| viXra.org > Number Theory > viXra:1512.0006 |
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| viXra.org > Number Theory > viXra:1512.0006 |
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Number Theory |
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Authors: Patrick Solé, Yuyang Zhu
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.
Comments: 6 Pages.
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[v1] 2015-12-02 03:13:54
[v2] 2016-01-04 08:02:02
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