Vol. 11, No. 4, 2017

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Gowers norms of multiplicative functions in progressions on average

Xuancheng Shao

Vol. 11 (2017), No. 4, 961–982
Abstract

Let μ be the Möbius function and let k 1. We prove that the Gowers Uk-norm of μ restricted to progressions {n X : n aq(modq)} is o(1) on average over q X12σ for any σ > 0, where aq(modq) is an arbitrary residue class with (aq,q) = 1. This generalizes the Bombieri–Vinogradov inequality for μ, which corresponds to the special case k = 1.

Keywords
multiplicative functions, Bombieri–Vinogradov theorem, Gowers norms
Mathematical Subject Classification 2010
Primary: 11P32
Secondary: 11B30, 11N13
Milestones
Received: 15 July 2016
Revised: 2 December 2016
Accepted: 29 January 2017
Published: 18 June 2017
Authors
Xuancheng Shao
Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
Oxford OX2 6GG
United Kingdom