Prime Obsession

Prime Obsession
	Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire
Author	John Derbyshire
Language	English
Subject	Mathematics, History of science
Genre	Popular science
Publisher	Joseph Henry Press
Publication date	2003
Publication place	United States
Pages	442
ISBN	0-309-08549-7

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis, named for Bernhard Riemann, and some of its applications. The book is written such that even-numbered chapters present historical elements related to the development of the conjecture, and odd-numbered chapters deal with the mathematical and technical aspects.

Overview

In chapter 1, "Card Trick", Derbyshire introduces the idea of an infinite series and the ideas of convergence and divergence of these series. He imagines that there is a deck of cards stacked neatly together, and that one pulls off the top card so that it overhangs from the deck. Explaining that it can overhang only as far as the center of gravity allows, the card is pulled so that exactly half of it is overhanging. Then, without moving the top card, he slides the second card so that it is overhanging too at equilibrium. As he does this more and more, the fractional amount of overhanging cards as they accumulate becomes less and less. He explores various types of series such as the harmonic series.

In chapter 3, the Prime Number Theorem (PNT) is introduced. The function which mathematicians use to describe the number of primes in N numbers, π(N), is shown to behave in a logarithmic manner, as so:

\pi (N)\approx {\frac {N}{\log(N)}}

where log is the natural logarithm. In chapter 5, the Riemann Zeta Function is introduced:

\zeta (s)=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}

In chapter 7, the sieve of Eratosthenes is shown to be able to be simulated using the Zeta function. With this, the following statement which becomes the pillar stone of the book is asserted: