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.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
View all posts by John Quintanilla

Published

One thought on “What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 11”

## One thought on “What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 11”