On digits of Mersenne numbers

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 $M_p$, $p\le X$, 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 $M_p$, $p\le X$. Previous results imply this only for about $(\log X{)}^{1+o(1)}$ rightmost digits.