Primes with an average sum of digits

Abstract

AbstractThe main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form $\sum _{p\le x} e(\alpha s_q(p))$ (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.