On the Mertens Conjecture for Function Fields

TL;DR

This paper investigates an analogue of the Mertens conjecture in function fields, revealing that most hyperelliptic curves do not satisfy the original conjecture, but a modified version holds for a positive proportion.

Contribution

It demonstrates that the classical Mertens conjecture fails for most hyperelliptic curves in function fields, but a modified conjecture with a larger constant is satisfied by many.

Findings

Most hyperelliptic curves do not satisfy the original Mertens conjecture.

A modified Mertens conjecture with a larger constant holds for a positive proportion of curves.

The work extends understanding of Mertens-type conjectures in the context of function fields.

Abstract

We study an analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves.