An asymptotic Robin inequality

TL;DR

This paper investigates the asymptotic behavior of a function related to Robin's inequality, providing new insights into its connection with the Riemann Hypothesis by analyzing the limit inferior of a normalized difference.

Contribution

It proves unconditionally that the normalized difference's limit inferior is zero and introduces a new criterion for the Riemann Hypothesis based on this limit.

Findings

Proves liminf_{n o } d(n) = 0 unconditionally.

Provides an effective estimate for Chebyshev's summatory function.

Derives a new criterion for the Riemann Hypothesis based on liminf_{n o } D(n).

Abstract

The conjectured Robin inequality for an integer $n>7!$ is $\sigma(n)<e^\gamma n \log \log n,$ where $\gamma$ denotes Euler constant, and $\sigma(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH). Writing $D(n)=e^\gamma n \log \log n-\sigma(n),$ and $d(n)=\frac{D(n)}{n},$ we prove unconditionally that $\liminf_{n \rightarrow \infty} d(n)=0.$ The main ingredients of the proof are an estimate for Chebyshev summatory function, and an effective version of Mertens third theorem due to Rosser and Schoenfeld. A new criterion for RH depending solely on $\liminf_{n \rightarrow \infty}D(n)$ is derived.