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On Robin’s criterion for the Riemann hypothesis. (English) Zbl 1163.11059
Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma(n):=\sum_{d\vert n} d<e^\gamma n\log\log n$ is satisfied for $n\geq 5041$, where $\gamma$ denotes the Euler(-Mascheroni) constant. The authors show by elementary methods that if $n\geq 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld they show, moreover, that $n$ must be divisible by a fifth power $>1$. As consequence the authors obtain that RH holds true if and only if every natural number divisible by a fifth power $>1$ satisfies Robin’s inequality.

MSC:
11M26Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
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