# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.