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An asymptotic Robin inequality

The conjectured Robin inequality for an integer $n>7!$ is $σ(n)<e^γn \log \log n,$ where $γ$ denotes Euler constant, and $σ(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH). Writing $D(n)=e^γn \log \log n-σ(n),$ 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.