On Robin’s criterion for the Riemann hypothesis

YoungJu Choie^[1]; Nicolas Lichiardopol^[2]; Pieter Moree^[3]; Patrick Solé^[4]

• [1] Dept of Mathematics POSTECH Pohang, Korea 790-784
• [2] ESSI Route des Colles 06 903 Sophia Antipolis, France
• [3] Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn, Germany
• [4] CNRS-I3S ESSI Route des Colles 06 903 Sophia Antipolis, France

Journal de Théorie des Nombres de Bordeaux (2007)

• Volume: 19, Issue: 2, page 357-372
• ISSN: 1246-7405

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Abstract

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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma \left(n\right):={\sum }_{d|n}d\<e^{\gamma }nloglogn$ 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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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 -

References

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8. S. Ramanujan, Highly composite numbers. Annotated and with a foreword by J.-L. Nicolas and G. Robin. Ramanujan J. 1 (1997), 119–153. Zbl0917.11043MR1606180
9. G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. (9) 63 (1984), 187–213. Zbl0516.10036MR774171
10. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. Zbl0122.05001MR137689
11. G. Tenenbaum, Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995. Zbl0831.11001MR1342300

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