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Zeros of the alternating zeta function on the line $\text{Re}\left(s\right)=1$. (English) Zbl 1187.11031
Summary: The alternating zeta function ${\zeta }^{*}\left(s\right)=1-{2}^{-s}+{3}^{-s}-···$ is related to the Riemann zeta function by the identity $\left(1-{2}^{1-s}\right)\zeta \left(s\right)={\zeta }^{*}\left(s\right)$. We deduce the vanishing of ${\zeta }^{*}\left(s\right)$ 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 ${\zeta }^{*}\left(s\right)$ 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:
 11M41 Other Dirichlet series and zeta functions 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$