# Euler–Mascheroni constant

limiting difference between the harmonic series and the natural logarithm

In mathematics, 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 ${\displaystyle \gamma }$ (gamma),[2] although Euler used the letters C and O instead.

It is not known yet whether the number is irrational (that is, cannot be written as a fraction with an integer numerator and denominator) or transcendental (that is, cannot be the solution of a polynomial with integer coefficients).[3] The numerical value of ${\displaystyle \gamma }$ is about ${\displaystyle 0.5772156649}$.[4][3] 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.[5]

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:[2][6]

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

It can also be written as an improper integral involving the floor function, a function which outputs the greatest integer less than or equal to a given number.[4]

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

The gamma constant is closely linked to the Gamma function [6], 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[6]

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

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. "Greek/Hebrew/Latin-based Symbols in Mathematics". Math Vault. 2020-03-20. Retrieved 2020-10-05.
3. Weisstein, Eric W. "Euler-Mascheroni Constant". mathworld.wolfram.com. Retrieved 2020-10-05.
4. "Euler-Mascheroni Constant | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-05.
5. Sandifer, Edward (October 2007). "How Euler Did It - Gamma the constant" (PDF). Retrieved 26 June 2017.
6. "The Euler Constant" (PDF). April 14, 2004. Retrieved June 26, 2017.