Irrationality of the Euler–Mascheroni constant

As of 2026, it remains unknown whether $\gamma$ is irrational. Unlike $\zeta(3)$ or $\ln 2$, $\gamma$ is not known to be a period in the sense of Kontsevich and Zagier, because it arises as the regularized constant term at the pole $s = 1$ of $\zeta(s)$ rather than as an integral of an algebraic differential form over a domain defined by algebraic inequalities. Padé approximants and linear forms in logarithms of rational numbers produce rational approximations $p_n/q_n$ to $\gamma$, but their denominators $q_n$ grow too fast relative to the remainder $|q_n \gamma - p_n|$ to cross the threshold $|q_n \gamma - p_n| \to 0$ required for an irrationality proof.