Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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Appendix: The Riemann Hypothesis in Song
Appendix: The Riemann Hypothesis in Song
Pages 393-404
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From page 393... ...
in 1955 and performed it at the Caltech Number Theory conference hel(1 in June of that year. Tom's original lyrics went only as far as Line 32; the last two stanzas were posted on a bulletin board at Cambridge University in 1973 by algebraic topologist Saunders MacLane.
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From page 394... ...
If square root of x times log x we could show, Then Riemann's conjecture would surely be so. Related to this is another enigma, Concerning the Lindelof function mu sigma, Which measures the growth in the critical strip; On the number of zeros it gives us a grip.
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From page 395... ...
"Villikens and his Dinah" was a favorite with Alice Liddell, the girl who inspired the books, and she actually did have a cat named Dinah.) If you had a British education that included membership in a school rugby club, you will most likely recognize the tune as that of the melancholy ballad beginning, "O Father, O Father, I've come to confess.
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From page 396... ...
what? We don't know." For big oh, see Chapter 15.ii-iii.
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From page 397... ...
For Lindelof the man, see Note 72. His hypothesis concerns the growth of the zeta function in a vertical direction that is, up a vertical line in the complex plane.
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From page 398... ...
398 ¢(-1 ~ ityl 75 50 25 ((1 it)
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From page 399... ...
for these graphs. lust from looking at them, you can guess the following: · For ~ =-1, -2, and -3, the graph looks as if it is big oh of some accelerating function of t, perhaps a power like t2 or t5, those powers seeming to get bigger as ~ heads west along the negative real axis.
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From page 400... ...
We also know the following, with mathematical certainty. · When ~ is less than or equal to zero, ~ ~ ~ ~ = 2 · When ~ is greater than or equal to 1, ,u ~ ~ ~ = 0.
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From page 401... ...
, but not vice versa. The LH is the weaker result.
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From page 402... ...
Line 31. "In order to strengthen the prime number theorem...." That is, in order to get the best possible big oh expression for the error term.
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From page 403... ...
Among the many alternative versions of Tom's lyrics to be found on the internet, I note that one ends with the line, "Use R.M.T., and you'll have better luck." This is a good-natured dig at the "physical" approach. "R.M.T." stands for"Random Matrix Theory."
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