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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

Citations:

Zbl 1282.11106
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References:

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