On the $p$-adic valuation of a hyperfactorial

Luca Onnis

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arXiv · 2109.05616

On the $p$-adic valuation of a hyperfactorial

Luca Onnis

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Abstract

In this document will be proved a formula to compute the $p$-adic valuation of a hyperfactorial. We call a hyperfactorial the result of multiplying a given number of consecutive integers from 1 to the given number,each raised to its own power. For example, the hyperfactorial of $n$ is equal to: $1^1 2^2 3^3\dots n^n$ . Lots of studies have been done about the hyperfactorial function, in particular two mathematicians: Glaisher and Kinkelin, who have found the asymptotic behaviour of this function as $n$ that approaches infinity (finding a costant, the Glaisher-Kinkelin costant, which has a lot of expressions using the Euler Gamma function and the Riemann Zeta function). In particular in this document I'll write about the p-adic valuation of this function, or rather the maximum exponent of $p$($p$ a prime integer) such that $p$ raised to that power divides the hyperfactorial of $n$. The formula which I will present uses the famous De-Polignac formula for the $p$-adic valuation of the simple factorial. Then I'll discuss about the asymptotic analysis of our result.

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Luca Onnis. 2021-09-12. On the $p$-adic valuation of a hyperfactorial. https://arxiv.org/abs/2109.05616

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