Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Sumaia Saad Eddin

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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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Proc. Japan Acad. Ser. A Math. Sci., Volume 93, Number 10 (2017), 120-123.

First available in Project Euclid: 30 November 2017

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Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11Y60: Evaluation of constants

The Laurent-Stieltjes constants Dirichlet $L$-function Riemann zeta function


Saad Eddin, Sumaia. Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 10, 120--123. doi:10.3792/pjaa.93.120.

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