Weak uniform distribution of certain arithmetic functions

Abstract: For any fixed integer $q$, it is a classical result (implicit in work of Landau, and perhaps known earlier) that Euler's function $\phi(n)$ is a multiple of $q$ asymptotically 100\% of the time. Thus, $\phi(n)$ is very far from being uniformly distributed mod $q$ in the usual sense (unless $q=1$ !). On the other hand, Narkiewicz has proved that $\phi(n)$ is weakly uniformly distributed mod $q$ whenever $q$ is coprime to 6; “weakly” means that every coprime residue class mod $q$ gets its fair share of values $\phi(n)$, from among the $n$ with $\phi(n)$ coprime to $q$. In fact, Narkiewicz proves this not just for $\phi$ but for a wide class of “polynomially-defined” multiplicative functions. In this talk, we will consider these weak uniform distribution problems with an eye towards obtaining wide ranges of uniformity in the modulus $q$.

This is joint work with Noah Lebowitz-Lockard and Akash Singha Roy.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)