Joint distribution in residue classes of families of polynomially-defined multiplicative functions

Akash Singha Roy

Published 2023-12-31Version 1

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, when the multiplicative functions can be controlled by the values of polynomials at the first few prime powers. We obtain complete uniform extensions of a criterion of Narkiewicz for families of such multiplicative functions, thus also generalizing and improving upon previous work for a single such function and establishing results that are optimal in most parameters and hypotheses. As a special case of our results, for any fixed $\epsilon>0$, the Euler totient function $\phi(n)$ and sum of divisors function $\sigma(n)$ are jointly asymptotically equidistributed among the reduced residue classes to moduli $q$ coprime to $6$ varying uniformly up to $(\log x)^{(1-\epsilon)\alpha(q)}$, where $\alpha(q) = \prod_{\ell \mid q} (\ell-3)/(\ell-1)$; furthermore, the coprimality restriction is necessary and the range of $q$ is essentially optimal.