Vol. 12, No. 6, 2018

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Generalized Fourier coefficients of multiplicative functions

Lilian Matthiesen

Vol. 12 (2018), No. 6, 1311–1400
Abstract

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function nδω(n), where δ {0} and where ω counts the number of distinct prime factors of n, as well as the function n|λf(n)|, where λf(n) denotes the Fourier coefficients of a primitive holomorphic cusp form.

For this class of functions we show that after applying a W-trick, their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green–Tao–Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalizes work of Green and Tao on the Möbius function.

Keywords
multiplicative functions, nilsequences, Gowers uniformity norms
Mathematical Subject Classification 2010
Primary: 11B30
Secondary: 11L07, 11N60, 37A45
Milestones
Received: 4 July 2016
Revised: 18 September 2017
Accepted: 30 October 2017
Published: 6 October 2018
Authors
Lilian Matthiesen
Department of Mathematics
KTH
Stockholm
Sweden