Are the Stieltjes constants irrational? Some computer experiments

TL;DR

This paper uses advanced theorems and high-precision computations to investigate the irrationality and normality of the Stieltjes constants, providing computational evidence supporting their transcendental nature.

Contribution

It applies Khinchin's and Gauss–Kuzmin theorems to extensive high-precision data to support the conjecture that Stieltjes constants are irrational and possibly transcendental.

Findings

High-precision computations suggest Stieltjes constants are irrational.

Evidence supports the conjecture that these constants are transcendental.

Study indicates potential normality of the constants.

Abstract

Khnichin's theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss--Kuzmin theorem to several thousand high precision (up to more than 53000 significant digits) initial Stieltjes constants $\gamma _{n}$, $n=0,1,...,5000$ in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We study also the normality of these important constants.