Explicit upper bounds for the Stieltjes constants

JOURNAL OF NUMBER THEORY

卷:133

Article

Eddin, Sumaia Saad

关键词: Stieltjes coefficients;    Dirichlet characters;    L-functions;    Gauss sums;    Functional equation;    Matsuoka's formula;   

DOI  :  10.1016/j.jnt.2012.09.001

来源: Elsevier

【 摘 要 】

Text. Let chi be a primitive Dirichlet character modulo q and let (-1)(n)gamma(n)(chi)/(n!) (for it larger than 0) be the n-th Laurent coefficient around z = 1 of the associated Dirichlet L-series. When chi is non-principal, (-1)(n)gamma(n)(chi) is simply the value of the n-th derivative of L(z, chi) at z = 1. In this paper we give an explicit upper bounds for vertical bar gamma(n)(chi)vertical bar For q <= pi/2 e(vertical bar n-1 vertical bar/2)/n+1. In particular, when q = 1 the explicit upper bound we get improves on earlier work. We conclude this paper by showing that we can altogether dispense in these proofs with the functional equation of L(z, chi). Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=q340UciEvAA. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free