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December 2017 Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series
Sumaia Saad Eddin
Proc. Japan Acad. Ser. A Math. Sci. 93(10): 120-123 (December 2017). DOI: 10.3792/pjaa.93.120

Abstract

The Laurent-Stieltjes constants $\gamma_{n}(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non-principal, $(-1)^{n}\gamma_{n}(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet $L$-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093$, with $0\leq \Re{(s)}\leq 1$. This work is a continuation of [24].

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Sumaia Saad Eddin. "Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series." Proc. Japan Acad. Ser. A Math. Sci. 93 (10) 120 - 123, December 2017. https://doi.org/10.3792/pjaa.93.120

Information

Published: December 2017
First available in Project Euclid: 30 November 2017

zbMATH: 06850986
MathSciNet: MR3732901
Digital Object Identifier: 10.3792/pjaa.93.120

Subjects:
Primary: 11M06
Secondary: 11Y60

Rights: Copyright © 2017 The Japan Academy

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Vol.93 • No. 10 • December 2017
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