Prime numbers in two bases
type: Item
instance of: scholarly article
title: Prime numbers in two bases
zbMATH Open document ID: 1484.11016
DOI: 10.1215/00127094-2019-0083
author: Michael Drmota
author: Christian Mauduit
author: Joël Rivat
published in: Duke Mathematical Journal
publication date: 2020-08-03T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1588903315
review text: For two coprime integer bases \(q_1, q_2 \ge 2\), let \(f\) be a proper strongly \(q_1\)-multiplicative function and \(g\) a strongly \(q_2\)-multiplicative function, for example exponentiated sum-of-digits functions in bases \(q_1\) and \(q_2\). The main result states that \[ \Big|\sum_{n\le x} \Lambda(n) f(n) g(n) \exp(2\pi i\vartheta n)\Big| \ll x \exp(-c \log x/\log \log x) \] uniformly for \(\vartheta \in \mathbb{R}\), where \(\Lambda\) denotes the von Mangoldt function. Here, the positive constant \(c\) and the implicit constant depend on \(f\) and \(g\). The same result holds for the Möbius function instead of the von Mangoldt function. In particular, \(f(n)g(n)\) is orthogonal to the Möbius function. Since these sequences are produced by zero-entropy dynamical systems, they provide a new class satisfying the Sarnak conjecture. For certain multiplicative functions, more explicit upper bounds and a prime number theorem are given. The proofs combine Fourier analysis, Diophantine approximation, and combinatorial arguments.
reviewed by: Wolfgang Steiner
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 11N60
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 7226651
zbMATH Keywords: prime numbers
zbMATH Keywords: \(q\)-additive functions
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
OpenAlex ID: W3021622884
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description: scientific article; zbMATH DE number 7226651