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

4 thoughts on “Math’s Bite: Euler-Mascheroni constant”

1. That this number isn’t even known to be rational or irrational is one of the most striking bits of mathematics, to me. It’s amazing to think a number this kind-of familiar is also so little known.

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1. Often in maths, the easiest questions to ask and understand are the hardest to answer, for example Fermat’s Last Theorem. To me, this is one of the fascinating things about maths..

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2. In case you haven’t read, you may enjoy reading this book: “Gamma: Exploring Euler’s Constant” by Julian Havil

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