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<h1 align=center> The maximal order of an element in the symmetric group
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Let $g(n)$ be the maximal order of a permutation of $n$ objects, $\omega(k)$ be the number of distinct prime divisors of the integer $k$ and $Li$ be the integral logarithm.

Then RH is equivalent to each of the following statements :

$$
\log g(n) < \sqrt{Li^{-1}(n)} \quad \text{for $n$ large enough};
$$

$$
\omega(g(n)) < Li (\sqrt{Li^{-1}(n)}) \quad \text{for $n$ large enough}.
$$

This is due to Massias, Nicolas and Robin [
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<a href="http://www.ams.org/msnmain?fn=130&s1=89i:11108&pg1=MR">MR 89i:11108</a>
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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