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2018 Generalized Fourier coefficients of multiplicative functions
Lilian Matthiesen
Algebra Number Theory 12(6): 1311-1400 (2018). DOI: 10.2140/ant.2018.12.1311

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 { 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.

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Lilian Matthiesen. "Generalized Fourier coefficients of multiplicative functions." Algebra Number Theory 12 (6) 1311 - 1400, 2018. https://doi.org/10.2140/ant.2018.12.1311

Information

Received: 4 July 2016; Revised: 18 September 2017; Accepted: 30 October 2017; Published: 2018
First available in Project Euclid: 25 October 2018

zbMATH: 06973914
MathSciNet: MR3864201
Digital Object Identifier: 10.2140/ant.2018.12.1311

Subjects:
Primary: 11B30
Secondary: 11L07 , 11N60 , 37A45

Keywords: Gowers uniformity norms , multiplicative functions , nilsequences

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.12 • No. 6 • 2018
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