Euler–Mascheroni Constant

Add up 1 + 1/2 + 1/3 + ... + 1/n and compare it to ln n. Both diverge, but their difference settles down to a fixed number: γ = 0.5772156649015329. Euler computed it to five places in 1734 and called it C; Lorenzo Mascheroni gave 32 digits in 1790, of which the first 19 turned out to be right, and introduced the symbol γ. It appears in the expansion of the gamma function, in the prime-counting estimates of analytic number theory, and in the expected value of the coupon-collector problem.

What makes γ remarkable is how little is known about it. π and e were proved irrational in the eighteenth century and transcendental in the nineteenth. For γ, after nearly three hundred years, nobody has proved even irrationality — it is not known whether γ is a ratio of two whole numbers. If it is rational, the denominator has been shown to exceed 10²⁴²⁰⁸⁰, so it very probably is not, but a computation of hundreds of billions of digits is not a proof. It is one of the most conspicuous open problems in mathematics precisely because the question is so easy to state.