Irrationality of the Euler–Mascheroni Constant

The Euler-Mascheroni constant $\gamma=\lim_{n\to\infty}(\sum_{k=1}^n 1/k-\log n)$ is among the most familiar constants of analysis, yet it is not known whether $\gamma$ is irrational. The question is surely old, but no reliably documented first explicit posing of the irrationality problem has been identified, so the problem carries no secure date of origin.

What is known falls well short of irrationality. Sondow established criteria that would yield the irrationality of $\gamma$ provided certain associated quantities behave as expected [Sondow2003Gamma], and Rivoal has studied the arithmetic nature of $\gamma$ alongside values of the gamma function and the Gompertz constant [Rivoal2009Gamma]. Strong rational approximations to $\gamma$ and related numbers have been constructed, and the general theory of irrationality measures for logarithms and allied constants, as surveyed by Zudilin [Zudilin2009Euler], supplies the natural toolkit; but none of these approaches has closed the gap. The problem remains completely open: no proof of irrationality is known, let alone of transcendence, and a resolution would need genuinely new arithmetic information about $\gamma$.