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).

 

 

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 17

Let \pi(n) denote the number of positive prime numbers that are less than or equal to n. The prime number theorem, one of the most celebrated results in analytic number theory, states that

\pi(x) \approx \displaystyle \frac{x}{\ln x}.

This is a very difficult result to prove. However, Gamma (page 172) provides a heuristic argument that suggests that this answer might be halfway reasonable.

Consider all of the integers between 1 and x.

  • About half of these numbers won’t be divisible by 2.
  • Of those that aren’t divisible by 2, about two-thirds won’t be divisible by 3. (This isn’t exactly correct, but it’s good enough for heuristics.)
  • Of those that aren’t divisible by 2 and 3, about four-fifths won’t be divisible by 5.
  • And so on.

If we repeat for all primes less than or equal to \sqrt{x}, we can conclude that the number of prime numbers less than or equal to x is approximately

\pi(x) \approx \displaystyle x \prod_{p \le \sqrt{x}} \left(1 - \frac{1}{p} \right).

From this point, we can use Mertens product formula

\displaystyle \lim_{n \to \infty} \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right)^{-1} = e^\gamma

to conclude that

\displaystyle \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right) \approx \displaystyle \frac{e^{-\gamma}}{\ln n}

if n is large. Therefore,

\pi(x) \approx x \displaystyle \frac{e^{-\gamma}}{\ln \sqrt{x}} = 2 e^{-\gamma} \displaystyle \frac{x}{\ln x}.

Though not a formal proof, it’s a fast way to convince students that the unusual fraction \displaystyle \frac{x}{\ln x} ought to appear someplace in the prime number theorem.

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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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 16

Let \pi(n) denote the number of positive prime numbers that are less than or equal to n. The prime number theorem, one of the most celebrated results in analytic number theory, states that

\pi(x) \approx \displaystyle \frac{x}{\ln x}.

This is a very difficult result to prove. However, Gamma (page 172) provides a heuristic argument that suggests that this answer might be halfway reasonable.

Consider all of the integers between 1 and x.

  • About half of these numbers won’t be divisible by 2.
  • Of those that aren’t divisible by 2, about two-thirds won’t be divisible by 3. (This isn’t exactly correct, but it’s good enough for heuristics.)
  • Of those that aren’t divisible by 2 and 3, about four-fifths won’t be divisible by 5.
  • And so on.

If we repeat for all primes less than or equal to \sqrt{x}, we can conclude that the number of prime numbers less than or equal to x is approximately

\pi(x) \approx \displaystyle x \prod_{p \le \sqrt{x}} \left(1 - \frac{1}{p} \right).

From this point, we can use Mertens product formula

\displaystyle \lim_{n \to \infty} \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right)^{-1} = e^\gamma

to conclude that

\displaystyle \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right) \approx \displaystyle \frac{e^{-\gamma}}{\ln n}

if n is large. Therefore,

\pi(x) \approx x \displaystyle \frac{e^{-\gamma}}{\ln \sqrt{x}} = 2 e^{-\gamma} \displaystyle \frac{x}{\ln x}.

Though not a formal proof, it’s a fast way to convince students that the unusual fraction \displaystyle \frac{x}{\ln x} ought to appear someplace in the prime number theorem.

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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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 14

I hadn’t heard of the worm-on-a-rope problem until I read Gamma (page 133). From Cut-The-Knot:

A worm is at one end of a rubber rope that can be stretched indefinitely. Initially the rope is one kilometer long. The worm crawls along the rope toward the other end at a constant rate of one centimeter per second. At the end of each second the rope is instantly stretched another kilometer. Thus, after the first second the worm has traveled one centimeter, and the length of the rope has become two kilometers. After the second second, the worm has crawled another centimeter and the rope has become three kilometers long, and so on. The stretching is uniform, like the stretching of a rubber band. Only the rope stretches. Units of length and time remain constant.

It turns out that, after n seconds, that the fraction of the band that the worm has traveled is H_n/N, where

H_n = \displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}

and N is the length of the rope in centimeters. Using the estimate H_n \approx \ln n + \gamma, we see that the worm will reach the end of the rope when

H_n = N

\ln n + \gamma \approx N

\ln n \approx N - \gamma

n \approx e^{N - \gamma}.

If N = 100,000 (since the rope is initially a kilometer long), it will take a really long time for the worm to reach its destination!

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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.

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.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 10

Suppose p_n is the nth prime number, so that p_{n+1} - p_n is the size of the nth gap between successive prime numbers. It turns out (Gamma, page 115) that there’s an incredible theorem for the lower bound of this number:

\displaystyle \limsup_{n \to \infty} \frac{(p_{n+1}-p_n) (\ln \ln \ln p_n)^2}{(\ln p_n)(\ln \ln p_n)(\ln \ln \ln \ln p_n)} \ge \displaystyle \frac{4 e^{\gamma}}{c},

where \gamma is the Euler-Mascheroni constant and c is the solution of c = 3 + e^{-c}.

