Generalized Fourier coefficients of multiplicative functions

Matthiesen, Lilian

doi:10.2140/ant.2018.12.1311

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Generalized Fourier coefficients of multiplicative functions

Lilian Matthiesen

Vol. 12 (2018), No. 6, 1311–1400

DOI: 10.2140/ant.2018.12.1311

Abstract

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto δ^{ω (n)}$ , where $δ \in ℝ ∖ {0}$ and where $ω$ counts the number of distinct prime factors of $n$ , as well as the function $n \mapsto | λ_{f} (n) |$ , where $λ_{f} (n)$ 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

Vol. 12, No. 6, 2018