Euler-Mascheroni gamma constant - Definition and Overview

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:

<math>\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<math>

Intriguingly, the constant is also given by the integral:

<math>\gamma = - \int_0^\infty { \ln(x) \over e^x }\,dx. <math>

Its value is approximately

γ ≈ 0.577215664901532860606512090082402431042159335 9399235988057672348848677267776646709369470632917467495...

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:

• a product formula for the gamma function
• calculations of the digamma function
• calculation of the Meissel-Mertens constant
• expressions involving the exponential integral
• the first term of the Taylor series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants.

It is named for the mathematicians Leonhard Euler and Lorenzo Mascheroni.

External link

• Euler-Mascheroni constant at MathWorld (http://mathworld.wolfram.com/Euler-MascheroniConstant.html)