两进制的质数

IF 2.3 1区数学 Q1 MATHEMATICS

Duke Mathematical Journal Pub Date : 2020-07-15 DOI:10.1215/00127094-2019-0083

M. Drmota, C. Mauduit, J. Rivat

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引用次数: 7

摘要

如果q1和q2是两个素数碱，f (p。G)强q1乘法(相对于;我们估计∑n≤x Λ(n)f(n)g(n) exp(2iπθn)和∑n≤x μ(n)f(n)g(n) exp(2iπθn))的和，其中Λ表示von Mangoldt函数(μ为Möbius函数)。这项工作的目标是引入一种新的方法来研究这些同时涉及两种不同基的和，结合傅里叶分析，丢番图近似和组合论证。我们从这些估计中推导出两个素数基中具有数字性质的整数序列的素数定理(和Möbius正交性)。

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Prime numbers in two bases

If q1 and q2 are two coprime bases, f (resp. g) a strongly q1-multiplicative (resp. strongly q2-multiplicative) function of modulus 1 and θ a real number, we estimate the sums ∑ n≤x Λ(n)f(n)g(n) exp(2iπθn) (and ∑ n≤x μ(n)f(n)g(n) exp(2iπθn)), where Λ denotes the von Mangoldt function (and μ the Möbius function). 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.

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