Derbyshire John. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

Washington, D.C.: Joseph Henry Press, 2003. — 447 p. With bookmarks.From amazon.com:
In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.
Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world.From the Prologue:
In August 1859, Bernhard Riemann was made a corresponding member of the Berlin Academy, a great honor for a young mathematician (he was 32). As was customary on such occasions, Riemann presented a paper to the Academy giving an account of some research he was engaged in. The title of the paper was: “On the Number of Prime Numbers Less Than a Given Quantity.” In it, Riemann investigated a straightforward issue in ordinary arithmetic. To understand the issue, ask: How many prime numbers are there less than 20? The answer is eight: 2, 3, 5, 7, 11, 13, 17, and 19. How many are there less than one thousand? Less than one million? Less than one billion? Is there a general rule or formula for how many that will spare us the trouble of counting them?
Riemann tackled the problem with the most sophisticated mathematics of his time, using tools that even today are taught only in advanced college courses, and inventing for his purposes a mathematical object of great power and subtlety. One-third of the way into the paper, he made a guess about that object, and then remarked: "One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation."
That casual, incidental guess lay almost unnoticed for decades. Then, for reasons I have set out to explain in this book, it gradually seized the imaginations of mathematicians, until it attained the status of an overwhelming obsession...Prologue
The Prime Number Theorem
Card Trick
The Soil, The Crop
The Prime Number Theorem
On The Shoulders Of Giants
Riemann’s Zeta Function
The Great Fusion
The Golden Key, And An Improved Prime Number Theorem
Not Altogether Unworthy
Domain Stretching
A Proof And A Turning Point
The Riemann Hypothesis
Nine Zulu Queens Ruled China
Hilbert’s Eighth Problem
The Argument Ant And The Value Ant
In The Grip Of An Obsession
Big Oh And Möbius Mu
Climbing The Critical Line
A Little Algebra
Number Theory Meets Quantum Mechanics
Turning The Golden Key
The Riemann Operator And Other Approaches
The Error Term
Either It’s True, Or Else It Isn’t
Epilogue
Notes
Appendix The Riemann Hypothesis In Song
Picture CreditsPictures