Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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14. In the Grip of an Obsession
14. In the Grip of an Obsession
Pages 223-237
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From page 223... ...
was famous, but only as a problem in integral functions; and all this took place in the long vacation when I had no access to literature, had I suspected there was any. (As for people better instructed, only some had heard of Hadamard's paper, and fewer still knew of de la Vallee Poussin's in a Belgian journal.
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in 1953. The other dramatic personae in the extract are the older English mathematician Godfrey Harold Hardy, 1877-1947, and the German Edmund Landau, 1877-1938.
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One of the papers read was "The Idea of Order and Absolute Position in Space and Time." Its author was a young British logician, also a Trinity man, named Bertrand Russell, who 10 years later, with Alfred North Whitehead, produced the classic of mathematical logic (to be more precise, of logicized mathematics) , Principia Mathematical The least abstract math, and the most, but the great middle ground of abstraction function theory, number theory, most of algebra was yiel(le(1 to the Continentals.
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Studying for his degree at Trinity in 1897, Hardy came across a famous textbook of the time, Cours d'Ana1/yse, by the French mathematician Camille Tordan. Tordan is familiar to students of complex variable theory for Tor(lan's Theorem, which says, basically, that a simple closed curve in the plane, for example a circle, has an inside and an outside.
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court tennis or jeu de paume) , a more difficult, more intellectually challenging game than culinary tennis.
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G.H. Hardy." If the boat sinks and Hardy drowns, everybody must believe that he has proved the Riemann Hypothesis.
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The story as his colleague Bela Bollobas tells it is that Littlewood, in his younger years, used to go for annual vacations with the family of a doctor in Cornwall, whose chil(lren grew up calling him"Uncle John." One of these children was named Ann; Littlewood referred to her as "my niece." However, after becoming close friends with Bollobas and his
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Landau was an instance of that uncommon phenomenon, the scion of a wealthy family who yet had a powerful work ethic and a record of great achievement in a non-commercial field. Landau's mother Tohanna, nee Tacoby, came from a rich banking family.
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From page 231... ...
Asked if she was not an instance of a great female mathematician, Landau replied: "I can testify that Emmy is a great mathematician, but that she is female, I cannot swear." His work ethic was legendary. It is said that when one of his junior lecturers was in hospital, recuperating from a serious illness, Landau climbed a ladder and pushed a huge folder of work through the poor man's window.
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From page 232... ...
It was certainly from the Handbuch that both Hardy and Littlewood became infected with the Riemann Hypothesis obsession. The first fruits came in 1914, not in the form of a collaboration, though they were collaborating by that time, but as two separate papers, both of major importance in the (levelopment of the theory.
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. Littlewoo(l's paper, also publishe(1 in the Paris Aca(lemy's Comptes Rendus of that year, was titled Sur 1/a distribution des nombres premiers.
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From page 234... ...
It was in focusing on the error term that mathematicians' attention was drawn to the Riemann Hypothesis, because Riemann's 1859 paper gave an exact expression for the error term. That expression, as I shall show in due course, involves all the non-trivial zeros of the zeta function, so the key to understanding the error term is hidden in among the zeros somehow.
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They are intimately connected, in a way I shall later show you in exact mathematical detail, with the error term. Although it is the relative error that the PNT speaks about, investigations in this area more often concentrate on the absolute error.
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That was the case with the first upper bound for the Littlewood violation. In 1933 Littlewood's student Samuel Skewes showed that if the Riemann Hypothesis is true, the crossover point must come be79 fore eee a number of about 10tenbilliontri~lion trillion digits Thatch not the number; that's the number of digits in the number.
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From page 237... ...
+ 0(4 log x) The equation is pronounced as, "Pi of x equals log integral of x plus big oh of root x log x." Now I have to explain the "big oh" notation.
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