if he or she can't go beyond a row picture of the facts, because it is natural. The point is simply that a complete explanation with all details, e.g. about the zeta function, would require a good part of an elementary course of complex variables, what is beyond the scope of thes articles. On the other hand, there is a minority, made of users with a mathematical culture, that go to a math article looking for a more technical, precise and usually very short information (tipically, in areas of mathematics different from their own). For them wikipedia has become quite a useful tool, so I think it should be a mistake to keep an article to the level of the mean users. Maybe a good thing should be to keep disjoint the general and the specialistic part... PMajer ( talk) 17:48, 8 September 2008 (UTC)
zeta(-2) = 1/1^-2 + 1/2^-2 + 1/3^-2 + ... = 1 + 2^2 + 3^2 + ... = infinity !=0
Someone is wrong - me, or the rest of the world. Who? (and why?) 84.108.198.129 ( talk) 23:43, 2 September 2008 (UTC)
How can a subset of a straight line have a fractal dimension greater than 1? Or does it mean that the Riemann hypothesis is false?-- Yecril ( talk) 23:08, 6 October 2008 (UTC)
Does anyone agree that the "Applications" section is a little silly? The connection to physics is sparse, and no connection to the use of the Riemann-Zeta function is actually motivated by it (would you actually need to use it to show that sum diverges? Surely you want to show something else). Njcnjc ( talk) 03:03, 21 October 2008 (UTC)
Can anyone elaborate on the "prime numbers" section? It says "this is a consequence..." without giving even a hint or flavour of why. -- Doradus 21:25, Sep 17, 2004 (UTC)
I've added what I believe to be a standard treatment of why the two expansions are equivalent. Do any real mathematicians want to do a better job? -- The Anome 23:27, 17 Sep 2004 (UTC)
I've turned the "vigorous handwaving" into an actual proof and have made a number of further changes and additions. Gene Ward Smith 08:42, 29 Jan 2005 (UTC)
There will always be a long standing question on the readability of this article; I feel the need to put my two cents in. I came to the page after reading probably 20-30 pages on abstract math, no formal training in mathematics beyond Differential Equations and Vector Calculus in college. I found it to be sufficiently readable that I was able to "get the gist" of what the Riemann Zeta function is and why it is important. No, I could not do a single calculation regarding it, but from what I gather, the masters of the topic have trouble doing more than the most basic proofs regarding the topic - so is it such a surprise that I have trouble?
Fundamentally, Riemann Zeta is not simple. Not by a long shot. But that doesn't mean its unimportant; it doesn't mean that mathematicians who can leverage the information should be denied such a tool. There's still truly trivial content on wikipedia; lets leave the profound content (like Riemann Zeta) in place. 199.46.245.230 ( talk) —Preceding undated comment was added at 17:04, 26 November 2008 (UTC).
I don't feel there is adequate discourse of using the reciprocal form. Specifically, readers will wonder why it is used when the sum of positive integers from 1 thru N is well established as N*(N+1)/2. Is the manipulation to be able to apply specialized analysis forms or ??-- Billymac00 ( talk) 03:17, 27 January 2009 (UTC)
The trivial zeros do not seem to yield zeros:
For example: ;
which yields infinity.
I will appreciate if someone lets me know where I am mistaken.
128.97.68.15 ( talk) 17:25, 1 February 2008 (UTC)
Based on a false premise, don't you think? Fergananim ( talk) 20:51, 5 February 2009 (UTC)
The following snippet appears to be meaningless. What is s? What is the significance of the variable x? A broader question is: is this formula significant enough to be in the article? If so, could somebody please correct it and give it enough context that it becomes meaningful and manifestly notable for inclusion in the article. Sławomir Biały ( talk) 03:05, 9 July 2009 (UTC)
here dN(x)/dx is just the derivative (as a distribution) of the number of zeros on the critical strip 0 < Re(s) < 1. A proof of this can be found on a work by Guo (see references).
- the derivatives are all respect to 'x'
- The formula is an expansion of the logarithm of on the critical strip, as you can see it reproduces all the poles of
- is just the expression , whith 'gamma' being a sum over the imaginary part of the zeros -- Karl-H ( talk) 09:45, 9 July 2009 (UTC)
this formula is just an expansion of the logarithmic derivative of Zeta function on critical strip, it is interesting (just my opinion) since it involves a sum over the poles of Riemann zeta, i think i found in the paper refereed before or in another context in papers talking about
Gutzwiller Trace formula and Riemann Hypothesis, perhaps this formula would suit better into an article about Riemann Hypothesis or Gutzwiller trace, as you wishes --
Karl-H (
talk)
20:19, 10 July 2009 (UTC)
If so, it wouldn't have had any zero on critical strip Re(s)=1/2, neither any trivial zero.
Clearly, '>' should be replaced by '<'.
See paragraph 1. —Preceding unsigned comment added by 151.53.136.190 ( talk) 21:02, 12 August 2009 (UTC)
The article says that all negative even integers are trivial zeroes, but the value of the zeta function is infinite for them, not 0.
