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

TL;DR

This paper establishes lower bounds on the tail distribution of the sum of digits of prime numbers, showing infinitely many primes with a high number of ones in their binary expansion.

Contribution

It provides new lower bounds for the tail distribution of digit sums of primes, extending previous asymptotic results to tail probabilities.

Findings

Lower bound on the tail distribution of prime digit sums.

Infinitely many primes have more than twice as many ones as zeros in binary.

Application of probabilistic and number-theoretic techniques to prime digit distributions.

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.