From: Gerry Myerson <gerry@mpce.mq.edu.au>
Subject: Re: Generalised Riemann Hypothesis
Date: Thu, 15 Feb 2001 09:51:45 +1000
Newsgroups: sci.math

In article <96eku4$knp$1@wanadoo.fr>, "Nico" <gerard.nin@wanadoo.fr> 
wrote:

> What is exactly the "Generalised Riemann Hypothesis" ?

According to Davenport, Multiplicative Number Theory, the GRH is 
"the hypothesis that not only zeta(s) but all the functions 
L(s, chi) have their zeros in the critical strip on the line 
sigma = 1/2 (this conjecture seems to have been first formulated 
by Piltz in 1884)." 

L(s, chi) is defined, for s > 1, by 

    L(s, chi) = sum_1^infty chi(n) n^(-s)

and then extended to the complex plane by analytic continuation 
(just as for the zeta function). 

chi here is any function of an integer variable n which is periodic 
with period q and multiplicative. q is an arbitrary integer exceeding 1. 

This is a "generalized" RH because you get RH when chi is identically 1. 

Gerry Myerson (gerry@mpce.mq.edu.au)