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Zeros of the alternating zeta function on the line Re(s)=1. (English) Zbl 1187.11031
Summary: The alternating zeta function ζ * (s)=1-2 -s +3 -s -··· is related to the Riemann zeta function by the identity (1-2 1-s )ζ(s)=ζ * (s). We deduce the vanishing of ζ * (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 ζ * (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. Am. Math. Soc. 126, No. 5, 1311–1314 (1998; Zbl 0890.11025).
MSC:
11M41Other Dirichlet series and zeta functions
11M06ζ(s) and L(s,χ)