Mean values of multiplicative functions over function fields

TL;DR

This paper studies the average behavior of multiplicative functions over function fields, adapting classical theorems and revealing unique features of the function field setting compared to integers.

Contribution

It provides a new proof of Halasz's theorem in the function field context and introduces Lipschitz estimates highlighting differences from the integer case.

Findings

Simpler proof of Halasz's theorem for function fields

Lipschitz estimates showing slow variation of mean values

Identification of features unique to function fields

Abstract

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.