The Distribution of $k$-Free Effective Divisors and the Summatory Totient Function in Function Fields

TL;DR

This paper studies the distribution and average behavior of error terms related to $k$-free divisors and totient functions in function fields, deriving their limiting distributions and unbiasedness under certain hypotheses.

Contribution

It provides explicit constructions of the limiting distributions of these error terms and analyzes their behavior across families of hyperelliptic curves, extending prior frameworks.

Findings

Error terms are unbiased, positive and negative equally often.

Explicit limiting distributions of error terms are constructed under the Linear Independence hypothesis.

Average behavior of error terms across hyperelliptic curve families is characterized.

Abstract

Motivated by the study of the summatory $k$-free indicator and totient functions in the classical setting, we investigate their function field analogues. First, we derive an expression for the error terms of the summatory functions in terms of the zeros of the associated zeta function. Under the Linear Independence hypothesis, we explicitly construct the limiting distributions of these error terms and compute the frequency with which they occur in an interval $[-\beta, \beta]$ for a real $\beta > 0$. We also show that these error terms are unbiased, that is, they are positive and negative equally often. Finally, we examine the average behavior of these error terms across families of hyperelliptic curves of fixed genus. We obtain these results by following a general framework initiated by Cha and Humphries.