Primes with an average sum of digits

Abstract

The main goal of this paper is to provide asymptotic expansions for the numbers #{p ≤ s: 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 ∑p≤xe(αsq(p))(where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors. Copyright © 2009 Foundation Compositio Mathematica.