Extrema of Luroth Digits and a zeta function limit relation

TL;DR

This paper explores the extremal properties of Luroth digits and establishes a probabilistic proof connecting the Riemann zeta function with Bernoulli triangles, also discussing trimmed sums and explicit formulas involving special functions.

Contribution

It introduces a novel probabilistic approach to relate Luroth digit extrema with the Riemann zeta function and provides explicit formulas through direct computations.

Findings

Probabilistic proof of a zeta function limit relation

Explicit formulas involving Luroth digits and special functions

Analysis of trimmed sums of Luroth digits

Abstract

We describe how certain properties of the extrema of the digits of Luroth expansions lead to a probabilistic proof of a limiting relation involving the Riemann zeta function and the Bernoulli triangles. We also discuss trimmed sums of Luroth digits. Our goal is to show how direct computations in this case lead to explicit formulas and some interesting discussions of special functions.