# Euler-Mascheroni constant

The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm:

[itex]\gamma = \lim_{n \rightarrow \infty } \left(

\sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx[itex]

## History

The constant was first defined by Swiss mathematician Leonhard Euler in a paper De Progressionibus harmonicus observationes published in 1735. Euler used the notation C for the constant, and initially calculated its value to 6 decimal places. In 1761 he extended this calculation, publishing a value to 16 decimal places. In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation γ for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made an error in the 20th decimal place.

## Properties

Intriguingly, the constant is also given by the integral:

[itex]\gamma = - \int_0^\infty { \ln(x) \over e^x }\,dx. [itex]

It can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:

[itex]\gamma = \sum_{n=2}^{\infty} \frac{(-1)^n\zeta(n)}{n}. [itex]

γ can also be calculated as a derivative of Euler's Gamma function:

[itex]\gamma = -\Gamma'(1).[itex]

Its value is approximately

γ ≈ 0.577215664901532860606512090082402431042159335 9399235988057672348848677267776646709369470632917467495...

The constant eγ is also important in number theory,

[itex]e^\gamma [itex] = 1.78107241799019798523650410310717954916964521430343...

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, its denominator has more than 10,000 digits.

The Euler-Mascheroni constant appears, among other places, in:

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy