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 φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n. We prove that the values φ(n), for n ≤ x, that are coprime to p are asymptotically uniformly distributed among the p−1 coprime residue classes modulo p, uniformly for 5 ≤ p ≤ (log x) (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 ≤ (log x). These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.

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