Register to reply 
Is this a route to the prime number theorem? 
Share this thread: 
#1
Oct610, 04:07 AM

PF Gold
P: 2,921

I notice that the trick to define Dirichlet Eta Function can be repeated for each prime number, let p a prime, and then
[tex]\eta_p= (1  p^{s}) \zeta(s)  p^{s} \zeta(s) = (1  2 p^s) \zeta(s)[/tex] So each prime p defines a function [tex]\eta_p[/tex] that adds a family of zeros at [itex]s= (\log 2 + 2 n i \pi) / \log p} [/itex]. Particularly, for p=2 it kills the only pole of zeta in s = 1. Repeating the trick for each prime, we seem to obtain a function [tex]\eta_\infty(s)[/tex] having the same zeros that the Riemann zeta plus families of zeros of density 1/log p placed at the lines r=log(2)/log(p) Or, if we dont like to crowd the critical strip, we can use only the left part of the eta, the [tex] \eta^F_p(s)= (1  p^{s}) \zeta(s) [/tex], and then for [tex]\eta^F_\infty(s)[/tex] we get all the families cummulated in the r=0 line. But on other hand it can be argued that [tex]\eta^F_\infty(s)=1[/tex], as we have removed all the factors in Euler product. So there should be some relation between the n/log(p) zeros in the imaginary line and the other zeros in the Riemann function 


Register to reply 
Related Discussions  
The prime number theorem  Linear & Abstract Algebra  1  
Question about the prime number theorem  General Math  1  
Prime number theorem  Linear & Abstract Algebra  6  
Prime Number Theorem  General Math  0  
A formula of prime numbers for interval (q; (q+1)^2), where q is prime number.  Linear & Abstract Algebra  0 