Holy cow, what a formula. Let’s take a look at just a small part of it.

Let’s look at the amazing function f(x) = \ln \ln \ln \ln x, iterating the natural logarithm function four times. This function has a way of converting really large inputs into unimpressive outputs. For example, the canonical “big number” in popular culture is the googolplex, defined as 10^{10^{100}}. Well, it takes some work just to rearrange \displaystyle f \left(10^{10^{100}} \right) in a form suitable for plugging into a calculator:

\displaystyle f \left(10^{10^{100}} \right) = \displaystyle \ln \ln \ln \left( \ln 10^{10^{100}} \right)

= \displaystyle \ln \ln \ln \left( 10^{100} \ln 10 \right)

= \displaystyle \ln \ln \left[ \ln \left(10^{100} \right) + \ln \ln 10 \right]

= \displaystyle \ln \ln \left[ 100 \ln 10 + \ln \ln 10 \right]

= \displaystyle \ln \ln \left[ 100 \ln 10 \left( 1 + \frac{\ln \ln 10}{100 \ln 10} \right) \right]

= \displaystyle \ln \left( \ln [ 100 \ln 10] + \ln \left( 1 + \frac{\ln \ln 10}{100 \ln 10} \right)\right)

\approx 1.6943

after using a calculator.

This function grows extremely slowly. What value of x gives an output of 0? Well:

\ln \ln \ln \ln x = 0

\ln \ln \ln x = 1

\ln \ln x = e

\ln x = e^e

x = e^{e^e} \approx 3,814,279.1

What value of x gives an output of 1? Well:

\ln \ln \ln \ln x = 1

\ln \ln \ln x = e

\ln \ln x = e^e

\ln x = e^{e^e}

x = e^{e^{e^e}}

\approx e^{3,814,279.1}

\approx 10^{3,814,279.1 \log_{10} e}

\approx 10^{1,656,420.367636}

\approx 2.3315 \times 10^{1,656,420}

That’s a number with 1,656,421 digits! At the rapid rate of 5 digits per second, it would take over 92 hours (nearly 4 days) just to write out the answer by hand!

Finally, how large does x have to be for the output to be 2? As we’ve already seen, it’s going to be larger than a googolplex:

\displaystyle f \left(10^{10^{x}} \right) = 2

\displaystyle \ln \ln \ln \left( \ln 10^{10^{x}} \right) = 2

\displaystyle \ln \ln \ln \left( 10^{x} \ln 10 \right) = 2

\displaystyle \ln \ln \left[ \ln \left(10^{x} \right) + \ln \ln 10 \right] = 2

\displaystyle \ln \ln \left[ x\ln 10 + \ln \ln 10 \right] = 2

\displaystyle \ln \ln \left[ x\ln 10 \left( 1 + \frac{\ln \ln 10}{x\ln 10} \right) \right] = 2

\displaystyle \ln \left( \ln [ x\ln 10] + \ln \left( 1 + \frac{\ln \ln 10}{x \ln 10} \right)\right) = 2

Let’s simplify things slightly by letting y = x \ln 10:

\displaystyle \ln \left( \ln y + \ln \left( 1 + \frac{\ln \ln 10}{y} \right)\right) = 2

\displaystyle \ln y + \ln \left( 1 + \frac{\ln \ln 10}{y} \right) = e^2

This is a transcendental equation in y; however, we can estimate that the solution will approximately solve \ln y = e^2 since the second term on the left-hand side is small compared to \ln y. This gives the approximation y = e^{e^2} \approx 1618.18. Using either Newton’s method or else graphing the left-hand side yields the more precise solution y \approx 1617.57.

Therefore, x \approx 1617.57 \ln 10 \approx 3725.99, so that

f \left(10^{10^{3725.99}} \right) \approx 2.

One final note: despite what’s typically taught in high school, mathematicians typically use \log to represent natural logarithms (as opposed to base-10 logarithms), so the above formula is more properly written as

\displaystyle \limsup_{n \to \infty} \frac{(p_{n+1}-p_n) (\log \log \log p_n)^2}{(\log p_n)(\log \log p_n)(\log \log \log \log p_n)} \ge \displaystyle \frac{4 e^{\gamma}}{c}.

And this sets up a standard joke, also printed in Gamma:

Q: What noise does a drowning analytic number theorist make?

A: Log… log… log… log…

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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.

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 2kth Bernoulli number.

 

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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.

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.

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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.

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).

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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.

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.

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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.