## Applications of the Laurent-Stieltjes constants for Dirichlet $$L$$-series.(English)Zbl 1430.11114

Summary: 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 $$|s-1|\leq2.2093$$, with $$0\leq\mathrm{Re}(s)\leq1$$. This work is a continuation of [the author, J. Number Theory 133, No. 3, 1027–1044 (2013; Zbl 1282.11106)].

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11Y60 Evaluation of number-theoretic constants

Zbl 1282.11106
Full Text:

### References:

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