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.

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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