From: Richard Pinch Newsgroups: sci.math Subject: Re: Generalized Riemann's Hypothesis Date: Sat, 05 Jul 1997 13:00:39 +0100 Harden wrote: > > What exactly is the Generalized Riemann's Hypothesis? I have heard it > implies many things, including a polynomial time nonprobabilistic > primality testing algorithm and the existence of small quadratic > nonresidues modulo primes( under 2(ln p)^2 where p is the prime ). A Dirichlet character chi is a map from the integers modulo m to roots of unity or zero which is totally multiplicative: chi(rs) = chi(r) chi(s) for all r, s. The L-series attached to chi is L(chi,s) = sum chi(n) n^{-s}. Like the zeta function it has an Euler product expansion prod_p (1 - chi(p)p^{-s})^{-1}, analytic continuation to the whole complex plane with a functional equation and so on (although if chi is non-trivial it does not have a pole at s=1). The GRH, apparently first written down by Piltz, 1884, is that every such function has its zeroes in the criticial strip 0 <= re.s <= 1 on the line re.s = 1/2. -- Richard Pinch Queens' College, Cambridge rgep@cam.ac.uk http://www.dpmms.cam.ac.uk/~rgep