Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Sumaia Saad Eddin¹ 

[1] Graduate School of Mathematics, Nagoya University

关键词: Mahler problem;    Liouville number;    transcendental function.;   

DOI  :  10.3792/pjaa.93.120

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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].

【 授权许可】

Unknown