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

,

producing Euler’s generalized constants. Alternatively (from Stieltjes), we could choose

.

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.

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