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

TL;DR

This paper proves that Euler's totient function and the sum of proper divisors are uniformly distributed in residue classes modulo primes p, even as p grows with x, extending previous results to larger moduli.

Contribution

It establishes the uniform distribution of these arithmetic functions modulo primes p that grow with x, a novel extension of prior work with fixed moduli.

Findings

Euler's totient values coprime to p are uniformly distributed mod p

Sum of proper divisors of composite numbers are uniformly distributed mod p

Results hold for primes p up to a power of log x

Abstract

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $\phi(n)$ and the sum-of-proper-divisors function $s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, 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 $x$.