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YoungJu Choie; Nicolas Lichiardopol; Pieter Moree; Patrick Solé
On Robin’s criterion for the Riemann hypothesis
Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 357-372, doi: 10.5802/jtnb.591
Article PDF | Reviews MR 2394891 | Zbl 1163.11059

Résumé - Abstract

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 $n\ge 5041$, where $\gamma $ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n\ge 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that $n$ must be divisible by a fifth power $>1$. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power $>1$ satisfies Robin’s inequality.

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