On digits of Mersenne numbers

TL;DR

This paper studies the distribution of q-ary digits in Mersenne numbers, providing new estimates on exponential sums that imply a higher degree of normality in their digit strings than previously known.

Contribution

It introduces improved bounds on rational exponential sums for Mersenne numbers, leading to stronger results on the normality of their q-ary digit distribution.

Findings

Demonstrates normality of about ()^{3/2+o(1)} rightmost digits of Mersenne numbers.

Provides new estimates on exponential sums involving Mersenne numbers.

Extends previous results from ()^{1+o(1)} to ()^{3/2+o(1)} digits.

Abstract

Motivated by recently developed interest to the distribution of $q$-ary digits of Mersenne numbers $M_p = 2^p-1$, where $p$ is prime, we estimate rational exponential sums with $M_p$, $p \leq X$, modulo a large power of a fixed odd prime $q$. In turn this immediately implies the normality of strings of $q$-ary digits amongst about $(\log X)^{3/2+o(1)}$ rightmost digits of $M_p$, $p \leq X$. Previous results imply this only for about $(\log X)^{1+o(1)}$ rightmost digits.