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

TL;DR

This paper analyzes the continued fraction expansions of the first 2600 nontrivial zeros of the Riemann zeta function, providing numerical evidence supporting their irrationality through statistical properties aligned with known constants.

Contribution

It offers the first extensive numerical investigation of the zeros' continued fractions, suggesting their irrationality based on convergence to Khinchin's and Khinchin-Levy constants.

Findings

Geometric means of denominators close to Khinchin's constant

Square roots of denominators near Khinchin-Levy constant

Supports the hypothesis that zeros are irrational

Abstract

We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin's constant, what 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 close to the Khinchin-Levy constant, again supporting the common believe that gamma_l are irrational.