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

2021/05/26 by Lebowitz-Lockard, Noah, Pollack, Paul, Roy, Akash Singha · 1 citation
#11N64 #FOS: Mathematics #Number Theory (math.NT) #Primary 11A25 #Secondary 11N36

paper · doi:10.48550/arxiv.2105.12850

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 ≤ (logx)^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≤ (logx)^A. These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.

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