Joint distribution of leftmost digits in positional notation and Schanuels's conjecture

Abstract

Assume that $n \geq 2$ and $B = (b_1,...,b_n)$ has distince integer entries $\geq 3.$ For $x > 0$ let $d_B(x) := (d_{b_1}(x),...,d_{b_n}(x))$ where $d_{b_i}(x) \in \{1,...,b_i-1\}$ is the leftmost digit in the base-$b_i$ positional notation representation of $x.$ We prove that if $d_B$ is surjective, then $\ln b_i$ and $\ln b_j$ are rationally independent whenever $i \neq j.$ We prove the converse for $n = 2,$ and for $n \geq 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers.