Algebra & Number Theory

Generalized Fourier coefficients of multiplicative functions

Lilian Matthiesen

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

Article information

Algebra Number Theory, Volume 12, Number 6 (2018), 1311-1400.

Received: 4 July 2016
Revised: 18 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 25 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11L07: Estimates on exponential sums 11N60: Distribution functions associated with additive and positive multiplicative functions 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]

multiplicative functions nilsequences Gowers uniformity norms


Matthiesen, Lilian. Generalized Fourier coefficients of multiplicative functions. Algebra Number Theory 12 (2018), no. 6, 1311--1400. doi:10.2140/ant.2018.12.1311.

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