Bounds For The Tail Distribution Of The Sum Of Digits Of Prime Numbers

Abstract

Let s_q(n) denote the base q sum of digits function, which for n<x, is centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper, they look at sum of digits of prime numbers, and provide asymptotics for the size of the set {p<x, p prime : s_q(p)=alpha(q-1)log_q x} where alpha lies in a tight range around 1/2. In this paper, we examine the tails of this distribution, and provide the lower bound |{p < x, p prime : s_q(p)>alpha(q-1)log_q x}| >>x^{2(1-alpha)}e^{-c(log x)^{1/2+epsilon}} for 1/2<alpha<0.7375. To attain this lower bound, we note that the multinomial distribution is sharply peaked, and apply results regarding primes in short intervals. This proves that there are infinitely many primes with more than twice as many ones than zeros in their binary expansion.