The Euler-Mascheroni constant is defined by
What I didn’t know, until reading Gamma (page 117), is that there are at least two ways to generalize this definition.
First, may be thought of as
and so this can be generalized to two dimensions as follows:
where is the radius of the smallest disk in the plane containing at least points so that and are both integers. This new constant is called the Masser-Gramain constant; like , the exact value isn’t known.
Second, let . Then may be written as
Euler (not surprisingly) had the bright idea of changing the function to any other positive, decreasing function, such as
When I researching for my series of posts on conditional convergence, especially examples related to the constant , the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.
Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.
In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.