Euler-Mascheroni constant

The Euler-Mascheroni constant, also known as Euler's constant or Mascheroni's constant, is the limit (as n goes to infinity) of


    1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n – ln n.


It is often denoted by a lower-case gamma, γ, and is approximately 0.5772156649... Even though over one million digits of this number have been calculated, it isn't yet known if it is a rational number (the ratio of two integers a/b). If it is rational, the denominator (b) must have more than 244,663 digits.


The constant γ crops in many places in number theory. For example, in 1898 the French mathematician Charles de la Vallée Poussin (who proved the prime number theorem) proved the following: Take any positive integer n and divide it by each positive integer m less than n. Calculate the average (mean) fraction by which the quotient n/m falls short of the next integer. The larger n gets, the closer the average gets to γ.