Generalized Fourier coefficients of multiplicative functions

TL;DR

This paper introduces a broad class of multiplicative functions, showing they become orthogonal to polynomial nilsequences after a '$W$-trick', which implies they have small uniformity norms and extends Green and Tao's work on the Möbius function.

Contribution

The paper generalizes Green and Tao's results by analyzing a wider class of multiplicative functions and establishing their orthogonality to polynomial nilsequences after a '$W$-trick'.

Findings

Functions become orthogonal to polynomial nilsequences after '$W$-trick

They have small uniformity norms of all orders

Generalizes previous work on the Möbius function

Abstract

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct prime factors of $n$, as well as the function $n \mapsto |\lambda_f(n)|$, where $\lambda_f(n)$ 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 generalises work of Green and Tao on the M\"obius function.