On Robin’s criterion for the Riemann hypothesis

Choie, YoungJu, et al. "On Robin’s criterion for the Riemann hypothesis." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 357-372. <http://eudml.org/doc/249981>.

@article{Choie2007,
abstract = {Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma (n):=\sum _\{d|n\}d&lt;e^\{\gamma \}n\log \log n$ 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 $&gt;1$. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power $&gt;1$ satisfies Robin’s inequality.},
affiliation = {Dept of Mathematics POSTECH Pohang, Korea 790-784; ESSI Route des Colles 06 903 Sophia Antipolis, France; Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn, Germany; CNRS-I3S ESSI Route des Colles 06 903 Sophia Antipolis, France},
author = {Choie, YoungJu, Lichiardopol, Nicolas, Moree, Pieter, Solé, Patrick},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Riemann hypothesis; Robin's criterion; Euler constant},
language = {eng},
number = {2},
pages = {357-372},
publisher = {Université Bordeaux 1},
title = {On Robin’s criterion for the Riemann hypothesis},
url = {http://eudml.org/doc/249981},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Choie, YoungJu
AU - Lichiardopol, Nicolas
AU - Moree, Pieter
AU - Solé, Patrick
TI - On Robin’s criterion for the Riemann hypothesis
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 2
SP - 357
EP - 372
AB - Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma (n):=\sum _{d|n}d&lt;e^{\gamma }n\log \log n$ 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 $&gt;1$. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power $&gt;1$ satisfies Robin’s inequality.
LA - eng
KW - Riemann hypothesis; Robin's criterion; Euler constant
UR - http://eudml.org/doc/249981
ER -