Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers - Hardcover

Rockmore, Daniel N.

 
9780375421365: Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers
Hardcover
ISBN 10: 037542136X ISBN 13: 9780375421365
Verlag: Pantheon Books, 2005

Inhaltsangabe

A leading mathematician presents a lively and accessible study of one of the great unsolved mysteries of mathematics, the mid-nineteenth-century Riemann hypothesis, which describes the occurrence of prime numbers, and of the quest to prove or disprove the theory from the time of Euclid to the statistics of solitaire, chaos theory, and quantum mechanics. 17,500 first printing.

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Über die Autorin bzw. den Autor

Dan Rockmore is a professor of mathematics and computer science at Darmouth College. He lives in New Hampshire with his wife, son, and golden retriever.


From the Trade Paperback edition.

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Prologue—It All Begins with Zero

It’s one of those slate-gray summer days that more properly belong to mid-August than late May, one of those days in New York City when it is barely clear where the city ends and the sky begins. The hard-edged lines and Euclidean-inspired shapes that are building, sidewalk, and pavement all seem to fuse into one huge melted mass that slowly dissolves into the humid, breezeless, torpid air. On mornings like this, even this irrepressible metropolis seems to have slowed a notch, a muffled cacophony more bass than treble, as the city that never sleeps stumbles and shuffles to work.

But here in Greenwich Village, at the corner of Mercer and West Fourth streets, where we find New York University’s Warren Weaver Hall, the hazy torpor is interrupted by a localized high-energy eddy. Here, deep in the heart of the artistic rain forest that is “the Village,” just across the street from the rock ’n’ rolling nightclub the Bottom line, a stone’s throw from the lofts and galleries that gave birth to

Jackson Pollock, Andy Warhol, and the Velvet Underground, is the home of the Courant Institute of Mathematical Sciences, where at this moment there is an excitement worthy of any gallery opening

in SoHo, or any new wave, next wave, or crest-of-the-wave musical performance.

The lobby and adjacent plaza are teeming with mathematicians, a polyglot and international group, abuzz with excitement. Listen closely, and amid the multilingual, every-accent mathematical jibber-jabber you’ll hear a lot of talk about nothing, or more properly a lot of talk about zero.

Zero is not an uncommon topic of conversation in New York, but more often than not it’s the “placeholder zeros” that are on the tip of the New Yorker’s tongue. These are the zeros that stand in for the orders of magnitude by which we measure the intellectual, cultural, and financial abundance that is New York: one zero to mark the tens of ethnic neighborhoods, two for the hundreds of entertainment options, three for the thousands of restaurants, six for the millions of people, and, of course, the zeros upon zeros that mark the billions or even trillions of dollars that churn through the city every day. These are not the zeros of void, but the zeros of plenty.

But, today, just one week past Memorial Day 2002, it’s a zero of a different flavor which has attracted this eclectic group to downtown New York City. Here some of the world’s greatest mathematicians are meeting to discuss and possibly, just possibly, witness the resolution of the most important unsolved problem in mathematics, a problem that holds the key to understanding the basic mathematical elements that are the prime numbers. The zeros that tip the tongues of these mathematical adventurers are zeta zeros,* and the air is electric with the feeling that perhaps this will be the day when we lay to rest the mystery of these zeros, which constitutes the Riemann hypothesis.

For over a century mathematicians have been trying to prove the Riemann hypothesis: that is, to settle once and for all a gently asserted conjecture of Bernhard Riemann (1826–1866), who was a professor of mathematics at the University of Göttingen in Germany. Riemann is perhaps best known as the mathematician responsible for inventing the geometrical ideas upon which Einstein built his theory of general relativity. But in 1859, for one brief moment, Riemann turned his attention to a study of the long-familiar prime numbers. These are numbers like two, three, five, and seven, each divisible only by one and itself, fundamental numerical elements characterized by their irreducibility. Riemann took up the age-old problem of trying to find a rule which would explain the way in which prime numbers are distributed among the whole numbers, indivisible stars scattered without end throughout a boundless numerical universe.

In a terse eight-page “memoir” delivered upon the occasion of his induction into the prestigious Berlin Academy, Riemann would revolutionize the way in which future mathematicians would henceforth study the primes. He did this by connecting a law of the primes to the understanding of a seemingly completely unrelated complex collection of numbers—numbers characterized by their common behavior under a sequence of mathematical transformations that add up to the Riemann zeta function. Like a Rube Goldbergesque piece of mathematical machinery, Riemann’s zeta function takes in a number as raw material and subjects it to a complicated sequence of mathematical operations that results in the production of a new number. The relation of input to output for Riemann’s zeta function is one of the most studied processes in all of mathematics. This attention is largely due to Riemann’s surprising and mysterious discovery that the numbers which seem to hold the key to understanding the primes are precisely the somethings which Riemann’s zeta function turns into nothing, those inputs into Riemann’s number cruncher that cause the production of the number zero. These are the zeta zeros, or more precisely the zeros of Riemann’s zeta function, and they are the zeros that have attracted a stellar cast of mathematicians to New York.

In his memoir, Riemann had included, almost as an aside, that it seemed “highly likely” that the zeta zeros have a particularly beautiful and simple geometric description. This offhand remark, born of genius and supported by experiment, is the Riemann hypothesis. It exchanges the confused jumble of the primes for the clarity of geometry, by proposing that a graphical description of the accumulation of the primes has a beautiful and surprisingly simple and precise shape. The resolution of Riemann’s hypothesis holds a final key to our understanding of the primes.

We’ll never know if Riemann had in mind a proof for this assertion. Soon after his brief moment of public glory, the ravages of tuberculosis began to take their toll on his health, leaving him too weak to work with the intensity necessary to tie up the loose ends of his Berlin memoir. Just eight years later, at the all too young age of thirty-nine, Riemann was dead, cheated of the opportunity to settle his conjecture.

Since then, this puzzling piece of Riemann’s legacy has stumped the greatest mathematical minds, but in recent years frustration has begun to give way to excitement, for the pursuit of the Riemann hypothesis has begun to reveal astounding connections among nuclear physics, chaos, and number theory. This unforeseen confluence of mathematics and physics, as well as certainty and uncertainty, is creating a frenzy of activity that suggests that after almost 150 years, the hunt might be over.

This is the source of the buzz filling the Courant Institute’s entryway. It is a buzz amplified by the fact that whoever settles the question of the zeta zeros can expect to acquire several new zeros of his or her own, in the form of a reward offered by the Clay Institute of Mathematics, which has included the Riemann hypothesis as one of seven “Millennium Prize Problems,” each worth $1 million. So the jungle of abstractions that is mathematics is now full of hungry hunters. They are out stalking big game—the resolution of the Riemann hypothesis—and it seems to be in their sights.

The Riemann hypothesis stands in relation to modern mathematics as New York City stands to the modern world, a crossroads and nexus for many leading figures and concepts, rich in unexpected and serendipitous conjunctions. The story of the quest to settle the Riemann hypothesis is one of scientific...

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Verlag
Pantheon Books
Erscheinungsdatum
2005
Sprache
Englisch
ISBN 10 
037542136X
ISBN 13 
9780375421365
Einband
Gebundene Ausgabe
Anzahl der Seiten
292
Kontakt zum Hersteller
PANTHEON
https://www.penguinrandomhouse.com/series/PGN/pantheon-graphic-library/

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