1+4+9+16+..., 1+16+81+256+..., etc. all diverge to infinity. —Preceding unsigned comment added by 75.28.53.84 ( talk) 14:51, 15 August 2009 (UTC)
Given the above endless confusion about the definition of the series, the definition of the function defined by the series and the definition of the zeta function itself, I've changed the wording of the definition to something that I hope is slightly clearer. I attempted to make clear what people on the talk page were saying: that the zeta function is the analytic continuation of the function defined by the series given. Please take note that I have NO UNDERSTANDING AT ALL of analytic continuations. I merely changed the definition to emphasize what has been repeatedly said on this talk page. If I got it wrong, please feel free to correct it. mkehrt ( talk) 09:03, 7 October 2009 (UTC)
The Euler product formula is from about a hundred years before Riemann was born. Is it known who was the first to consider this function? It seems Chebyshev's work connecting prime numbers to this function was before Riemann. Why is it called the Riemann zeta function? Was Riemann the first to define it for complex numbers? A section on history would be nice. Regards, Shreevatsa ( talk) 14:57, 18 March 2009 (UTC)
The article states that the function was named after Riemann because "he introduced it"... Is that the case, or is it as Shreevatsa says, that he was the first to define it for complex numbers? —Preceding unsigned comment added by 24.12.13.8 ( talk) 19:07, 15 October 2009 (UTC)
The argument: (harmonic diverges) -> (euler's formula predicts infinitely many primes) is flawed since Euler's formula depends on there being infinitely many primes to sieve the infinite sum of the riemman-zeta function to '1' —Preceding unsigned comment added by 99.73.17.9 ( talk) 06:17, 7 April 2010 (UTC)
Only the original formula of Riemann zeta function is in the article. I believe it's very important to also include its analytic continuation formula too. It is a crucial piece of information. Also it would be nice to add in the article how this analytic continuation was found. I'm searching for this information myself so if somebody could do it it would be very appreciated. SmashManiac 20:15, 19 April 2007 (UTC)
Thanks for the history of derivation. Can you also provide the actual analyiticall continued formula for complex plane.
-Subhash —Preceding unsigned comment added by 12.144.36.2 ( talk) 19:09, 24 May 2010 (UTC)
Can some insert the actual RZF for complex plane. I found some formula at [1]. But I do not think I am competetive enought to validate the formula. -Subhahsh —Preceding unsigned comment added by 12.144.36.2 ( talk) 19:59, 24 May 2010 (UTC)
For the given functional equation
it follows that
but
Why are these values different? 67.185.99.246 02:40, 11 February 2007 (UTC)
Speaking of that formula, it actually doesn't help find riemann zeta function since it uses circular definition to define it (if you don't know riemann zeta of n or 1-n you can't use it). It can be used to define factorial if you isolate (-n)! and let n=-x. Note: letting x=0 or -1 will result in zero times infinity in the formula, so here is the heads up on those: 0!=1 -1!=+-infinity srn347 —Preceding unsigned comment added by 68.7.25.121 ( talk) 06:33, 19 November 2008 (UTC)
I was likewise confused about the meaning of the functional equation. It says it's valid for all s (except 0 and 1)... but since the gamma function has poles for negative integers, the functional equation is actually meaningless for s = 2, 3, ..., _unless_ it is understood to involve limits in that case. Since the average reader might not be aware of this shorthand, I mentioned it in the article. If you can find a better way to say what I said, go for it. Kier07 ( talk) 03:20, 24 August 2010 (UTC)
Disclosure: I am the son of a world-famous mathematician -- indeed, in her later life a number theorist. However, she mated with a whiskey-soaked advertising copywriter and I was raised on a farm by psychopathic semi-deaf-mutes shunned by their neighbors for the worldly sin of using electricity and automobiles. Mother did not rescue me from this Appalachian pastoral idyll for many years, whereupon she made my bedtime tales out of graph theory.
So, I have only half the genes and half the nurture. I have managed to stagger through a computer engineering career with nothing higher than the calculus -- which only served me once, and that indirectly. On the brighter side, I'm generally able to comprehend intelligent explanations, often in polynomial time.
As it stands, I find this article to be incomprehensible and without merit here. Wikipedia is a general reference work. For content to be included here, it has to pass the Joe test: If you had unlimited time in which to explain it (in far more detail than here) to Joe, the Everyman; if Joe was completely cooperative, intelligent, and patient; if eventually he understood all you intend, then could he imagine any possible way in which this subject might be of interest? This topic fails the Joe test.
This article bears a disturbing resemblance to Graveler. It is an isolated swatch of factoplasm lifted out of a highly specialized context, meaningless outside that context. Nothing has been done to show that it has any application to the Real World. While the subject may be a vital part of its own bubble universe, nothing in the article connects it to anything outside.
To be sure, this is longer than Graveler, and arguably factual; but I think that by the time you get past definitions of terms, it is equally valid to say that it is very like a playing card in a mathematical game. If there is a connection to anything concrete, that needs to be shown.
So, I suggest the article be downsized to a bare description of the function and merged into its parent article. (It does at least have some relationship to a larger mathematical topic, does it not?)
Another possibility is to open a new WikiBook entitled "Higher Mathematics" and expand this article into a whole chapter -- there.
I hold out one other route for improvement. I almost began to see a glimmer of light in Applications. Perhaps a diligent effort could actually find some application of this function to something tangible. If so, rewrite this section and move it to the introduction. MBAs and fools like me can read the 4 or 5 sentences that place the function in context, and then we can graze on.
— Xiong 熊 talk * 05:11, 2005 September 12 (UTC)
Aside from psychoanalysis of this person (is he (or she) really Chinese), he has some point. But he is questioning how wikipedia should be written, and this is probably not a place to do. There are many, too many, hopelessly technical pages already. But the consensus is we should try to revise it so that laymen can understand it, not that eliminate or downsize it from wikipedia. -- Taku 03:31, 14 September 2005 (UTC)
And no matter how uninteresting "Joe" might find this, im sure he's interested by the fact that the Riemann Hypothesis one of the Millenium Problems. As for being useless, this problem is closely related to the primes. Both the distribution of primes and primality testing are two of the biggest problems in number theory, both of which are closely related to cyptanalysis, which is certainly useful. -- He Who Is[ Talk ] 01:58, 29 June 2006 (UTC)
a) This article does not fail the "Joe test" as characterized above. b) Even if it did, the "Joe test" does not reflect Wikipedia policy or intent. -- 98.108.202.144 ( talk) 01:56, 31 August 2010 (UTC)
I have read, reread, and then reread this article. I am not a mathematician, but I am also not ignorant of higher mathematical concepts. This article is so technically oriented that it is virtually impossible to comprehend for a layman. In fact, I would wager that unless one already understood the Riemann zeta function that this article would be so complex that it would be useless. Please don't attack me for saying this, I only wish to improve an article that others have obviously worked very hard on. I hope someone will take up the challenge. —Preceding unsigned comment added by 70.121.7.89 ( talk • contribs)
You may be right that it's impossible for non-mathematicians to understand most of it. But that may perhaps be true of nearly any article that could be written on this sort of topic.
But I think you're wrong to say that only those who already understand the zeta function can understand this article. I think most mathematicians not familiar with the zeta function would understand it.
Michael Hardy
01:30, 28 January 2007 (UTC)
Looking at it now, I'd say the "Definition" section, the "Relationship to primes numbers" section, and the "Specific values" subsection would be readily understood by non-mathematicians (even if not by those who simply dislike math and never study math at any level). So I think you're being a bit alarmist. Certainly there are some things here that few besides mathematicians will understand, but far from everything. Michael Hardy 23:20, 4 February 2007 (UTC)
This paragraph is not formal at all, and it surely quite confusing. Who says that being coprime with a prime and with another one are mutually indipendent? This kind of trick doesn't always work and often one has to introduce a correction factor. Also, what does "s randomly selected integers" mean? What kind of probabilty distribution are we using on the set of natuaral numbers?-- Sandrobt ( talk) 16:17, 7 December 2010 (UTC)
please insert this astonishing new informations from Marcus du Sautoy (" The Music of the Primes"): "the moments of the Riemann zeta function...We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42 ... Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence"
188.46.217.137 ( talk) 21:13, 12 December 2010 (UTC)
Stumbled upon this article from 97, http://arxiv.org/abs/physics/9705021 claiming a new formula for \zeta(2n+1). Money is on it being a crank, however, if its actually right then it seemed like something to throw on the page. -- Differentiablef ( talk) 16:14, 13 April 2011 (UTC)
That formula for zeta(2*n+1) seems interesting since i don't think anyone was able to 'translate' it to any of the other formulas we know... and i don't mean his definition of B(s) := -s*zeta(1-s) and B'(s)=-zeta(1-s)+s*zeta'(1-s) and then zeta(2*n+1) in terms of B'(s) - since that is just a reformatting of the functional equation of zeta and the derivative (of the functional equation). I mean the formula for B(s) in terms of how Woon calls it 'analytic continuation of an operator'. The closest _looking_ formla to Woons formula for B(s) that we have is i think this http://en.wikipedia.org/wiki/Hurwitz_zeta_function#Series_representation and http://en.wikipedia.org/wiki/Bernoulli_polynomials#Another_explicit_formula. They look very similar to Woons - but still slightly different. But the common wisdom is that there are no _easy_ formulas for zeta(2*n+1) - there are definitly ones we can discount because they are not easy - so maybe that can also be applied to Woons formula. But for example what about zeta(n) = C_n*int(B_n(x)*cot(pi*x),x,0,1) for odd integer n? Here B_n(x) is the nth bernoulli polynomial, cot(x) = cos(x)/sin(x), int(f(x),x,0,1) is the integral for x over [0,1] and C_n is expressible in powers of pi and factorial(n) (see further 'Abramowitz and Stegun: Handbook of Mathematical Functions' http://people.math.sfu.ca/~cbm/aands/page_807.htm). Is that worth mentioning? — Preceding unsigned comment added by Petersheldrick ( talk • contribs) 00:09, 8 July 2011 (UTC)
