Generalized Fourier coefficients of multiplicative functions

Abstract

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto δ^{ω(n)}$, where $δ\neq 0$ 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 generalises work of Green and Tao on the Möbius function.