Open Access

Mean values of multiplicative functions over function fields

  • Andrew Granville1Email author,
  • Adam J. Harper2 and
  • Kannan Soundararajan3
Research in Number Theory20151:25

https://doi.org/10.1007/s40993-015-0023-5

Received: 29 April 2015

Accepted: 28 August 2015

Published: 21 December 2015

Abstract

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors’ new proof of Halász’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.

Keywords

Multiplicative functionsHalász’s theoremFunction fields