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The formula in the following referenced assertion, which has been recently inserted in the subsection « Other results » of « Zeros, the critical line, and the Riemann hypothesis » is problematic, mainly because of the use by 178.219.5.13 of the symbol , which is infamous for its mathematical imprecision. It is true that this symbol is used at different places in the paper (as well as ), but on the one hand it is never defined there, and on the other hand the well-known symbol , whose rigorous definition is very consensual amongst mathematicians, is also used, seemingly interchangeably with — and even with !— (for instance the authors outrageously state the prime number theorem as , and mention the « asymptotic behaviour » (!), although they constantly use the same symbol when rounding up numerical value to various decimals). Moreover the formula in the following referenced assertion is in fact not given under this form in the cited paper. What the authors state is that the ordinates of zeros on the critical line are given by a certain equation ((13) in the paper). Then, by ignoring the limit term in (13) (although pointing it is usually not zero), they consider a simplified equation (62), whose solutions (63)
they state are approximate solutions for the ordinates of the Riemann zeros. They produce tables of computed values of and showing these values look indeed close to each other, but they never rigorously say what they mean by « approximate ». All of this is extremely sloppy, and the shortcut adopted below makes things even sloppier.
(Reproduced insert, needing clarification/modification/(suppression?):
" The estimated imaginary part of the n-th zero on the critical line has this evaluation: [1]
Sapphorain ( talk) 14:06, 24 January 2023 (UTC)
References
All important results are based on the sieves. These are hard to do and there is only computational result. So to be honest the density of the prime numbers vanishes in the limit of all positive integers. But it does this only in the limit and for any finite and so big integer there is a finite prime number density in the positive integers! SteJaes ( talk) 20:12, 16 March 2023 (UTC)
The surface of the Riemann zeta function looks like this in the critical stripe:
All the zeros look like touches to the surface 0, like that there is a pencil pointing to 0, like dip, the curves look like roots or potency functions from the zero on the critical line to the borders of the stripe. I can offer some pictures showing that exemplary and fundamental behavior. On the borders of the critical stripe there are special behaviors. On the critical line the zero of the real and imaginary part coincide. For y=1 the real part does not have any zeros and the absolute function does not either. For y=0 the situation is different. It is like that the real and imaginary part do a schwebung and the absolute function is the upper limit of the schwebung without a zero possibly. This is hard to prove. The absolute function of the zeta function diverges at most.
This together gives reasons for the idea that the nontrivial zeros are isolated touches on the surface 0 in the Riemann zeta function.
SteJaes ( talk) 20:37, 16 March 2023 (UTC)
Picture of the absolute function of the Riemann zeta function with the first two nontrivial zeros shown.
I suggest to improve the article with my picture of the first two nontrivial zeros. This shows exemplary and fundamental the behavior on the critical line in the critical stripe in the complex plane. This supports but does not prove the Riemann conjecture. It suggests on the other hand the path to a prove. SteJaes ( talk) 20:48, 16 March 2023 (UTC)
I suppressed the (unsourced; and by the way the only appeal to Titchmarsh’s book is wrong) « Proof 2 » of the functional equation, which is incorrect for several reasons. In particular the series in line 9 does in general not converge for (neither to nor to anything else). Sapphorain ( talk) 23:54, 21 December 2023 (UTC)
There's nothing about the finite sum . Wouldn't that be a useful addition? Where would it fit into the article? Renerpho ( talk) 22:55, 18 June 2024 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
This page has archives. Sections older than 150 days may be automatically archived by Lowercase sigmabot III when more than 5 sections are present. |
The formula in the following referenced assertion, which has been recently inserted in the subsection « Other results » of « Zeros, the critical line, and the Riemann hypothesis » is problematic, mainly because of the use by 178.219.5.13 of the symbol , which is infamous for its mathematical imprecision. It is true that this symbol is used at different places in the paper (as well as ), but on the one hand it is never defined there, and on the other hand the well-known symbol , whose rigorous definition is very consensual amongst mathematicians, is also used, seemingly interchangeably with — and even with !— (for instance the authors outrageously state the prime number theorem as , and mention the « asymptotic behaviour » (!), although they constantly use the same symbol when rounding up numerical value to various decimals). Moreover the formula in the following referenced assertion is in fact not given under this form in the cited paper. What the authors state is that the ordinates of zeros on the critical line are given by a certain equation ((13) in the paper). Then, by ignoring the limit term in (13) (although pointing it is usually not zero), they consider a simplified equation (62), whose solutions (63)
they state are approximate solutions for the ordinates of the Riemann zeros. They produce tables of computed values of and showing these values look indeed close to each other, but they never rigorously say what they mean by « approximate ». All of this is extremely sloppy, and the shortcut adopted below makes things even sloppier.
(Reproduced insert, needing clarification/modification/(suppression?):
" The estimated imaginary part of the n-th zero on the critical line has this evaluation: [1]
Sapphorain ( talk) 14:06, 24 January 2023 (UTC)
References
All important results are based on the sieves. These are hard to do and there is only computational result. So to be honest the density of the prime numbers vanishes in the limit of all positive integers. But it does this only in the limit and for any finite and so big integer there is a finite prime number density in the positive integers! SteJaes ( talk) 20:12, 16 March 2023 (UTC)
The surface of the Riemann zeta function looks like this in the critical stripe:
All the zeros look like touches to the surface 0, like that there is a pencil pointing to 0, like dip, the curves look like roots or potency functions from the zero on the critical line to the borders of the stripe. I can offer some pictures showing that exemplary and fundamental behavior. On the borders of the critical stripe there are special behaviors. On the critical line the zero of the real and imaginary part coincide. For y=1 the real part does not have any zeros and the absolute function does not either. For y=0 the situation is different. It is like that the real and imaginary part do a schwebung and the absolute function is the upper limit of the schwebung without a zero possibly. This is hard to prove. The absolute function of the zeta function diverges at most.
This together gives reasons for the idea that the nontrivial zeros are isolated touches on the surface 0 in the Riemann zeta function.
SteJaes ( talk) 20:37, 16 March 2023 (UTC)
Picture of the absolute function of the Riemann zeta function with the first two nontrivial zeros shown.
I suggest to improve the article with my picture of the first two nontrivial zeros. This shows exemplary and fundamental the behavior on the critical line in the critical stripe in the complex plane. This supports but does not prove the Riemann conjecture. It suggests on the other hand the path to a prove. SteJaes ( talk) 20:48, 16 March 2023 (UTC)
I suppressed the (unsourced; and by the way the only appeal to Titchmarsh’s book is wrong) « Proof 2 » of the functional equation, which is incorrect for several reasons. In particular the series in line 9 does in general not converge for (neither to nor to anything else). Sapphorain ( talk) 23:54, 21 December 2023 (UTC)
There's nothing about the finite sum . Wouldn't that be a useful addition? Where would it fit into the article? Renerpho ( talk) 22:55, 18 June 2024 (UTC)