Distribution of sums involving Dirichlet characters over the $k$-free integers

TL;DR

Under certain hypotheses, the paper proves the existence of a limiting distribution for normalized sums involving Dirichlet characters over $k$-free integers and refines a related conjecture on their magnitude.

Contribution

It establishes a logarithmic limiting distribution for these sums assuming GRH and bounds on zeta function moments, and refines previous conjectures on their size.

Findings

Existence of a logarithmic limiting distribution under GRH.

Refinement of the conjecture on the order of magnitude of partial sums.

Results apply to quadratic and modified Dirichlet characters over $k$-free integers.

Abstract

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial sums $x^{-\frac{1}{2k}}\sum_{n\leq x}f(n)$, where $f$ is either a quadratic Dirichlet character or a modified Dirichlet character, restricted to the $k$-free integers. Moreover, we strengthen a conjecture made by Aymone, Medeiros and the author (cf. Ramanujan J. 59(3):713-728, 2022) concerning the precise order of magnitude for these partial sums.