Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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12. Hilber's Eighth Problem
12. Hilber's Eighth Problem
Pages 184-200
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From page 184... ...
Mathematicians refer to this kind of thing as an "existence proof' Hilbert used the following everyday example in his lectures. "There is at least one student in this class let us name him 'X' for whom the following statement is true: no other student in the class has more hairs on his head than X
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From page 185... ...
The names "Hilbert" and "Gottingen" are yoked together in the minds of modern mathematicians as closely as, in other spheres, are "Joyce" an(l"Dublin," or"Tohnson" an(l"Lon(lon." Hilbert an Gottingen (lominate(1 mathematics (luring the first thir(1 of the twentieth century not merely German mathematics, but all mathematics. The Swiss physicist Paul Scherrer, arriving at Gottingen as a student in 1913, reported finding there "an intellectual life of unsurpassed intensity." An astonishing proportion of important mathematicians and physicists of the first half of the century had studied either at Gottingen, or under someone who had studied there.
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From page 186... ...
This was particularly unfortunate in Hilbert's case because Franz, his only child, was afflicted with serious mental problems. Unable to learn anything much, or to hold down any kind of job, Franz also suffered occasional lapses into paranoia, following which he had to be kept in a mental hospital for a while.
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From page 187... ...
Knowing the pleasure Hilbert took in strolls in the countryside while talking mathematics, Courant invited him for a walk. Courant managed matters so that the pair walked through some thorny bushes, at which point Courant informed Hilbert that he had evidently torn his pants on one of the bushes.
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From page 188... ...
Bright blue eyes looked innocently but firmly out from behind shining lenses. Hilbert delivered his address, in German, in a stuffy lecture hall at the Sorbonne.
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From page 189... ...
=1+—+ 1 + 1 +.., 2s as 4s all have the realpart 2 ~ except the well-known negative integral real zeros. As soon as this proof has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially, to decide whether the difference between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order notgreater than 2 in x.
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From page 190... ...
Bernhard Riemann's 1859 paper had not, of course, proved the PNT, but it had mightily suggested that it should be true, and even further had suggested an expression for the error term. That expression involved all the non-trivial zeros of the zeta function.
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From page 191... ...
"Critical strip" and "critical line" are common terms of art in discussions of the Riemann Hypothesis, and from now on I shall use them quite freely. The Riemann Hypothesis (stated geometrically)
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From page 192... ...
If one side of that formula is zero, the other side must be too. Leaving aside integer values of s, where other terms in the formula misbehave or go to zero, this formula says that if ~ (s)
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From page 193... ...
The world of any given mathematical specialty is, too, a small one, with its own heroes, forbore, and oral traditions binding the community together in both time and space. From speaking with living mathematicians to gather material for this book, I came to feel that the twentieth century was not such a very long span of time after all, the great names of its early years almost within hailing distance.
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From page 194... ...
Here is the beginning of the first proposition in Alain Connes' Noncommutative Geometry (1990) , a pretty typical higher-math text of the later twentieth century.
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From page 195... ...
. A typical product of the twentieth century, by contrast, was "functional analysis," where the fun(lamental object of stu(ly is sequences of functions, which might or might not converge, and where a function is itself liable to be treated as a "point" in a space of infinitely many (limensions.
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From page 196... ...
David Hilbert, as I have already described, listed the Riemann Hypothesis eighth in his list of 23 problems for mathematicians of the twentieth century to concentrate their efforts on.
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From page 197... ...
There was an algebraic thread, started by Emil Artin in 1921, attempting to take the Riemann Hypothesis by a flanking movement through an algebraic topic called Field Theory. Later in the century, as a result of a remarkable encounter I shall write about in (lue course, a physical!
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From page 198... ...
In covering these developments I shall try to make it clear at every point which thread I am talking about, though sometimes skipping carelessly from one to another to maintain the overall chronological narrative. Let me begin with a brief introductory remark about the computational thread, since that is the easiest for a nonmathematician to understand.
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From page 199... ...
HILBERT'S EIGHTH PROBLEM 199 Figure 12-2. His list, which contained some slight inaccuracies in the right-most (ligits, begins - + 14.134725 i, - + 21.022040 i, - + 25.010856 i ....
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From page 200... ...
In the next part of my historical narrative I shall take the reader to England, in the high Edwardian summer of her imperial glory. But first let me show you what the zeta function actually looks like.
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