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April 2007 Robin's inequality and the Riemann hypothesis
Marek Wójtowicz
Proc. Japan Acad. Ser. A Math. Sci. 83(4): 47-49 (April 2007). DOI: 10.3792/pjaa.83.47


Let $f(n)=\sigma(n)/e^\gamma n\log\log n$, $n=3,4,\ldots$ , where $\sigma$ denotes the sum of divisors function. In 1984 Robin proved that the inequality $f(n)>1$, for all $n\ge 5041$, is equivalent to the Riemann hypothesis. Here we show that the values of $f$ are close to $0$ on a set of asymptotic density $1$. Similarly, an inequality by Rosser and Schoenfeld of 1962, dealing with the Euler totient function $\varphi$, is essential only on "thin" subsets of $\mathbf{N}$.


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Marek Wójtowicz. "Robin's inequality and the Riemann hypothesis." Proc. Japan Acad. Ser. A Math. Sci. 83 (4) 47 - 49, April 2007.


Published: April 2007
First available in Project Euclid: 30 April 2007

zbMATH: 1142.11064
MathSciNet: MR2326201
Digital Object Identifier: 10.3792/pjaa.83.47

Primary: 11M06 , 11N37

Keywords: Asymptotic density , Riemann hypothesis , Robin's inequality

Rights: Copyright © 2007 The Japan Academy

Vol.83 • No. 4 • April 2007
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