Background

The Euler–Mascheroni constant $\gamma$ is defined as the limiting difference between the harmonic series and the natural logarithm. That is $\gamma =\underset{\mathrm{n\to \infty }}{lim}{\sum }_{\mathrm{k=1}}n\frac{1}{k}-\ln \left(n\right)$. This constant appears in analytic number theory and the study of the gamma function.

Although $\gamma$ has not been proven irrational, it arises in integrals and series expansions such as the Taylor series of $\log$ and the asymptotics of the factorial via Stirling's approximation. Its numerical value is approximately 0.57721.

This calculator estimates $\gamma$ by computing the $n$-th harmonic number $H_n={\sum }_{\mathrm{k=1}}^n\frac{1}{k}$ and subtracting $\ln \left(n\right)$. As $n$ grows, the result approaches $\gamma$.