Beatty sequences and prime numbers with restrictions on strongly q-additive functions

Abstract

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes p = αn +β and f ( p) ≡ a(mod b), with n ≥ N , where α, β are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer k ≥ 3, all sufficiently large N ≡ k(mod 2) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.