2021/09/12 by Luca Onnis, Onnis, Luca
Mathematics · #11S99 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2109.05616
openalex publication_date 2021/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: 11 22 33… nn . 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.