Two arguments that the nontrivial zeros of the Riemann zeta function are irrational. II

TL;DR

This paper extends computational experiments on the first 40,000 nontrivial zeros of the Riemann zeta function, providing numerical evidence supporting their irrationality through continued fraction analysis.

Contribution

It significantly increases the dataset size and accuracy of zeros analyzed, strengthening the numerical evidence for their irrationality.

Findings

Continued fraction denominators' means approach Khinchin's constant.

Square roots of denominators approach Khinchin-Levy constant.

Results support the hypothesis that the zeros are irrational.

Abstract

We extend the results of our previous computer experiment performed on the first 2600 nontrivial zeros $\gamma_l$ of the Riemann zeta function calculated with 1000 digits accuracy to the set of 40000 first zeros given with 40000 decimal digits accuracy. We calculated the geometrical means of the denominators of continued fractions expansions of these zeros and for all cases we get values very close to the Khinchin's constant, which suggests that $\gamma_l$ are irrational. Next we have calculated the $n$-th square roots of the denominators $Q_n$ of the convergents of the continued fractions obtaining values very close to the Khinchin---L{\'e}vy constant, again supporting the common opinion that $\gamma_l$ are irrational.