Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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17. A Little Algebra
17. A Little Algebra
Pages 265-279
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From page 265... ...
Field theory is important because it has allowed something very much like the Riemann Hypothesis to actually be proved. Many researchers believe that field theory offers the most promising line of attack on the original, classical Riemann Hypothesis.99 Operator theory became important following the remarkable and rather romantic (developments I shall (lescribe in the next chapter.
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From page 266... ...
Each has an infinite number of elements, of course. Other infinite fields are easy to construct.
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From page 267... ...
For other finite fields, the arithmetic is more subtle. Figure 17-1, for example, shows the clock arithmetic addition and multiplicationfor a clock with four hours marked (i.e., O
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From page 268... ...
A, and (C, in which no amount of adding 1 to itself will ever produce zero, are said to have characteristic zero. (You might think that characteristic infinity would be more logical, and you might be right, but there are good reasons for choosing zero instead.)
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From page 269... ...
Thus an infinite field can have a finite characteristic. It is not very helpful to ask what x represents in these last two examples.
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From page 270... ...
Conjectures." In 1973 the Belgian mathematician Pierre Deligne, in a sensational achievement that won him a Fields Medal, proved the Weil Conjectures, essentially completing the program initiated by Artin. Whether the techniques developed to prove these analogues of the Riemann Hypothesis for these very abstruse fiel(ls can be used to
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From page 271... ...
For the crux of the matter, go back to the second paragraph in this section, where I said that one of these analogue zeta functions is associated with a certain kind of field. For the classical zeta function, the one to which the original Riemann Hypothesis applies the one this book is mainly about the equivalent associated field is Q
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From page 272... ...
A matrix can be any size, though: 3 x 3, 4 x 4, 120 x 120, and so on. It can even be infinite in size, though the rules change slightly for infinite matrices.
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From page 273... ...
If I look at it closely, I can see the months and days of the traditional Chinese lunar calen(lar marked, each month beginning with a new moon. The numbers are all different from the solar numbers, but they represent the same celestial events, the same passage of time, the same actual days.
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From page 274... ...
(And complex conjugates of each other which is always the case for a polynomial with real coefficients.) That is quite normal, even when, as here, ah the numbers in the home matrix are real num
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From page 275... ...
They had developed, so to speak, a taxonomy of matrices, in which the entire family of Nx N matrices referred to by mathematicians as "the general linear group for N." and symbolize(1 by"GLN' was organize(1 into species an(1 genera. I am going to pluck just one species out of that great matrix zoo, the Hermitian matrix, named after the great French mathematician Charles Hermite, whom we met briefly in Chapter l O.v.
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From page 276... ...
at the time, that is, when David Hilbert was just embarking on his investigation of integral equations, in which the study of operators played a key role. Other mathematicians some independently, some inspired by Hilbert's work also spent the early years of the twentieth century absorbed in the stu(ly of operators.
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From page 277... ...
With both the eigenvalues of a Hermitian matrix and the non-trivial zeta zeros, we have a list of unexpectedly real numbers emerging from the key property of an essentially complex object. Hence, The Hi1/bert-Po1/ya Conjecture The non-trivial zeros of the Riemann zeta function correspond to the eigenvalues of some Hermitian operator.
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From page 278... ...
and, says Peter Sarnak, so far as he knows the only written evidence for the Hilbert-Polya Conjecture having been conjectured consists of a letter Polya wrote to Andrew Odlyzko 60 years later, part of which is shown in Figure 17-3. In it, Polya said that he had been asked the following question by Edmund Landau: "Can you think of any physica11 reason why the Riemann Hypothesis might be true?
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From page 279... ...
The most exceptional individuals were awarded the title "Geheimrat" by the German government a rank roughly equivalent to a British knighthood. The correct form of address was then "Herr Geheimrat," though Hilbert himself did not care for this level of formality.
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