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 , 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 so that (page 23).
Part 2: Except for a couple select values of , the sum 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 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 (page 68).
Part 8: The series for quickly computing to high precision (page 89).
Part 9: An observation about the formulas for (page 81).
Part 10: A lower bound for the gap between successive primes (page 115).
Part 11: Two generalizations of (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 th 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).