# Math’s Bite: Euler-Mascheroni constant

The Euler-Mascheroni Constant appears in analysis and number theory, and is denoted by γ. It is defined as the limit of the difference between the harmonic series and the natural logarithm:

\begin{align} \gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)\\ &=\int_1^\infty\left(\frac1{\lfloor x\rfloor}-\frac1{x}\right)\,dx. \end{align}

Note that the ⌊x⌋ represents a floor function.

It has not been shown whether γ is algebraic or transcendental. In fact, it is not even known whether it is irrational. If γ is a simple fraction a/b, it is known that  (a result given by T. Papanikolaou).

Hilbert said that the irrationality of γ is an unsolved problem that seems “unapproachable” and in front of which mathematicians stand helpless.

### History

This constant was first defined by Euler in a paper published in 1734 entitled ‘De Progressionibus harmonicis observationes’. In the paper, Euler used the letters C or O for the constant. In 1790, Mascheroni used the notations A or a for this constant. It is said that the symbol γ was chosen at a later time due to its relation to the gamma function; γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1:

$-\gamma = \Gamma'(1) = \Psi(1).$

### Approximations

The constant has a numerical value of 0.57721566490153286060651209008240243104215933593992 to 50 decimal places.

A beautiful approximation to is γ given by

which is correct to three decimal places.

An approximation given by P Galliani, involving unit fractions:

is good to 12 decimal places.

Furthermore, Barbosa gave the pandigital approximation:

which is good to 13 decimal places!

Hope you liked today’s post. M x