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

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post.

When I was 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 along with the page numbers in the book — while giving the book a very high recommendation.

Part 1: The smallest value of n so that 1 + \frac{1}{2} + \dots + \frac{1}{n} > 100 (page 23).

Part 2: Except for a couple select values of m<n, the sum \frac{1}{m} + \frac{1}{m+1} + \dots + \frac{1}{n} is never an integer (pages 24-25).

Part 3: The sum of the reciprocals of the twin primes converges (page 30).

Part 4: Euler somehow calculated \zeta(26) without a calculator (page 41).

Part 5: The integral called the Sophomore’s Dream (page 44).

Part 6: St. Augustine’s thoughts on mathematicians — in context, astrologers (page 65).

Part 7: The probability that two randomly selected integers have no common factors is 6/\pi^2 (page 68).

Part 8: The series for quickly computing \gamma to high precision (page 89).

Part 9: An observation about the formulas for 1^k + 2^k + \dots + n^k (page 81).

Part 10: A lower bound for the gap between successive primes (page 115).

Part 11: Two generalizations of \gamma (page 117).

Part 12: Relating the harmonic series to meteorological records (page 125).

Part 13: The crossing-the-desert problem (page 127).

Part 14: The worm-on-a-rope problem (page 133).

Part 15: An amazingly nasty formula for the nth prime number (page 168).

Part 16: A heuristic argument for the form of the prime number theorem (page 172).

Part 17: Oops.

Part 18: The Riemann Hypothesis can be stated in a form that can be understood by high school students (page 207).



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