Without wishing to enter into a debate over the modern convention for defining the function, I would like to point out that Riemann himself defined it as a contour integral, not by analytical continuation. It is evident from Riemann's paper that this is so. There is quite a good discussion of this point also in section 1.4 of Edwards' book, which you include in the References section. — Preceding unsigned comment added by 58.168.69.120 ( talk) 00:22, 13 October 2011 (UTC)
The first diagram presents a color-based encoding of the values of ζ(s) in the complex plane. The last sentence of the caption to that diagram says: "Positive real values are presented in red." I'm wondering if that should more precisely read: "Positive real parts are presented in red", especially as the word "value" is used previously to refer to the value of ζ(s), not to the [real] value of its real or imaginary part. — Preceding unsigned comment added by 210.9.140.245 ( talk) 09:53, 18 March 2012 (UTC)
So all values of the zeta function in the entire half-plane Re(s) > 1 are real and positive? Equivalently, the Dirichlet series of which the zeta function is the analytic continuation converges to a positive real number for any s satisfying Re(s) > 1? — Preceding unsigned comment added by 122.110.54.252 ( talk) 03:43, 19 March 2012 (UTC)
Is there some mathematician who can clear up my confusion? If I'm reading the caption correctly, the red expanse in the first diagram in the article indicates that ζ(s) is everywhere real and positive in the half-plane Re(s) > 1. The function obviously takes real values for real s > 1 but is it actually real and positive elsewhere in that half-plane? Or am I misunderstanding the diagram? — Preceding unsigned comment added by 210.9.140.245 ( talk) 20:21, 21 March 2012 (UTC)
Still no answer to the previous question but in the meantime I've made a slight correction to the series definition of ζ(s) in the Definition section, where it had a set membership sign (read as "σ belongs to the real part of s > 1") that was obviously meant to be an equals sign. — Preceding unsigned comment added by 210.9.140.245 ( talk) 07:45, 6 April 2012 (UTC)
Thanks for your response but I'm still confused. I think the caption needs to be rewritten for clarity. The relationship between color and hue is quite unclear, and I can't tell whether they're meant to be independent attributes of each pixel. There seems to be a "parent" attribute (color) with two child attributes, "dark color" and "hue", which makes no logical sense to me.
If there are two independent attributes of each pixel that constitute the encoding, then the significance of each needs to be clearly defined, and with no interaction between them. If I knew enough about the RZF I could probably work out what the caption is trying to say and could redraft it myself. But I don't, so I can't. As someone who understands the rudiments of complex-valued functions of a complex variable (which is all the knowledge that the diagram presupposes), I find the caption in its current form very confusing. I appreciate that my difficulty may not be comprehensible to someone with a good understanding of the RZF but I presume that the article, and certainly that diagram, is written for the likes of myself, not for RZF specialists. — Preceding unsigned comment added by 210.9.140.245 ( talk) 10:18, 9 April 2012 (UTC)
I wasn't and thank you. If I'm understanding things correctly, the diagram uses two of the three HSL attributes, viz., hue (to represent the argument) and lightness (to represent distance from zero, or equivalently the modulus).
Actually I don't think it's either necessary or desirable to require or assume familiarity of the reader with the HSL model, or to have to point the reader to any explanation of it. In any case, since hue is referred to explicitly (with a link to its Wikipedia entry) I would expect the same for lightness, which as far as I can tell is referred to only implicitly (in the phrase "dark colors"). The fact is, the information that needs to be conveyed in the caption is simple enough that informal English can be used for greater clarity and without any loss of precision. Thus, I would suggest use "color" and avoid "hue" entirely, and perhaps use "intensity" to capture the lightness attribute. In that way, the caption would be immediately intelligible to any reader regardless of their familiarity with HSL and without the reader having to wonder about the difference between "color" and "hue", or indeed whether there is any (which was another source of my confusion).
I'm still a little confused by the statement "Positive real values are presented in red", which doesn't seem to sit well with either the previous use of the term "values" ("dark colors denote values [of the function] close to zero") or the HSL model. If those values are of the argument (as previously explained by sandrobt) it seems a weird thing to say, as arguments are always real anyway. This was the source of my original confusion. I would therefore suggest rewriting the last sentence of the caption to avoid the word "real" and to make it clear that red represents values of the function that have a positive argument. I find the use of the unqualified word "values" in the last sentence ambiguous and confusing.
