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Generalized Fourier
coefficients of multiplicative functions
Lilian Matthiesen |
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Vol. 12 (2018), No. 6, 1311–1400
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Abstract |
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We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function , where and where counts the number of distinct prime factors of , as well as the function , where denotes the Fourier coefficients of a primitive holomorphic cusp form. For this class of functions we show that after applying a -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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Keywords
multiplicative functions, nilsequences, Gowers uniformity
norms
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Mathematical Subject Classification 2010
Primary: 11B30
Secondary: 11L07, 11N60, 37A45
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Milestones
Received: 4 July 2016
Revised: 18 September 2017
Accepted: 30 October 2017
Published: 6 October 2018
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Authors |
| Lilian Matthiesen | |
| Department of Mathematics KTH Stockholm Sweden |
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