On Robin’s criterion for the Riemann Hypothesis
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Published:2021-12
Issue:4
Volume:27
Page:15-24
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ISSN:1310-5132
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Container-title:Notes on Number Theory and Discrete Mathematics
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language:
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Short-container-title:NNTDM
Author:
Aoudjit Safia, ,Berkane Djamel,Dusart Pierre, ,
Abstract
Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.
Publisher
Prof. Marin Drinov Publishing House of BAS (Bulgarian Academy of Sciences)
Cited by
1 articles.
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1. Robin’s criterion on divisibility;The Ramanujan Journal;2022-05-03