Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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8. Not Altogether Unworthy
8. Not Altogether Unworthy
Pages 118-136
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From page 118... ...
The 1830s, when spirits had revived after the exhaustion of the long wars, were an unsettled time, marked by the July revolution in France, a nationalist uprising in Poland (at that time part of the Russian empires) , agitation among
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From page 119... ...
In the following decade darker forces stirred, culminating in 1848, "the year of revolutions," whose disturbances, as we saw in Chapter 2, penetrated for a moment even the deep reserve of Bernhard Riemann. Gottingen was for all this period a provincial backwater illuminated mainly by the presence of Gauss.
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From page 120... ...
The physicist Wilhelm Weber, one of the Gottingen Seven cashiered in 1837, had returned to the university to teach, the political climate having thawed considerably. An old friend and colleague of Gauss's the two of them had together invented the electric telegraph Weber taught a course in experimental physics, which Riemann attended.40 II.
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From page 121... ...
The paper also contains the first sketches of the theory of Riemann surfaces, a fusion of function theory with topology the latter topic so new at the time there was really no coherent body of knowledge about it, only some scattered results going back to Euler's time.4i Riemann's doctoral thesis is, in short, a masterpiece. Both Riemann and Dedekind then embarked on the second leg of the academic marathon to which they had committed themselves, the habilitation thesis and trial lecture required for a teaching position at the university.
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From page 122... ...
In Kazan, the university boasted the presence of the great mathematician Nikolai Lobachevsky, who served as Rector until his dismissal in 1846. Lobachevsky was the inventor of non-Euclidean geometry, of which I shad have more to say shortly.43 And now, in 1849-1850, 25 years into the reign of Nicholas I, intellectual life in Russia was enduring another spell of repression, as Nicholas reacted to the 1848 revolutions in Europe.
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From page 123... ...
Says the hagiographer of this Pafnuty: "He was a virgin and an ascetic, and, because of this, a great wonderworker and seer." (In the middle of writing this chapter I got an e-mail from a reader of my web column asking me to suggest a name for her new dog. There is now a Pafnuty chasing squirrels somewhere in the Midwest.)
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From page 124... ...
He brought the full power of complex function theory to bear on the issue he was investigating. The results he got were so striking that other mathematicians followe(1 him, and the PNT was prove(1 at last using Riemann's non-elementary methods.
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From page 125... ...
'real variable' proof of the prime number theorem, that is to say a proof not involving explicitly or implicitly the notion of an analytic function of a complex variable, has never been discovered, and we can now understand why this should be so.... ,, Then, to everybo(ly's astonishment, such a proof was (liscovere in 1949 by Atle Selberg, a Norwegian mathematician working at the Institute for Advanced Study in Princeton, New Tersey.44 There was much controversy over the result, because Selberg had communicated some of his preliminary ideas to the eccentric Hungarian mathematician Paul Erdo's, who used them to create a proof of his own at the same time.
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From page 126... ...
V In the fall of 1852, the first year of work on his habilitation thesis, Riemann met Dirichlet again.
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From page 127... ...
The thesis itself it is titled"On the representability of a Unction by a trigonometric series" is a landmark paper, giving the world the Riemann integral, now taught as a fundamental concept in higher calculus courses. The habilitation lecture, however, far surpassed the thesis.
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From page 128... ...
Riemann's starting point was some ideas Gauss had put forward in an 1827 paper titled "A General Investigation into Curved Surfaces." Gauss ha(1 been employe(1 for the previous few years in carrying out a detailed topographical survey of the Kingdom of Bavaria (during which, by the way, he invented the heliotrope, a device for making long-distance observations by reflecting flashes of sunlight from an arrangement of mirrors)
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From page 129... ...
(In one of his few recorded negative comments about anyone at all, Riemann noted that the Berlin mathematician Gotthold Eisenstein "stopped at formal computation.") What, then, was a function?
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From page 130... ...
It is, when you think about it, very odd that inquiries about the infinitesimal neighborhoods of points and numbers should give us the power to explain the large global properties of functions and spaces. This is especially apparent in General Relativity theory, where you start off by analyzing microscopic regions of space-time and en(1 by contemplating the shape of the universe and the death throes of galaxies.
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From page 131... ...
1 am not qualified to Judge whether this is true. I can, however, give wholehearted assent to another remark of Freudenthal's: "Riemann's style, influenced by philosophical reading, exhibits the worst aspects of German syntax; it must be a mystery to anyone who has not mastered German." I confess that, though I possess a copy of Riemann's collected works in the original German it is a single volume of 690 pages and have done my best with his actual words, where he departs from straightforward mathematical expositionas, for example, in the habilitation lecture I have approached his tremendous thoughts mainly through translations and secondary sources.46 ~ , 1 .
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From page 132... ...
This might be true; but surviving letters byRiemann's students suggest that as late as 1861 "His thoughts frequently failed him and he was unable to explain the simplest things." Riemann's own take on the matter is, as usual, rather touching. Writing to his father after his first lecture, which was on October 5, 1854, he says "I hope that in half a year I shall feel easier about my lectures, and the thought of them will not spoil my stay in Quickborn and my being together with you, as last time." This was a desperately shy man.
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From page 133... ...
He threw himself into work, and in 1857 produced the landmark paper on function theory that I mentione(1 in Chapter 1, the paper that ma(le his name known. The effort, however, combine(1 with grief, precipitate(1 a nervous breakdown.
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From page 134... ...
a heart attack while lecturing in Switzerland and was brought back to Gottingen only with much difficulty. While he was lying gravely ill, his wife died suddenly of a stroke.
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From page 135... ...
In the second sentence he showed the Golden Key. In the third he named the zeta function.
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From page 136... ...
136 PRIME OBSESSION The Riemann Hypothesis, which appears on the fourth page of that paper, asserts a certain fact about the zeta function. To advance in our understanding of the Hypothesis, we must now go deeper into the zeta function.
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