In mathematics there are a few famous constants; maybe the most well known is pi but you might have also heard about the Euler constant, the golden ratio, or the imaginary unit. Today I want to talk about a constant that is less well known, but that appears frequently in different parts of analysis and number theory. This constant, also appears sporadically in unexpected places like in quantum information theory or in quantum field theory. We are referring to the Euler-Mascheroni constant or gamma constant. Just to be clear on notation, this constant should be written with a lowercase Greek gamma character and should not be confused with the gamma function which is represented with the uppercase Greek gamma character. Interestingly,  the gamma constant and the gamma function are related as we can see from the following representation of the gamma function described by Weierstrass:

Please notice that indeed the gamma function can be expressed using the gamma constant highlighted here in red in the figure. In fact, probably the gamma constant took its name because of this relationship with the gamma function which was not immediately realized as neither Euler nor Mascheroni use this notation in their papers.

But what is the value of this constant anyway? The gamma constant is approximately equal to 0.57721, etc. Like many other mathematical constants this is a number with an infinite number of decimal places. Euler who discover or defined the constant calculated this constant up to 16 decimal places. Mascheroni did just a very small contribution and extend this calculation to 32 decimal places. In fact, this is one case in which we have given too much credit to a mathematician. In all honesty, this should be call only the Euler constant, but maybe because we already have another more important Euler constant “e” as well as many other things in mathematics named after Euler, the mathematical community has decided to recognize also Mascheroni here as well. Anyway, after the calculation of Mascheroni, many others extended this number. Currently with the use of high performance computers and dedicated algorithms we have extended the known decimal values of the gamma constant to more than half a trillion.

But how do we get to this number? How is the gamma constant defined? See below the expression for the gamma constant. Here we can see that the gamma constant is defined as the difference between the harmonic series and the natural logarithm.

The harmonic series is the sum of the reciprocals of all positive integers and is a series that diverges slowly to infinity. In fact, as you can see this series diverges following a growth pattern that closely resembles a logarithmic growth. However, as we will see, the growth of the harmonic series follows closely, but not exactly a logarithmic growth. This mismatch is the basis to define the gamma constant.

Let’s slow down here a second to make this point clear as this is in fact is related to the way in which the gamma constant was initially described by Euler. The strategy to define the gamma constant is not difficult to visualize, let me give you some intuition about it. I am not say that this intuition was the same way Euler arrived to the constant but surely he must some of this though process to prove it. Imagine that you start from noticing the similarity in the shape and growth between the harmonic series and the logarithmic function. As you can see from the plot both functions grow in a similar. The harmonic series has values that are slightly higher than the logarithmic function for any given “n”. So there is a gap between the two functions.

Now we can imagine that, for simplicity, as the functions grow to infinite, three possible outcomes are possible:

1. The gap between the functions grow also to infinity.
2. The gap between the functions collapses to zero and the two functions touch each other at infinity (or maybe cross each other at a very large value of “n”),
3. And finally that the gap converges to a constant as we approach to infinity.

This last option was the one explored and proved by Euler. He assumed that the gap or difference between these two functions was the sum of a constant (that we will call gamma) and another factor epsilon that has a dependency with “n”. In fact, Euler showed that epsilon is inversely proportional to “n” and therefore epsilon vanishes to zero as n goes to infinity. From this also follows that the difference between the harmonic series and the natural logarithm converges to the constant gamma as “n” goes to infinity. The actual proof is of course much more convoluted but in this short video I was only aiming to give a basic intuitions about the definition of the gamma constant.

The Euler-Mascheroni constant is quite ubiquitous in several areas of mathematics and physics. One would assumed that because of this the gamma constant would probably be very well understood. However this is not the case and there is still work even for pure mathematicians regarding this constant. Particularly two issues come to mind:

1. We don’t yet know if the gamma constant is a transcendental number or not. As a reminder, a transcendental number is a number that IS NOT a root of any polynomial equation with integer coefficients. Pi and the Euler constant “e” are famous transcendental numbers.
2. Secondly, we don’t even know if the gamma constant is rational or irrational. Again as a reminder, an irrational number is a real number that cannot be expressed as a fraction of integer numbers.

As you see, there is still space for pure fundamental research regarding the Euler-Mascheroni constant… By now you should know the basic ideas regarding this constant, hopefully you will even be intrigue to dig a little deeper into this subject.