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

The Euler-Mascheroni  constant \gamma is defined by

\gamma = \displaystyle \lim_{n \to \infty} \left( \sum_{r=1}^n \frac{1}{r} - \ln n \right).

What I didn’t know, until reading Gamma (page 117), is that there are at least two ways to generalize this definition.

First, \gamma may be thought of as

\gamma = \displaystyle \lim_{n \to \infty} \left( \sum_{r=1}^n \frac{1}{\hbox{length of~} [0,r]} - \ln n \right),

and so this can be generalized to two dimensions as follows:

\delta = \displaystyle \lim_{n \to \infty} \left( \sum_{r=2}^n \frac{1}{\pi (\rho_r)^2} - \ln n \right),

where \rho_r is the radius of the smallest disk in the plane containing at least r points (a,b) so that a and b are both integers. This new constant \delta is called the Masser-Gramain constant; like \gamma, the exact value isn’t known.

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Second, let f(x) = \displaystyle \frac{1}{x}. Then \gamma may be written as

\gamma = \displaystyle \lim_{n \to \infty} \left( \sum_{r=1}^n f(r) - \int_1^n f(x) \, dx \right).

Euler (not surprisingly) had the bright idea of changing the function f(x) to any other positive, decreasing function, such as

f(x) = x^a, \qquad -1 \le a < 0,

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

f(x) = \displaystyle \frac{ (\ln x)^m }{x}.

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When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, 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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