Finally, a question: At the scale at which the diagram is drawn, would no significant variation in the lightness of the red expanse be visible? In other words, over the red-colored half-plane does the function take only values whose (close) distance from zero can't be visually distinguished on the scale of the diagram? I'm assuming that the red is meant to be a "dark" version of that color (= hue) and thereby denotes proximity to zero as per the caption ("dark colors denote values close to zero").
This may all sound somewhat pedantic but I'm speaking from the point of view of the audience to whom I believe the article is addressed, viz., someone literate in English and partially literate in complex functions. It's a nice diagram, which uses a powerful visual technique to capture the reader's attention and illustrate some salient aspects of the behavior of the function. I think it's therefore reasonable to expect a comparable level of clarity in the wording of the caption as in the mathematical discussion that it accompanies.
Thanks for the further clarification. So it's true that positive real values are encoded in red — as well as a lot of other values! I was mistakenly (though I think understandably) reading the sentence as a definition of "red" and thus inferring the unintended converse of your statement. Your new wording is much better. I would suggest one more improvement for maximum clarity: "Values with arguments close to zero (in particular, positive real values on the real half-line) are encoded in red". This just makes it absolutely clear that the red encoding applies to positive reals by virtue of a more inclusive encoding criterion, and also tells the reader where those positive real values actually occur among that vast expanse of red. — Preceding unsigned comment added by 210.9.140.245 ( talk) 21:06, 11 April 2012 (UTC)
It's clear now, thank you. — Preceding unsigned comment added by 210.9.140.245 ( talk) 09:49, 14 April 2012 (UTC)
That diagram is not explained very well ... where is the "white spot" at s1. First of all, why isn't "S" labelled on the axis. Which Axis? If you label an axis "S" then we could find S1 but you didn't and then I don't see any "white spot" anyways. So, if you could label which axis is S and then maybe make the "white spot" maybe bigger so we can actually see it because I don't see a white spot do you?
Ty
173.238.43.211 ( talk) 18:53, 6 May 2012 (UTC)
Ya well it shouldn't take a lot of ingenuity to figure out THERES NO WHITE SPOT ON THAT GRAPH. Its bad enough you think a reader can just assume which axis you are talking about but its even worse that you claim theres a "white spot" and then not even label which axis its on. I can only assume that kinda black/green/white "bubble" area in the middle of the chart is supposed to be a "white spot" but it happens to have black and green in it also so no idea where this "white spot" is or which is s1 so does anyone know who made this graph so he can explain it so we get it? Do you mean the black egg shaped spot in the centre?? Is that black spot the "white spot"?
173.238.43.211 ( talk) 08:24, 7 May 2012 (UTC)
Ok, thank-you. Perhaps someone else that doesn't see the "obvious white spot on the unmarked axis consisting of blue/green/ and yellow colors" will hopefully read through this talk page to find out that the blue/green/yellow spot is the "white spot". Do you see how small that "white spot" is? Its a pin prick. If only the supposed "white spot" area was circled then noone would ever question where this "white spot" that has blue, green, and yellow in it actually is. That was only the point of me asking where it was. It was where I kind of guessed it is but wasn't sure since the diagram in no way makes it easy to know for sure was all I wanted to get across. Sorry if my question bothered you. We all win in the end.
173.238.43.211 ( talk) 18:19, 9 May 2012 (UTC)
Oh and this article and graph about the zeta function is less problematic to understand in my opinion;
http://simple.wikipedia.org/wiki/Riemann_hypothesis
173.238.43.211 ( talk) 12:31, 10 May 2012 (UTC)
Maybe calling it a "singularity" instead of "white spot" at s1 would be better since its really blue/green/ and yellow and the axis isnt labelled so we could know for sure - just my opinion
173.238.43.211 ( talk) 22:39, 10 May 2012 (UTC)
The link to Möbius function leads to a function defined only for integers. I suspect that this is not the function intended. Klausok ( talk) 08:13, 23 October 2012 (UTC)
Hi, I think this should be mentioned at the original page
Log[2/3/E^(5 (Pi/2)) + E^(7 Pi)/(7/2/E^(7 (Pi/2)) + 5/2/E^(5 (Pi/2)) + 3/2/E^(3 (Pi/2)) + E^(5 (Pi/2)) + 2 Pi)] = 14.134725141734373744769652011837891376454997031685134...
But the first zero on the critical line is Zeta(1)=
14.134725141734693790457251983562470270784257115699...
Have a nice day, Dietmar — Preceding unsigned comment added by 217.81.127.212 ( talk) 16:08, 5 November 2012 (UTC)
How is the zeta function of a half calculated? In fact, how is the zeta function of any number between zero and one calculated?! — Preceding unsigned comment added by 86.136.129.187 ( talk) 19:12, 3 December 2012 (UTC)
Would it be possible for me to re-write the article from scratch, and then to go from there?
The article is in a deplorable state. It is too technical and in addition the technical parts are not central to the current (or past) research on $\zeta(s)$. The emphasis on "Selberg's conjecture" (not a conjecture in the first place but more of a remark in Selberg's paper) is badly misleading.
