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 ${\sum }_{n\le N}{\rho }_1(n!)$.

arXiv comment33 pages. Minor corrections, change of title, and reordering of the sections to match the accepted version