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Generalized Fourier coefficients of multiplicative functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, Dept Math, Stockholm, Sweden..ORCID iD: 0000-0002-0301-3320
2018 (English)In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 12, no 6, p. 1311-1400Article in journal (Refereed) Published
Abstract [en]

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function n bar right arrow delta(omega(n)), where delta is an element of R \ {0} and where omega counts the number of distinct prime factors of n, as well as the function n bar right arrow vertical bar lambda(f)(n)vertical bar, 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 generalizes work of Green and Tao on the Mobius function.

Place, publisher, year, edition, pages
MATHEMATICAL SCIENCE PUBL , 2018. Vol. 12, no 6, p. 1311-1400
Keywords [en]
multiplicative functions, nilsequences, Gowers uniformity norms
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-238169DOI: 10.2140/ant.2018.12.1311ISI: 000447301000001Scopus ID: 2-s2.0-85062677047OAI: oai:DiVA.org:kth-238169DiVA, id: diva2:1261206
Funder
Swedish Research Council, 2016-05198
Note

QC 20181106

Available from: 2018-11-06 Created: 2018-11-06 Last updated: 2019-05-16Bibliographically approved

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Matthiesen, Lilian

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