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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 - log n.

It is often denoted by a lower-case gamma, gamma, 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 gamma 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 gamma.

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