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2023•The Michigan Mathematical JournalOpen access

Distribution mod p of Euler’s Totient and the Sum of Proper Divisors

Noah Lebowitz-Lockard, Paul Pollack, Akash Singha Roy

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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 of 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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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 of 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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Available 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 of 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.

Key concepts: Euler's totient function, Modulo, Mathematics, Coprime integers, Primitive root modulo n, Residue (chemistry), Combinatorics, Distribution (mathematics)

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