The irrationality conjecture for Euler's constant

Let $\gamma$ denote Euler's constant, defined by

$$\gamma=\lim_{n\to\infty}\left(\sum_{j=1}^n\frac{1}{j}-\log n\right).$$

Euler's constant irrationality conjecture. Euler's constant is irrational. This is a long-standing open problem concerning the arithmetic nature of Euler's constant.