Arithmetic functions at factorial arguments

TL;DR

This paper investigates the behavior of various arithmetic functions evaluated at factorials and their sums, revealing new asymptotic formulas and phenomena related to divisor functions and maximum divisor counts.

Contribution

It provides new asymptotic expansions for divisor functions at factorials and explores the sum of maximum divisor counts at factorials, uncovering novel behaviors and relationships.

Findings

New asymptotic formulas for d(n!) and σ(n!)

Asymptotic value for sum of ρ₁(n!) over n

Insights into the behavior of arithmetic functions at factorial arguments

Abstract

For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have ${f(n!)=\sum_{p\le n}f(p)}$ (where the sum runs over all the primes $p\le n$). Also, for some $f$, their minimum order coincides with $\lim_{n\to \infty}f(n!)$, for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of $f(n!)$ and of $\sum_{n\le N}f(n!)$ for a wide variety of arithmetical functions $f$. In particular, letting $d(n)$ and $\sigma(n)$ stand respectively for the number of positive divisors of $n$ and the sum of the positive divisors of $n$, we obtain new accurate asymptotic expansions for $d(n!)$ and $\sigma(n!)$. Furthermore, setting $\rho_1(n):=\max\{d\mid n:d\le \sqrt n\}$ and observing that no one has yet obtained an asymptotic value for $\sum_{n\le N} \rho_1(n)$ as $N\to \infty$, we show how one can obtain the asymptotic value of $\sum_{n\le N} \rho_1(n!)$.