It seems that somebody very fond of Karatsuba keeps putting these here. It is not normal to have 8 references to Karatsuba and 1 to Shanker (never heard of him) and no references to Hardy-Littlewood, Ingham, Bohr-Landau, Titchmarsh and more recently to for example Conrey, Soundararajan, Iwaniec, etc.
Anyway, if this is not possible then I will go away and the article will remain in its deplorable state... — Preceding unsigned comment added by Karatsuba ( talk • contribs) 07:50, 9 June 2013 (UTC)
Basically everything after and including the section "Zeros, the critical line, ..." needs to be purged and re-written. There is too much misleading emphasis (i.e Karatsuba, identities which nobody uses) there and too much original research (i.e Shanker).
The Section "Estimates of the maximum of the modulus of the zeta function" contains a typo: "The case was studied by Ramachandra" should read "The case was studied by Ramanujan". I'd not go as far as editing the article as I would not dare to touch an article on mighty maths, just a suggestion. — Preceding unsigned comment added by 152.66.244.111 ( talk • contribs) 15:13, 13 May 2011 (UTC)
No the result is due to Ramachandra and not Ramanujan. — Preceding unsigned comment added by Karatsuba ( talk • contribs) 07:56, 9 June 2013 (UTC)
Recently text on the so-called Selberg conjecture was removed by an anonymous IP editor. This removal was reverted (on somewhat superficial grounds). It seems to me that the IP has a legitimate point. The material under discussion at best seems like undue weight, and at worst original research. (See the comments also in the preceding section, which pertain to the same material). May I suggest that we leave this material out unless the case for its inclusion is made more clear. Sławomir Biały ( talk) 01:55, 11 June 2013 (UTC)
There should be more on zero density theorems here or in the article on the Riemann hypothesis. — Preceding unsigned comment added by 86.181.154.200 ( talk) 12:55, 28 December 2013 (UTC)
The recent addition of an external link to Brady Haran's video on the zeta function was reverted as Added link does not add any new information about the subject (just mentions it). The subject of the video was "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12". This brings up three points.
I'm not interested in an edit war, so I'll neither put the link back nor remove the deminishment of Euler. But somebody should. user:JMOprof ©¿©¬ 18:22, 10 January 2014 (UTC)
There seems to be an edit war going on at the moment, and we should resolve it here on the talk page rather than by a bunch of different editors reverting and re-reverting.
Is Zeta(-1) = 1 + 2 + 3 + 4 .... ? Until we can come to consensus and reliably source this as settled, we probably shouldn't put it in the article, or if we do we need to be careful to provide context so that it is not misunderstood by the lay reader. My understanding of wiki policy is to leave it out until a consensus is reached.
A couple of observations:
There's a quite misleading video making the rounds, and many users are coming here to "verify" whether it's true or not. We need to provide a solid article. Looking forward to comments. Mr. Swordfish ( talk) 19:03, 21 January 2014 (UTC)
As of today, the formula
is back in the article. Are we going to just keep removing and restoring it on a weekly basis, or are we going to try to come to some consensus here on the talk page? Let's hear the reasons for including it. Mr. Swordfish ( talk) 16:33, 17 February 2014 (UTC)
Such a statement obviously presents philosophical difficulties. Namely, one is forced to ask how the “sum” of a divergent series of entirely positive terms can be negative. Yet the manipulations involved in our determination of s are no more outlandish than those used in determining 1 − 1 + 1 − 1 + ··· = 1/2. We will see later that in a very precise sense, −1/12 is the correct value of 1 + 2 + 3 + 4 +···.
With 64 threads, some of which are approaching their second decade, how about implementing auto archive? I will volunteer to do it if there is consensus. Mr. Swordfish ( talk) 16:55, 18 February 2014 (UTC)
You say that the Riemann zeta function can also be defined by the integral:
But you never stated whether this definition is valid everywhere on the complex plane (wherever the function exists). People just learning about Zeta function regularization of what I like to call the Ramanujan (1+2+3+...) are interested in the analytic continuation of the Riemann zeta function over the entire complex plane. It seems trivial to me that the aforementioned definition is valid over the entire plane, but I might be wrong. You need to either state that this equation is valid for the entire complex plane, or you need to help people like me who will look at the equation, do a bit of sloppy mental math and conclude (falsely?) that the expression does define the function everywhere that it exists.-- guyvan52 ( talk) 15:18, 21 December 2014 (UTC)
The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function, following Riemann.)
The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:
The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Eqivalentlyreference?, it can also be defined everywhere that it exists on the complex plane by the integral
-- guyvan52 ( talk) 15:18, 21 December 2014 (UTC)
Under Various properties and Reciprocal is the statement "the claim that this expression is valid". This expression? Which one? John W. Nicholson ( talk) 04:54, 15 March 2015 (UTC)
Someone changed the equation into: (i.e.: minus 1 changed into plus 1 in the denominator)
This seems suspicious, since there is no explanation, and no other edits from that IP. Can someone looks into it to verify if it is legitimate? — Preceding unsigned comment added by Dhrm77 ( talk • contribs) 05:20, 25 March 2016 (UTC)
This paragraph first states "it was first proved by blah in 19xx" but then says "Nevertheless, none of the proofs above are definitive or complete, since a definitive proof would imply that Riemann's Hypothesis is true."
Is it proved or not proved? Whats missing? This is confusing to me. — Preceding unsigned comment added by 32.215.35.33 ( talk) 18:54, 21 June 2016 (UTC)
Well yes, it is not written exactly clearly. But idea is, that someone proved something while considering that Riemann Hypothesis is true, so basically if Riemann Hypothesis is true then whatever that someone proved will be also true, but if it is not, then it might not be. So basically they proved something, but took an unproven hypothesis as basis for their proof. Trimutius ( talk) 15:40, 22 June 2016 (UTC)
That whole section from that point down starts assuming that the reader knows things that have not yet been explained or defined. I would say the entire page needs to be reworked, starting at the top with definitions. 71.48.255.210 ( talk) 20:56, 14 July 2016 (UTC)
i understand nothing to this notion of «ζ(x) zero», so if true, i think we should ad it to explain — Preceding unsigned comment added by BeKowz ( talk • contribs) 12:59, 3 December 2016 (UTC)
The article mentions the following relationship between and under the heading "Mellin transform".
I was hoping to explore the relationship above using the Fourier series representation for which is an infinite series of Fourier series where each Fourier series consists of an infinite series of Sine terms (see Fourier Series Representations of Prime Counting Functions). Unfortunately a term of the form resists integration. However, I have been able to explore the relationship above using the relationship below was derived from the original relationship above via integration by parts.
I haven't been able to explore the second relationship above using the Fourier series representation for because a term of the form also resists integration, but the second relationship above can be evaluated by writing the integral as a sum as follows.
Using the sum formula for the second relationship above, I confirmed the the real part of converges to some degree for the two relationships above for , but the imaginary part of diverges significantly using the sum formula above which makes me suspect the validity of the two relationships above.
StvC ( talk) 01:04, 5 November 2016 (UTC) StvC ( talk) 00:49, 15 December 2016 (UTC) StvC ( talk) 05:15, 16 December 2016 (UTC) StvC ( talk) 16:46, 16 December 2016 (UTC)
In the Riemann zeta function page it is written that:
and that:
.
However looking at the Gamma function definition in Wikipedia one can see that:
.
Correspondingly .
More looking at Riemann's paper he has the s-1 also.
However Riemann instead of Gamma used factorial .
I tried to fix the page but it was rejected as vandalism.
Please correct it or explain to me what am I wrong about.
Adikatz ( talk) 06:52, 26 July 2017 (UTC)
In the definition of the zeta function the definition of the Gamma function is wrong. It should be s-1 instead of s. The s-1 should also be in the integral representing the gamma function. I have tried to fix but it was revoked as vandalism.
. Or . But not . Sapphorain ( talk) 07:57, 26 July 2017 (UTC)
Sorry. I was mistaken since I did not pay attention to the fact that the dx was divided by x. Adikatz ( talk) 14:21, 26 July 2017 (UTC)
Here are some more representations, but I don't know what subsection to put them in, or what commentary to give on them:
for positive integer n
for 0 < Re(s) < 1, where frac is the fractional part
for Re(s) > 1
(Reference: http://mathworld.wolfram.com/RiemannZetaFunction.html)
-- AndreRD ( talk) 16:29, 27 July 2017 (UTC)
If the complex number s = 1/2 + 0j is put into the Riemann functional equation after the equation is rearranged so the left hand side reads E{s}/E{1-s} then the left hand side with a value of 1 does not equal the right hand side and this means that the complex part is not zero and so there can be no solution on the 1/2,0 coordinate of the complex plane as Riemann says there is. Have I got this right?
Soopdish ( talk) 12:48, 8 August 2017 (UTC)Soopdish
The Integral sections shows a formula for the zeta function that is super wrong. The correct formula can be found here
Abel–Plana formula
It should look like this:
The provided link #24 is also wrong. It links to an arc length calculus page.
