Prime numbers in two bases

Abstract

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.