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The Riemann hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums. (English) Zbl 0890.11025
Define \[ \alpha_m(s)= \sup\left\{a: \sum^N_{n=1} {\log^m(n/N) \over (n/N)^s}= \int^1_0 {\log^mx \over x^s} dx+O (N^{-a}) \right\}. \] It is shown that \(\alpha_0(s)\) is continuous on \(\mathbb{C}\setminus \{0,1\}\), except at the nontrivial zeros of \(\zeta(s)\). Moreover \(\alpha_1(s)\) is continuous on \(\mathbb{C}\setminus \{1\}\), except at possible multiple zeros of \(\zeta(s)\). For the proof one uses the Euler-Maclaurin summation formula to give a full asymptotic series for the sum above. When \(m=0\), for example, there is a term \(\zeta(s) N^{s-1}\), which dominates in the critical strip, except when \(\zeta(s)=0\).

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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References:
[1] H. M. Edwards, Riemann’s zeta function, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 58. · Zbl 0315.10035
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