What is the the sum of orders of all elements of $S_n$?

combinatorics group-theory finite-groups permutations symmetric-groups

According to “The average order of a permutation” by Richard Stong, the sum of the orders of all elements of $S_n$ has the following asymptotic:

$$n!e^{C\sqrt{\frac{n}{\log(n)}} + O\left(\frac{\sqrt{n}\log(\log(n))}{\log(n)}\right)}$$

where $C \approx 2.99047$ is a constant.

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