A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime

Christian Berg

paper

A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime

arXiv:2402.08378

Abstract

Let $f_{r} (x) = lo g (1 + r x) / lo g (1 + x)$ for $x > 0$ . We prove that $f_{r}$ is a complete Bernstein function for $0 \leq r \leq 1$ and a Stieltjes function for $1 \leq r$ . This answers a conjecture of David Bradley that $f_{r}$ is a Bernstein function when $0 \leq r \leq 1$ .

12 pages, 2 figures. Section 5 added. To appear in Expositiones Mathematicae