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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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