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Zeros of the alternating zeta function on the line $$\text{Re}(s)=1$$. (English) Zbl 1187.11031
Summary: 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. Am. Math. Soc. 126, No. 5, 1311–1314 (1998; Zbl 0890.11025).

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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