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 $n$ 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).

Computing e to Any Power: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series examining one of Richard Feynman’s anecdotes about mentally computing $e^x$ for three different values of $x$ .

Part 1: Feynman’s anecdote.

Part 2: Logarithm and antilogarithm tables from the 1940s.

Part 3: A closer look at Feynman’s computation of $e^{3.3}$ .

Part 4: A closer look at Feynman’s computation of $e^{3}$ .

Part 5: A closer look at Feynman’s computation of $e^{1.4}$ .

A Natural Function with Discontinuities: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on a natural function that nevertheless has discontinuities.

Part 1: Introduction

Part 2: Derivation of this piecewise function, beginning.

Part 3: Derivation of the piecewise function, ending.

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.

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

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

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 8

I had always wondered how the constant $\gamma$ can be computed to high precision. I probably should have known this already, but here’s one way that it can be computed (Gamma, page 89):

$\gamma = \displaystyle \sum_{k=1}^n \frac{1}{k} - \ln n - \sum_{k=1}^{\infty} \frac{B_{2k}}{2k \cdot n^{2k}}$ ,

where $B_{2k}$ is the $2k$ th Bernoulli number.

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 5

Check out this lovely integral, dubbed the Sophomore’s Dream, found by Johann Bernoulli in 1697 (Gamma, page 44):

$\displaystyle \int_0^1 \frac{dx}{x^x} = \displaystyle \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4} + \dots$ .

I’ll refer to either Wikipedia or Mathworld for the derivation.

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.

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.

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 3

At the time of this writing, it is unknown if there are infinitely many twin primes, which are prime numbers that differ by 2 (like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc.) However, significant progress has been made in recent years. However, it is known (Gamma, page 30) the sum of the reciprocals of the twin primes converges:

$\displaystyle \left( \frac{1}{3} + \frac{1}{5} \right) + \left( \frac{1}{5} + \frac{1}{7} \right) + \left( \frac{1}{11} + \frac{1}{13} \right) + \left( \frac{1}{17} + \frac{1}{19} \right) = 1.9021605824\dots$ .

This constant is known as Brun’s constant (see also Mathworld). In the process of computing this number, the infamous 1994 Pentium bug was found.

Although this sum is finite, it’s still unknown if there are infinitely many twin primes since it’s possible for an infinite sum to converge (like a geometric series).

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 2

Let’s define partial sums of the harmonic series as follows:

$H(m,n) = \displaystyle \frac{1}{m} + \frac{1}{m+1} + \frac{1}{m+2} + \dots + \frac{1}{n-1} + \frac{1}{n}$ ,

where $m < n$ are positive integers. Here are a couple of facts that I didn’t know before reading Gamma (pages 24-25):

$H(m,n)$ is never equal to an integer.
The only values of $n$ for which $H(1,n)$ is an integer are $n = 2$ and $n=6$ .

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 1

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.

It is well known the harmonic series diverges:

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots = \infty$ .

This means that, no matter what number $N$ you choose, I can find a number $n$ so that

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n} > N$ .

What I didn’t know (p. 23 of Gamma) is that, in 1968, somebody actually figured out the precise number of terms that are needed for the sum on the left hand side to exceed 100. Here’s the answer:

15,092,688,622,113,788,323,693,563,264,538,101,449,859,497.

With one fewer term, the sum is a little less than 100.