Denoted γ, has the numerical value:

γ = 0.577215664901532860606512090082402431042...

The Greek letter γ was first used by Leonhard Euler to denote this value in 1781, so it is also known as Euler's constant. He also computed its value to 16 digits.

As of 2002, it is unknown whether the Euler-Mascheroni constant is even an irrational number let alone a transcendental number. If γ is rational, expressible as a fraction a/b, then b must be greater than 10^{244,663}, as shown by Thomas Papanikolaou in 1998.

Some representations include:

∞
γ = ∫ exp(-x) ln(x) dx
0
∞
= 1 + ∑ (1/k + ln((k-1)/k))
k=2
= lim H - ln(n)
n→∞ n
∞ n ζ(n)
= ∑ (-1) ----
n=2 n

where H_{n} is a harmonic number and ζ(n) is the Riemann zeta function.