The sum of digits of primes

It is relatively easy to show that the average number of non-zero binary digits of primes < x is almost the same as the average number of non-zero binary digits of all natural numbers < x, namely (1/2) \log_2 x + O(1). The main purpose of this talk is to provide asymptotic expansions for the number of primes < x with precisely k non-zero binary digits for k close to (1/2) \log_2 x. The proof is based on a thorough analysis of exponential sums involving the sum-of-digits function (that is related to a recent solution of problem of Gelfond) and a refined central limit theorem for the sum-of-digits function of primes. Interestingly this result answers a question that is attributed to Ben Green whether for every given k there exists a prime with k non-zero binary digits. There is also a very nice relation to the Thue-Morse sequence. This is joint work with Christian Mauduit and Joel Rivat.