Zeros of the alternating zeta function on the line R(s)=1

Jonathan Sondow

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Zeros of the alternating zeta function on the line R(s)=1

Jonathan Sondow

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Abstract

The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s} without using the identity. Instead, we use a formula connecting the partial sums of the series for zeta*(s) to Riemann sums for the integral of x^{-s} from x=1 to x=2. We relate the proof to our earlier paper "The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums," Proc. Amer. Math. Soc. 126 (1998) 1311-1314.

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Jonathan Sondow. 2003-03-19. Zeros of the alternating zeta function on the line R(s)=1. https://arxiv.org/abs/math/0209393

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