Proceedings of the Japan Academy, Series A, Mathematical Sciences

Robin's inequality and the Riemann hypothesis

Marek Wójtowicz

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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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Proc. Japan Acad. Ser. A Math. Sci., Volume 83, Number 4 (2007), 47-49.

First available in Project Euclid: 30 April 2007

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Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11N37: Asymptotic results on arithmetic functions

Riemann hypothesis Robin's inequality asymptotic density


Wójtowicz, Marek. Robin's inequality and the Riemann hypothesis. Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 4, 47--49. doi:10.3792/pjaa.83.47.

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