An asymptotic Robin inequality

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 ${\mathrm{liminf}}_{n\to \infty }d(n)=0.$ 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 ${\mathrm{liminf}}_{n\to \infty }D(n)$ is derived.

5 pages