Explicit upper bounds for |L(1, \chi)|
by
Stephane Louboutin
IML, Luminy, France

The aim of this talk is to present the proof and variuos applications of the following result of us which has just appeared in Quart. J. Math. 55:

Let S be a given finite set of pairwise distinct rational primes. Then, for any primitive Dirichlet character \chi of conductor q_\chi > 1 we have

ê ê ì í î Õ p in S  (1-  \chi(p) p ) ü ý þ L(1, \chi) ê ê <=  1 2 ì í î Õ p in S  (1-  1 p ) ü ý þ æ è logq\chi +\kappa\chi +\omegalog4+2 å p in S   logp p-1 ö ø +o(1),

where

\kappa\chi = ì ï í ï î \kappaeven = 2+\gamma-log(4\pi) = 0.046191 ... if \chi(-1)=+1 \kappaodd = 2+\gamma-log\pi = 1.432485 ... if \chi(-1)=-1,

where \omega >= 0 is the number of primes p in S which do not divide q_\chi, and where o(1) is an explicit error term which tends rapidly to zero when q_\chi goes to infinity. Moreover, if S = \emptyset or if S = {2}, then this error term o(1) is always less than or equal to zero, and if none of the prime in S divides q_\chi then this error term o(1) is less than or equal to zero for q_\chi large enough.

Date received: March 23, 2004