Beyond the Erd\H{o}s discrepancy problem in function fields

TL;DR

This paper investigates the behavior of partial sums of multiplicative functions over function fields, revealing distinctions based on interval types and confirming conjectures about extremal functions, with implications for the Erdős discrepancy problem.

Contribution

It characterizes the discrepancy behavior of multiplicative functions in function fields, confirming conjectures and classifying counterexamples for modified Dirichlet characters.

Findings

Short interval sums are bounded iff the function matches a modified Dirichlet character.

Lexicographic discrepancy is always infinite for completely multiplicative sequences.

Classified growth rates of discrepancy for modified Dirichlet characters.

Abstract

We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative $f:\mathbb{F}_q[t]\to\{-1,+1\}$ with $q$ odd has bounded short interval sums if and only if $f$ coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over $\mathbb{Z}$ that such modified characters are extremal with respect to the growth rate of partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of $\mathbb{F}_{q}[t]$. This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erd\H{o}s discrepancy problem admits infinitely many completely multiplicative counterexamples on $\mathbb{F}_q[t]$. Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers.