Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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10. A Proof and a Turning Point
10. A Proof and a Turning Point
Pages 151-166
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From page 151... ...
. His actual words, after making a statement that is equivalent to the Hypothesis, were 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 (einigenlquchtigen vergeblichen Versuchen)
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From page 152... ...
The usual example of an extremely logical mathematician is the German analyst Karl Weierstrass (1815-1897) , who did his great work in the third quarter of the nineteenth century.
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From page 153... ...
Then, in a space of less than 10 years, Hadamard, van Mangoldt and de la Vallee Poussin succeeded in proving both Riemann's main formula for ~r (x) and the prime number theorem, as well as a number of other related theorems.
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From page 154... ...
2 + 3 + 5 + 7 + I! + 13 + 17 + + p + · In 1885 the Dutch mathematician Thomas Stieltjes claimed to have a proof of the Riemann Hypothesis.
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From page 155... ...
It was then plain that if a certain theorem much weaker than the Riemann Hypothesis could be proved, the application of the result to van MangolUt's formula would prove the PNT. · In 1896 two mathematicians working independently, the aforementioned Jacques Hadamard and the Belgian Charles de la Vallee Poussin, proved that weaker result and, therefore, the PNT.
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From page 156... ...
I am content to say, along with the whole world of mathematics, that in 1896, Jacques Hadamard of France and Charles de la Vallee Poussin of Belgium, working independently, proved the PNT.
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From page 157... ...
, then being proved at the end of the next (Hadamard and de la Vallee Poussin, 18961. Once that theorem had been disposed of, the attention of mathematicians turned to the Riemann Hypothesis, which occupied them for the following century which came to its end without any proof being arrived at.
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From page 158... ...
This tipped the officer class toward the old "Throne and Altar" style of French conservatism, and to some (legree cut it off from the mainstream of French life in these decades. The mainstream was all in the direction of a bustling, openminded, commercial, and industrial republic, a leader in the arts and sciences, a center of brilliance, wit, and gaiety the wonderful, glittering France of the Belle Epoque, one of the great high points of western civilization.
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From page 159... ...
He was a mathematician of considerable scope, producing original work in several different areas. Undergraduate students of math generally first encounter his name attached to the Three Circles Theorem in complex function theory, a result Hadamard obtained in 1896, and which you can look up in any good encyclopedia of mathematics.52 You will see it written that Hadamard was the last of the universal mathematicians the last, that is, to encompass the whole of the subject, before it became so large that this was impossible.
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From page 160... ...
. From 1882 onward Hermite had been conducting a mathematical correspondence with a younger mathematician, a Dutchman named Thomas Stieltjes.53 In 1885, Stieltjes published a note in the Comptes Ren~us54 of the Paris Academy of Sciences, claiming to have proved my Theorem 15-1 a result stronger than the Riemann Hypothesis, from which, if Stieltjes had indeed proved it, the truth of the Hypothesis would follow (but whose falsehood would not disprove the Hypothesis see Chapter 15.v)
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From page 161... ...
Both Charles de la Vallee Poussin at the University of Louvain in BeIgium and Jacques Hadamard in Bordeaux took up the lesser challenge and soon got the result. They proved the PNT.
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From page 162... ...
In March 1896 Colonel George Picquart of French military intelligence happened to notice that the document that had been the principal item of evidence against Dreyfus was in fact in the handwriting not of Dreyfus but of another officer, Major Esterhazy, a man of erratic character and extravagant habits, chronically beset by gambling debts. Picquart informed his superiors.
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From page 163... ...
An anti-Semitic polemical book, La France Juive, hall been a huge publishing success in 1886, and an anti-Semitic newspaper, La Libre Parolle, was wi(lely read. The Affair brought all this to the surface and made French Jews wonder if they had been living in a fool's paradise.
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From page 164... ...
Hadamard's third son, born in February 1899, was named Mathieu-Georges, "Mathieu" after Dreyfus's brother and most tireless champion, "Georges" after the remarkable Colonel Picquart, whose iron integrity an(1 quiet insistence on telling the truth were key factors in the eventual vindication of Dreyfus (whom Picquart personally detested)
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From page 165... ...
VII. In concentrating on Jacques Ha(lamar(l, I have in(lulge(1 my personal fondness for an attractive personality and fine mathematical talent, intending no disrespect to the other mathematicians who participated in the clarification of Riemann's great paper and the proof of the PNT.56 By the later nineteenth century the world of mathematics had passed out of the era when really great strides could be ma(le by a single min(1 working alone.
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From page 166... ...
On the morning of August 8, in a lecture hall at the Sorbonne, Hilbert stood up before the 200-odd delegates to the Congress, Jacques Ha(lamar(1 among them, and (lelivere(1 an a(l(lress on "Mathematical Problems." His aim was to concentrate the minds of his fellow mathematicians on the challenges facing them in the new century. To effect this goal, he directed their attention to a handful of the most important topics needing investigation, and problems needing solution.
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