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 $n\mapsto {\delta }^{\omega (n)}$, where $\delta \ne 0$ 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 Möbius function.

95 pages; final version

References in corpus (1)

• Uniformity in the Wiener-Wintner theorem for nilsequences

Cited by in corpus (5)

• Linear correlations of multiplicative functions
• Higher uniformity of arithmetic functions in short intervals I. All intervals
• Partition regularity of Pythagorean pairs
• Weyl sums with multiplicative coefficients and joint equidistribution
• Smooth numbers are orthogonal to nilsequences