Editing Prime Obsession (section)

==Overview==
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.<ref name=redmond>{{cite journal|title=Review of ''Prime Obsession''|journal=Mathematical Reviews|first=Don|last=Redmond|year=2004|mr=1968857}}</ref> Despite the title, the book provides biographical information on many iconic mathematicians including [[Euler]], [[Gauss]], and [[Lagrange]].<ref name=graham>{{cite web|title=Review of ''Prime Obsession''|work=MAA Reviews|first=S. W.|last=Graham|date=August 2003|url=https://www.maa.org/press/maa-reviews/prime-obsession-bernhard-riemann-and-the-greatest-unsolved-problem-in-mathematics}}</ref>

In chapter 1, "Card Trick", Derbyshire introduces the idea of an infinite series and the ideas of [[convergence (mathematics)|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 [[:wikt:equilibrium|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 (mathematics)|harmonic series]].

In chapter 2, [[Bernhard Riemann]] is introduced and a brief historical account of [[Eastern Europe]] in the 18th Century is discussed.

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:
:<math> \pi(N) \approx \frac{N}{\log(N)} </math>
where ''log'' is the [[natural logarithm]].

In chapter 4, Derbyshire gives a short biographical history of [[Carl Friedrich Gauss]] and [[Leonard Euler]], setting up their involvement in the [[Prime Number Theorem]].

In chapter 5, the [[Riemann Zeta Function]] is introduced:
:<math> \zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots = \sum_{n = 1}^\infty \frac{1}{n^s} </math>

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:
:<math> \zeta(s) = \prod_{p\ \mathrm{prime}} \frac{1}{1 - {p^{-s}}}</math>

Following the derivation of this finding, the book delves into how this is manipulated to expose the PNT's nature.