Tsukuba Journal of Mathematics

Upper bound for sum of divisors function and the Riemann hypothesis

Aleksander Grytczuk

Full-text: Open access

Abstract

Let $\sigma(n)$ denote the sum of divisors function. We prove that if $(2, m) = 1$ and $2m \gt 3^{9}$ then $1^{0} \sigma(2m) \lt \frac{39}{40}e^{\gamma}2m \log \log 2m$, and for all odd integers $m \gt \frac{3^{9}}{2}$, we have $2^{0} \sigma(m) \lt e^{\gamma}m \log \log m$. Moreover, we show that if $\sigma(2m) \lt \frac{3}{4}e^{\gamma}2m \log \log 2m$, for $m \gt m_{0}$ and $(2, m) = 1$, then the inequality $\sigma(2^{\alpha}m) \lt e^{\gamma}2^{\alpha}m \log \log 2^{\alpha}m$ is true for all integers $\alpha \geq 2$ and $m \gt m_{0}$. Robin criterion implies that the Riemann hypothesis is true for these cases.

Article information

Source
Tsukuba J. Math., Volume 31, Number 1 (2007), 67-75.

Dates
First available in Project Euclid: 30 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1496165115

Digital Object Identifier
doi:10.21099/tkbjm/1496165115

Mathematical Reviews number (MathSciNet)
MR2337120

Zentralblatt MATH identifier
1138.11042

Citation

Grytczuk, Aleksander. Upper bound for sum of divisors function and the Riemann hypothesis. Tsukuba J. Math. 31 (2007), no. 1, 67--75. doi:10.21099/tkbjm/1496165115. https://projecteuclid.org/euclid.tkbjm/1496165115


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