Question of the Week, 11.5.2011
The Euler-Mascheroni constant γ plays a major role in numerous fields of mathematics and yet, it is not known whether it is rational or irrational, algebraic or transcendental. γ often appears as value or limit of special functions and is especially closely related to the exponential integral, but may be formulated as definite integral as well. Recent numerical computations provided up to several billion digits of γ and suggest that it is irrational. However, a rigorous mathematical proof is yet to be presented.
See for a detailed analysis: J. Sondow, “Criteria for irrationality of Euler’s constant,<http://arXiv.org/abs/math.NT/0209070>” Proceedings of the American Mathematical Society 131 3335-3344.