Explicit upper bounds for \(|L(1,\chi)|\) for primitive even Dirichlet characters (Q2773314)

The author proves estimates of the shape NEWLINE\[NEWLINE \biggl|\prod_{1\leq i\leq r}(1-\chi(p_i) p_i^{-1})L(1,\chi)\biggr|\leq \tfrac{1}{2}\prod_{1\leq i\leq r}(1- p_i^{-1})\log f_\chi+\kappa NEWLINE\]NEWLINE for \textit{even} primitive Dirichlet characters of conductor \(f_\chi\), in particular improving on the values of \(\kappa\) given in \textit{S. Louboutin} [J. Reine Angew. Math. 419, 213-219 (1991; Zbl 0721.11049)]. The results are too numerous and various for us to state them all. Here are two typical corollaries~: if \(\chi\) is even then NEWLINE\[NEWLINE |(1-\chi(2)/2)L(1,\chi)|\leq(\log f_\chi+5)/4 NEWLINE\]NEWLINE and if \(\chi\) is even with an even conductor \(f_\chi\) then NEWLINE\[NEWLINE |(1-\chi(3)/3)L(1,\chi)|\leq(\log f_\chi+6)/6. NEWLINE\]NEWLINE Such bounds are especially effective when \(\chi\) is of low order and we can hope special methods will apply in case of characters of higher order as noted recently by \textit{A. Granville} and \textit{K. Soundararajan} [J. Am. Math. Soc. 14, 365-397 (2001; Zbl 0983.11053)]. NEWLINENEWLINENEWLINEThe main novelty in the proofs is from the use of a (corrected version of a) lemma of Hua which enables the author to handle in an efficient way character sums with non-primitive characters. An upper bound for the residue in 1 of the Dedekind zeta-function of a real abelian field is derived from these results. Further uses in an algebraic context are in preparation with Y.-S. Park and S.-H. Kwon.