Algebra & Number Theory

Generalized Fourier coefficients of multiplicative functions

Lilian Matthiesen

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1540432828

Digital Object Identifier
doi:10.2140/ant.2018.12.1311

Mathematical Reviews number (MathSciNet)
MR3864201

Zentralblatt MATH identifier
06973914

Subjects
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]

Keywords
multiplicative functions nilsequences Gowers uniformity norms

Citation

Matthiesen, Lilian. Generalized Fourier coefficients of multiplicative functions. Algebra Number Theory 12 (2018), no. 6, 1311--1400. doi:10.2140/ant.2018.12.1311. https://projecteuclid.org/euclid.ant/1540432828


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