When the Robin inequality does not hold

20 September 2021, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n)5040$ if and only if the Riemann Hypothesis is true. Let $n>5040$ be $n=r\timesq$, where $q$ denotes the largest prime factor of $n$. If $n>5040$ is the smallest number such that Robin inequality does not hold, then we show the following inequality is also satisfied: $\sqrt[q]{e}+\frac{\log\log r}{\log\log n}>2$.

Keywords

Riemann hypothesis
Robin inequality
sum-of-divisors function
prime numbers

Supplementary weblinks

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