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arXiv · 2105.12850

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 $\phi(n)$ 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 $x$.

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Noah Lebowitz-Lockard, Paul Pollack, Akash Singha Roy. 2021-05-26. Distribution mod $p$ of Euler's totient and the sum of proper divisors. https://arxiv.org/abs/2105.12850

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