Distribution mod $p$ of Euler's totient and the sum of proper divisors

Abstract

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\ddot{I}•(n)$ and the sum-of-proper-divisors function $s(n):=\ddot{I}ƒ(n)-n$. We prove that the values $\ddot{I}•(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5\le p\le (\log x{)}^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log x{)}^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.

24 pages