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Exponential sums over Möbius convolutions with applications to partitions

Part of: Exponential sums and character sums Additive number theory; partitions

Published online by Cambridge University Press:  09 January 2025

Debmalya Basak ,
Nicolas Robles  and
Alexandru Zaharescu
Debmalya Basak*
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA
Nicolas Robles
Affiliation:
RAND Corporation, Engineering and Applied Sciences, 1200 S Hayes, Arlington, VA 22202, USA e-mail: nrobles@rand.org
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA; and Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, Bucharest, Ro-70700, Romania e-mail: zaharesc@illinois.edu
*
e-mail: dbasak2@illinois.edu
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Abstract

We establish bounds for exponential sums twisted by generalized Möbius functions and their convolutions. As an application, we prove asymptotic formulas for certain weighted chromatic partitions by using the Hardy–Littlewood circle method. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.

Keywords

Exponential sumsMöbius convolutionsHardy-Littlewood circle methodweighted partitionsRiemann zeta-function

MSC classification

Primary: 11L07: Estimates on exponential sums 11P55: Applications of the Hardy-Littlewood method
Secondary: 11L03: Trigonometric and exponential sums, general 11P82: Analytic theory of partitions

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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