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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
##### Keywords:
Riemann hypothesis; Robin’s criterion; Euler constant
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##### References:
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