#### Vol. 12, No. 6, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Generalized Fourier coefficients of multiplicative functions

### Lilian Matthiesen

Vol. 12 (2018), No. 6, 1311–1400
##### Abstract

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function $n↦{\delta }^{\omega \left(n\right)}$, where $\delta \in ℝ\setminus \left\{0\right\}$ and where $\omega$ counts the number of distinct prime factors of $n$, as well as the function $n↦|{\lambda }_{f}\left(n\right)|$, where ${\lambda }_{f}\left(n\right)$ 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.

##### Keywords
multiplicative functions, nilsequences, Gowers uniformity norms
##### Mathematical Subject Classification 2010
Primary: 11B30
Secondary: 11L07, 11N60, 37A45