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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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