What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 4

For s > 1, Riemann’s famous zeta function is defined by

\zeta(s) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}.

This is also called a p-series in calculus.

What I didn’t know (Gamma, page 41) is that, in 1748, Leonhard Euler exactly computed this infinite series for s = 26 without a calculator! Here’s the answer:

\displaystyle 1 + \frac{1}{2^{26}} + \frac{1}{3^{26}} + \frac{1}{4^{26}} + \dots = \frac{1,315,862 \pi^{26}}{11,094,481,976,030,578,125}.

I knew that Euler was an amazing human calculator, but I didn’t know he was that amazing.

green line

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.

One thought on “What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 4

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