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

Abstract

Assume that $n\ge 2$ and $B=(b_1,...,b_n)$ has distince integer entries $\ge 3.$ 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 $i\ne j.$ We prove the converse for $n=2,$ and for $n\ge 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers.

arXiv comment3 pages An error was corrected by changing Proposition 1 to: Conjecture 1 implies conjecture 2