A pseudo zeta function and the distribution of primes
AUTOR(ES)
Chernoff, Paul R.
FONTE
The National Academy of Sciences
RESUMO
The Riemann zeta function is given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\zeta}(s)={ \,\substack{ ^{{\infty}} \\ {\sum} \\ _{n=1} }\, }\hspace{.167em}1/n^{s}={\prod_{p\hspace{.167em}{\mathrm{prime}}}}\hspace{.167em} \left( 1-\frac{1}{p^{s}} \right) ^{-1}\hspace{.167em}{\mathrm{for\hspace{.167em}Re\hspace{.167em}}}s>1.\end{equation*}\end{document} ζ(s) may be analytically continued to the entire s-plane, except for a simple pole at s = 0. Of great interest are the complex zeros of ζ(s). The Riemann hypothesis states that the complex zeros all have real part 1/2. According to the prime number theorem, pn ≈ n logn, where pn is the nth prime. Suppose that pn were exactly nlogn. In other words, in the Euler product above, replace the nth prime by nlogn. In this way, we define a pseudo zeta function C(s) for Re s > 1. One can show that C(s) may be analytically continued at least into the half-plane Re s > 0 except for an isolated singularity (presumably a simple pole) at s = 0. It may be shown that the pseudo zeta function C(s) has no complex zeros whatsoever. This means that the complex zeros of the zeta function are associated with the irregularity of the distribution of the primes.
