# 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.