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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 \(\sigma(n):=\sum_{d| n} d<e^\gamma n\log\log n\) 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:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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