Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
Currently Skimming:
Notes
Notes
Pages 365-392
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 365... ...
He is the leading authority on the life and work of Bernhard Riemann and has edited Riemann's letters. I have made use of his researches in this book.
|
From page 366... ...
The term is hardly used nowadays. I shall mention multivalued functions in Chapter 3, complex function theory in Chapter 13, and the inverting of integrals in Chapter 21.
|
From page 367... ...
and "negative" to mean "bad." Adding a positive number means sending some good people into the auditorium, which obviously increases the net quantity of goodness in there. Adding a negative number means sending some bad people in, which decreases the net goodness.
|
From page 368... ...
Gracious Queen, we thee implore To go away and sin no more; But if this effort be too great, To go away, at any rate. One of the Duke's maternal aunts married a Holy Roman Emperor and begat Maria Theresa, the great Hapsburg empress.
|
From page 369... ...
It is said to have inspired the tragic Evariste Galois the narrator in Tom Petsinis's novel The French Mathematician to take up a career in mathematics. More relevant to the present narrative, his book Theory of Numbers the renamed third edition of the Essay mentioned in the text was lent by a schoolmaster to the adolescent Bernhard Riemann, who returned it in less than a week with the comment, "This is truly a wonderful book; I know it by heart." The book has 900 pages.
|
From page 370... ...
22. Euler's Latin is a stripped-down, racing version of the language, designed not to show off the writer's superb grasp of Augustan style (which Euler probably could have done if he had wanted to he knew the Aeneid by heart)
|
From page 371... ...
Gauss's great classic on number theory was titled Disquisitiones Arithmeticae (1801~. It seems to have been sometime in the later nineteenth century that "arithmetic" was definitely reserved for the basic manipulations learned in elementary school, with "number theory" used for the deeper researches of
|
From page 372... ...
It's a nice story, but Nobel was not married. Felix's first cousin, Ottilie, married the great German mathematician Eduard Kummer; their grandson, Roland Percival Sprague, was co-creator of"Sprague-Grundy Theory," in twentieth-century Game Theory....
|
From page 373... ...
So are the following terms for the two parts, "the Dirichlet series" for the infinite sum, and "the Euler product" for the infinite product. Strictly speaking, the left-hand side is a Dirichlet series and the right-hand side is an Euler product.
|
From page 374... ...
46. I should perhaps explain that mathematicians have their own particular approach to the learning of foreign languages.
|
From page 375... ...
Hollond, recorded the following note about him in 1972: "In his 87th year he is still working long hours at a stretch, writing papers for publication and helping mathematicians who send their problems to
|
From page 376... ...
Though a fine mathematician, he appears to have been active in no other sphere. I mentioned this to Atle Selberg, the only mathematician I have spoken with who might have known both men.
|
From page 377... ...
Edwards, "reasonable, not vitriolic." 59. In German, Wer con uns wurde nichtgern den Schleier luiten, unter dem die ZuLunft verborgen liegt, um einen Thick zu werfen auf die bevorstehenden Fortschritte unserer Wissenschaft und in die Geheimnisse ihrer Entwickelung wdhrend der kunitigen lahrhunderte?
|
From page 378... ...
This small and obscure nation produced an astonishing proportion of the world's finest mathematicians: Bollobas, Erdelyi, Erdo's, Fejer, Haar, Kerekjarto, two Ko'nigs, Kurschak, Lakatos, Rado, Renyi, two Rieszes, Szasz, Szego', Szokefalvi-Nagy, Turan, van Neumann, and I have probably missed a few. There is a modest literature attempting to explain this phenomenon.
|
From page 379... ...
69. Mathematicians working with functions of a complex variable generally say "the z plane" and "the w plane," it being understood that "z" is the generic argument and "w" the generic value in complex function theory.
|
From page 380... ...
is violated infinitely many times, but that this is also true of Chebyshev biases. For some very fascinating recent insights on this topic, see the paper "Chebyshev's Bias," by Michael Rubinstein and Peter Sarnak, in Experimental Mathematics, Vol.3, 1994 (pp.
|
From page 381... ...
(It had previously been described by another mathematician, Johann Listing, also in 1858. Listing published, and Mobius didn't, so according to the academic rules it should really be called "the Listing strip." There is no justice in this world.)
|
From page 382... ...
If a human being 11 feet tall were discovered living in the remote highlands of New Guinea, then the existence of that person would prove Theorem 15-1 to be false. The Riemann Hypothesis, however, would still be open, since the giant is not a U.S.
|
From page 383... ...
89. In the world of mathematics another instance was Ludwig Bieberbach, author of a famous conjecture in complex function theory (proved in 1984 by Louis de Branges)
|
From page 384... ...
94. The Fields Medal, first awarded 1936, was the idea of Canadian mathematician John Charles Fields (1863-1932~.
|
From page 385... ...
Patterson, in his book, An Introduction to the Theory ofthe Riemann Zeta-Function, §5.11, wrote: "The most convincing reason that has so far been evinced for the validity of the Riemann Hypothesis is that an analogous statement is valid for the zeta-functions attached to curves over finite fields. The formal similarities are so striking that it is difficult to believe that they do not lead to even more farreaching coincidences." (My italics.)
|
From page 386... ...
The German for "the Riemann Hypothesis," by the way, is Die Riemannsche Vermutung, from the verb vermuten "to surmise." 110. Professor of Physics at Bristol University in England.
|
From page 387... ...
112. The earliest reference I have been able to track down to the Montgomery-Odlyzko Law thus named is in a paper by Nicholas Katz and Peter Sarnak published in 1999.
|
From page 388... ...
Reviewing Connes 1999 paper"Trace Formulae in Noncommutative Geometry and the Zeros of the Riemann Zeta Function," Peter Sarnak (who is neither of my mathematicians X and Y) noted: "The analogies and calculations in the paper and its appendices are suggestive, pleasing and intricate and for these reasons this appears to offer more than just another equivalence of RH.
|
From page 389... ...
127. It has no direct bearing on the argument here, but I can't resist adding, as a matter of interest, that one of the most famous theorems in complex function theory concerns entire functions.
|
From page 390... ...
Keating has worked closely with Sir Michael Berry on the physical aspects of the RH.
|
From page 391... ...
One of the professional mathematicians who looked over my manuscript expressed frank disbelief at this, though. The idea that one might be able to make money by doing mathematics is extremely difficult for mathematicians to take seriously.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.