PRIME NUMBERS IN TWO BASES

摘要

We estimate the sums ∑ n≤x Λ(n) f (n) g (n) exp(2iπvn) and ∑ n≤x μ(n) f (n) g (n) exp(2iπvn), where Λ denotes the von Mangoldt function (and μ the M?bius function) whenever q_1 and q_2 are two coprime bases, f (resp., g) is a strongly q1-multiplicative (resp., strongly q_2-multiplicative) function of modulus 1, and v 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 M?bius orthogonality) for sequences of integers with digit properties in two coprime bases.