Are the Stieltjes constants irrational? Some computer experiments

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.