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

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 Is |s - 1| = 2.2093, with 0 = R(s) = 1. This work is a continuation of [24].