Prime numbers in two bases

Abstract

We estimate the sums $${\sum }_{n\le x}\Lambda \left(n\right)f\left(n\right)g\left(n\right)exp\left(2i\pi \vartheta n\right)$$ and $${\sum }_{n\le x}\mu \left(n\right)f\left(n\right)g\left(n\right)exp\left(2i\pi \vartheta n\right),$$ where $\Lambda$ denotes the von Mangoldt function (and $\mu$ the Möbius function) whenever $q_1$ and $q_2$ are two coprime bases, $f$ (resp., $g$) is a strongly $q_1$-multiplicative (resp., strongly $q_2$-multiplicative) function of modulus $1$, and $\vartheta$ 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.