Euler-Mascheroni Constant

[Ed. note: Moved here from Gamma.]

Used in mathematics as a symbol for the Euler-Mascheroni Constant, which is defined as

lim       (1+1/2+1/3+1/4+...+1/n) - ln(n)
n->inf

It's approximately 0.5772156649.... The series converges very slowly, but it's pretty interesting that it converges at all. Most people know that sum of reciprocals of the integers to n tends to ln(n) as n goes to infinity, but that there should be a limit to the difference is pretty fascinating. There are faster ways to compute it, of course.

There is no reason to suspect this number is not transcendental, and there are so many more trancendental numbers than algebraic numbers, so it almost certainly is... and yet, there is no proof that it's even irrational! If it is rational, at least we know its denominator is quite large.

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.