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15 July 2020 Prime numbers in two bases
Michael Drmota, Christian Mauduit, Joël Rivat
Duke Math. J. 169(10): 1809-1876 (15 July 2020). DOI: 10.1215/00127094-2019-0083

Abstract

We estimate the sums nxΛ(n)f(n)g(n)exp(2iπϑn) and nxμ(n)f(n)g(n)exp(2iπϑn), where Λ denotes the von Mangoldt function (and μ the Möbius function) whenever q1 and q2 are two coprime bases, f (resp., g) is a strongly q1-multiplicative (resp., strongly q2-multiplicative) function of modulus 1, and ϑ is a real number. The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation, and combinatorial arguments. We deduce from these estimates a prime number theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.

Citation

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Michael Drmota. Christian Mauduit. Joël Rivat. "Prime numbers in two bases." Duke Math. J. 169 (10) 1809 - 1876, 15 July 2020. https://doi.org/10.1215/00127094-2019-0083

Information

Received: 19 July 2018; Revised: 5 November 2019; Published: 15 July 2020
First available in Project Euclid: 8 May 2020

zbMATH: 07226651
MathSciNet: MR4118642
Digital Object Identifier: 10.1215/00127094-2019-0083

Subjects:
Primary: 11A63
Secondary: 11L03, 11L20, 11N05, 11N60

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 10 • 15 July 2020
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