The Mertens and Pólya conjectures in function fields

Abstract

The Mertens conjecture on the order of growth of the summatory function of the M{u00F6}bius function has long been known to be false. We formulate an analogue of this conjecture in the setting of global function fields, and investigate the plausibility of this conjecture. First we give certain conditions, in terms of the zeroes of the associated zeta functions, for this conjecture to be true. We then show that in a certain family of function fields of low genus, the average proportion of curves satisfying the Mertens conjecture is zero, and we hypothesise that this is true for any genus. Finally, we also formulate a function field version of P{u00F3}lya's conjecture, and prove similar results.