From: Jan-Christoph Puchta <jcp@arcade.mathematik.uni-freiburg.de>
Subject: Re: a question in number theory
Date: Tue, 16 May 2000 19:38:03 +0200
Newsgroups: sci.math.research
Summary: [missing]

qcheng wrote:
> 
> Hi, I have a question about A(q,1) in following
> formula.
> 
> \sum_{p\leq x, p\equiv 1 (mod q)} \phi(q)/p
> = \log \log x + A(q,1) +O( 1/\log x)
> 
> What is behavior of A(q,1) when q goes to infinity?
> Is A(q,1) bounded from above?

A(q, 1) = \sum_{\chi\neq\chi_0} L(1, \chi) - C, where C does not depend
on q. Using the technic from Montgomery and Masley (J Reine Angew Math
286/287) one can show that 1/log q << A(q, 1) << log q, in an article in
the recent Arch. Math. it is shown that this can be improved to
1/loglog^2 q << A(q, 1) << loglog^2 q, provided there is now Siegel
zero. In general the problem is similar to the asymptotic behaviour of
the class number of the p-th cyclotomic field, and Kummer's conjecture
(which is equivalent to A(q, 1) - A(q, -1) -> 0) seams to be wrong.
There has been an article by A. Granville (I think) giving heuristics
that A(q, 1)- A(q, -1) is not bounded.


JCP