The maximal order of an element in the symmetric group

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 :


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


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

This is due to Massias, Nicolas and Robin [ MR 89i:11108].



Back to the main index for The Riemann Hypothesis.