# Euler–Mascheroni constant

The Euler-Mascheroni constant is a number that appears in analysis and number theory. It first appeared in the work of Swiss mathematician Leonhard Euler in the early 18th century.[1] It is usually represented with the Greek letter, ${\displaystyle \gamma }$, although Euler used the letters C and O instead. It is not known yet whether the number is irrational (which would mean that it cannot be written as a fraction with an integer numerator and denominator) and/or transcendental (which would mean that it is not the solution of a polynomial with integer coefficients). The numerical value of ${\displaystyle \gamma }$ is about ${\displaystyle 0.5772156649}$. Italian mathematician Lorenzo Mascheroni also worked with the number, and tried, unsuccessfully, to approximate the number to 32 decimal places, making mistakes on five digits. [2]

It is significant because it links the divergent harmonic series with the natural logarithm. It is given by the limiting difference between the natural logarithm and the harmonic series [3]:

${\displaystyle \gamma =\lim _{t\to \infty }\left(\sum _{n=1}^{t}{\frac {1}{n}}-\log(t)\right)}$

It can also be written as an improper integral involving the floor function, which gives the greatest integer less than or equal to a given number.

${\displaystyle \gamma =\int _{1}^{\infty }\left({\frac {1}{\lfloor t\rfloor }}-{\frac {1}{t}}\right)\mathrm {d} t}$

The gamma constant is closely linked to the Gamma function [3], specifically its logarithmic derivative, the digamma function, which is defined as

${\displaystyle \mathrm {\Psi } _{0}(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\log(\Gamma (x))={\frac {\Gamma '(x)}{\Gamma (x)}}}$

For ${\displaystyle x=1}$, this gives us[3]

${\displaystyle \mathrm {\Psi } _{0}(1)=-\gamma }$

Using properties of the digamma function, ${\displaystyle \gamma }$ can also be written as a limit.

${\displaystyle -\gamma =\lim _{t\to 0}\left(\mathrm {\Psi } _{0}(t)+{\frac {1}{t}}\right)}$

## References

1. Euler, Leonhard (1735). De Progressionibus harmonicus observationes (PDF). pp. 150–161.
2. Sandifer, Edward (October 2007). "How Euler Did It - Gamma the constant" (PDF). Retrieved 26 June 2017.
3. "The Euler Constant" (PDF). April 14, 2004. Retrieved June 26, 2017.