--
EmpCarnivore (
talk)
21:19, 7 September 2018 (UTC)
Is there a formula out there that relates the zeta function to the perfect powers (1, 4, 8, 9, 16, 25, 27, 32, etc) in a similar way that Euler's product formula does for primes? The reason I ask is because I discovered a very simple one a while back, but I can't find information about any other such formulas. Thanks. -- Vagodin 14:47, August 21, 2005 (UTC)
The 'globally convergent series' found by Hasse appears to be essentially just the Euler transform applied to the Dirichlet eta function. — Preceding unsigned comment added by Carifio24 ( talk • contribs) 23:22, 4 July 2007 (UTC)
I programmed the second of the two Hasse series, and the second one seems to be wrong. It returns large incorrect values. If there's a bug, I must be blind:
The first series (which works correct) uses the following code:
double hasse0(double s, uint64_t limit) {
double u = 0; for (uint64_t n=0; n<limit; ++n) { double r = 0; for (uint64_t k=0; k<=n; ++k) { double t = choose(n,k) / pow(k+1,s); if (k%2) r -= t; else r += t; } u += r/pow(2,n+1); } return u/(1-pow(2,1-s));
}
The second series (which does not work) uses the exact same template:
double hasse1(double s, uint64_t limit) {
double u = 0; for (uint64_t n=0; n<limit; ++n) { double r = 0; for (uint64_t k=0; k<=n; ++k) { double t = choose(n,k) / pow(k+1,s-1); if (k%2) r -= t; else r += t; } u += r/(n+1); } return u/(s-1);
} — Preceding unsigned comment added by Pcp071098 ( talk • contribs) 23:07, 27 August 2013 (UTC)
Under "Specific values", the graph seems to be of three functions, only one of which is the Zeta function. The other two seem to be based on a finite number of terms of the infinite series in the definition of the Zeta function. — Preceding unsigned comment added by 79.79.29.112 ( talk) 08:15, 28 July 2017 (UTC)
Also, for we have:
where, in both expressions, refers to the
Euler-Mascheroni constant. — Craciun Lucian.
In the section on the proof of the functional equation, the text keeps mentioning criteria on sigma, like sigma > 0, sigma > 1. But then the actual computations have no sigma. They're all in terms of s, is sigma meant to be s here? - lethe talk + 19:04, 29 January 2020 (UTC)
Why use , not , as ? I think is more “natural”. Three similar functions are , and . — Preceding unsigned comment added by Xayahrainie43 ( talk • contribs) 13:54, 24 September 2018 (UTC)
The following anonymous edit comes from an IP with a very checkered history. It needs vetting:
Thanks. -- Wetman 13:02, 18 Apr 2005 (UTC)
This section seems suspect since 1) the reference [21] is just a reference to the cauchy integral theorem, and 2) the series presented does not appear to be a dirichlet series. Can someone confirm if this is even true? — Preceding unsigned comment added by 208.38.59.163 ( talk) 20:19, 23 September 2020 (UTC)
Note: it does give the correct value of zeta at 2, but i am still not seeing where it comes from? — Preceding unsigned comment added by 208.38.59.163 ( talk) 20:29, 23 September 2020 (UTC)
Under "Globally convergent series", I think that Knopp's conjecture was made in 1926. — Preceding unsigned comment added by 213.48.238.18 ( talk) 13:42, 3 November 2020 (UTC)
Under "Generalizations", the Clausen function is mentioned, ungrammatically. — Preceding unsigned comment added by 79.77.163.188 ( talk) 13:01, 26 April 2021 (UTC)
This article uses the picture /info/en/?search=Riemann_zeta_function#/media/File:Zeta_polar.svg
Note that it is not a polar graph in any sense, it is a regular Cartesian graph, with and , . It should be corrected and renamed; the current version is misleading. A1E6 ( talk) 18:17, 21 July 2021 (UTC)
There is of course -s function equation and alternating series form but what about actual closed form? Valery Zapolodov ( talk) 10:13, 4 January 2022 (UTC)
On Riemann sphere that is? Valery Zapolodov ( talk) 13:50, 4 January 2022 (UTC)
@ Jomu5221, Tanfer Tanriverdi, and JayBeeEll: I just removed a recently added result from the article because it is not notable:
The recent addition of a claim about "the first work that led to the derivation of the formula" is even less suitable for the article:
Cheers, -- Macrakis ( talk) 00:28, 17 January 2022 (UTC)
I am fairly certain that this article uses log() to denote logarithms in base e rather than base 10. If so, please confirm this and I will add the following template: Toadspike ( talk) 16:32, 12 May 2022 (UTC)
The first series under the "Representation" section does not seem to converge correctly (e.g. near zero, it converges to zero when it should converge to -1/2). Other representations seem to work fine; indeed the very next series converges to -1/2 as expected. Moreover, I chased the given reference, and did not see (either) series listed in the given location. It is possible, perhaps likely, that I have made an error. — Preceding unsigned comment added by Geometrian ( talk • contribs) 03:26, 2 June 2022 (UTC)
https://en.wikipedia.org/?title=Riemann_zeta_function&oldid=prev&diff=1102657060
Oh I guess I should've linked to this
/info/en/?search=Convergence_tests#p-series_test
instead of that
but still?
Thewriter006 ( talk) 07:27, 6 August 2022 (UTC)
A formula for involving the Thue-Morse sequence was recently added. While this result is interesting, I have my doubts that it is sufficiently notable for an encyclopedia article on the Riemann zeta function. I note that the same editor added similar content to the article on Apery's constant. 74.111.98.156 ( talk) 16:57, 18 December 2022 (UTC)