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