## Algebra & Number Theory

### Generalized Fourier coefficients of multiplicative functions

Lilian Matthiesen

#### 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 $δ ∈ ℝ ∖ { 0 }$ 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
Revised: 18 September 2017
Accepted: 30 October 2017
First available in Project Euclid: 25 October 2018

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

#### 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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