×

Extreme values of Dirichlet \(L\)-functions at 1. (English) Zbl 0942.11040

Győry, Kálmán (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter. 1039-1052 (1999).
In 1928, J. E. Littlewood [ Proc. Lond. Math. Soc. 27, 349-357 (1929; JFM 54.0367.03)] proved that if the Generalized Riemann Hypothesis (GRH) is true, then there are infinitely many \(q\) such that \[ L(1,\chi)\sim \pi^2 e^{-\gamma}/(6 \log \log q)\tag \(*\) \] where \(\chi\) is a real non-principal character modulo \(q\) and \(\gamma\) is Euler’s constant. In 1949, S. Chowla [ Proc. Lond. Math. Soc. (2) 50, 423-429 (1949 Zbl 0032.11006)] showed unconditionally that \((*)\) holds for infinitely many \(q\) with \(\chi\) a real primitive character modulo \(q\). R. C. Vaughan [Analytic number theory. Vol. II, Prog. Math. 139, 755-766 (1996; Zbl 0853.11073)] showed that the phenomenon in \((*)\) happens quite frequently.
In this paper, the authors develop a probabilistic model for extremal values of \(L(1,\chi)\), and they use this model to make a series of conjectures on such extremal values and the frequency with which values close to the extremal are attained. They also prove the following theorem: Suppose that \(\eta:\mathbb{R}_{\geq 1} \to \mathbb{R}_{\geq 1}\) is such that \(\eta(Q)\to 0\) as \(Q\to \infty\). Let \({\mathcal K}(Q)\) denote the set of real primitive characters \(\chi\) with modulus \(q\leq Q\). Then there is a subset \({\mathcal E}(Q)\) of \({\mathcal K}(Q)\) such that \(\text{card } {\mathcal E}(Q)\ll Q^{1-\eta(Q)}\) and for each \(\chi \in \) \({\mathcal K}(Q)\backslash {\mathcal E}(Q)\) we have \[ {\pi^2 e^{-\gamma} \over 6 \log\log Q}(1+\eta_1(Q)) < L(1,\chi) < e^\gamma(\log\log Q)(1+\eta_2(Q)) \] where \(\eta_j(Q) \ll \eta(Q)+ (\log\log\log Q/\log\log Q).\)
P. D. T. A. Elliott [ Litov. Mat. Sb. 10, 189-197 (1970; Zbl 0213.06501)] obtained a result similar to Theorem 1 in which the bounds for \(L(1,\chi)\) are more precise, but the upper bound for \({\mathcal E}(Q)\) is larger.
For the entire collection see [Zbl 0911.00018].

MSC:

11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
PDFBibTeX XMLCite