On the $p$-adic valuation of a hyperfactorial

TL;DR

This paper derives a formula for calculating the p-adic valuation of hyperfactorials, extending classical factorial valuation results and analyzing their asymptotic behavior.

Contribution

It introduces a novel formula for the p-adic valuation of hyperfactorials, building on De-Polignac's formula for factorials, and explores their asymptotic properties.

Findings

Derived a formula for p-adic valuation of hyperfactorials.

Connected hyperfactorial valuation to De-Polignac's factorial formula.

Analyzed the asymptotic behavior of the valuation.

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.