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Is there such a thing as a generalized totient function? Specifically, a function φi(n) that is the count of positive integers less than n with exactly i prime divisors. This means that Euler's totient function φ(n) is φ1(n) in the more general form (i.e., it is the number of positive integers less than n with only one prime divisor). Likewise, φ2(n) is the number of positive composite integers less than n with only two prime divisors, e.g., numbers from the set {4,6,9,10,14,15,21,22,25,...,n}, which includes all the squares of primes. φ3(n) includes {8,12,18,27,28,30,...,n}, and so forth. I dimly recall seeing something about Ramanujan studying something similar to this(?). Perhaps such a thing might also be related to the Riemann hypothesis? — Loadmaster ( talk) 17:24, 30 May 2013 (UTC)
With your help, I would like to find out a mathematical relationship between complete elliptic integrals of the first kind
all of which are known to possess the following property
where
It also goes on without saying that the factorial of every positive number is the gaussian integral of its reciprocal or multiplicative inverse
— 79.118.171.165 ( talk) 18:33, 30 May 2013 (UTC)
Mathematics desk | ||
---|---|---|
< May 29 | << Apr | May | Jun >> | May 31 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Is there such a thing as a generalized totient function? Specifically, a function φi(n) that is the count of positive integers less than n with exactly i prime divisors. This means that Euler's totient function φ(n) is φ1(n) in the more general form (i.e., it is the number of positive integers less than n with only one prime divisor). Likewise, φ2(n) is the number of positive composite integers less than n with only two prime divisors, e.g., numbers from the set {4,6,9,10,14,15,21,22,25,...,n}, which includes all the squares of primes. φ3(n) includes {8,12,18,27,28,30,...,n}, and so forth. I dimly recall seeing something about Ramanujan studying something similar to this(?). Perhaps such a thing might also be related to the Riemann hypothesis? — Loadmaster ( talk) 17:24, 30 May 2013 (UTC)
With your help, I would like to find out a mathematical relationship between complete elliptic integrals of the first kind
all of which are known to possess the following property
where
It also goes on without saying that the factorial of every positive number is the gaussian integral of its reciprocal or multiplicative inverse
— 79.118.171.165 ( talk) 18:33, 30 May 2013 (UTC)