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

by Derbyshire, John

• ISBN13: 9780309085496
• ISBN10: 0309085497
• Format: Hardcover
• Copyright: 2003-04-01
• Publisher: Natl Academy Pr

Summary

In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark the Riemann Hypothesis that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis

Author Biography

John Derbyshire is a mathematician and linguist by education, a systems analyst by profession, and a celebrated writer in his spare time. His work appears frequently in the National Review and The New Criterion. Born and raised in England, he has made his home in the United States for the past 15 years. He currently lives in Huntington, New York, with his wife and two children

Table of Contents

Prologue	ix
Part I The Prime Number Theorem
Card Trick	3	(16)
The Soil, the Crop	19	(13)
The Prime Number Theorem	32	(16)
On the Shoulders of Giants	48	(15)
Riemann's Zeta Function	63	(19)
The Great Fusion	82	(17)
The Golden Key, and an Improved Prime Number Theorem	99	(19)
Not Altogether Unworthy	118	(19)
Domain Stretching	137	(14)
A Proof and a Turning Point	151	(18)
Part II The Riemann Hypothesis
Nine Zulu Queens Ruled China	169	(15)
Hilbert's Eighth Problem	184	(17)
The Argument Ant and the Value Ant	201	(22)
In the Grip of an Obsession	223	(15)
Big Oh and Mobius Mu	238	(14)
Climbing the Critical Line	252	(13)
A Little Algebra	265	(15)
Number Theory Meets Quantum Mechanics	280	(16)
Turning the Golden Key	296	(16)
The Riemann Operator and Other Approaches	312	(15)
The Error Term	327	(23)
Either It's True, or Else It Isn't	350	(12)
Epilogue	362	(3)
Notes	365	(28)
Appendix: The Riemann Hypothesis in Song	393	(12)
Picture Credits	405	(2)
Index	407