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August 23, 2007

Comments

j.

Yes. It is unfortunate how hard it is to explain RH. I'd be great if it was like the one you mentioned by Fermat.

Anyways, the idea is that you have a function, "the zeta function" (put some echo here), defined on the complex plane (this is, the function takes complex numbers and gives you complex numbers). This "zeta function" is an "extension" of a real infinite series (one of those infinite sums Ramanujan loved) but I guess we can skip that detail by now.

The point is that if you do z(-2) or z(-4) or z(-6) (and so on, any negative even number), then you get 0. That's more or less easy to see once you know the formula for the zeta function. When they say "trivial zeros", they mean these guys ("the negative even numbers"). However, it was clear the zero function had more zeros (the so-called non-trivial ones) and some properties were known about them.

Parenthesis: To explain these properties, it is probably good to remember a little bit about the complex numbers. A complex number is something of the form a+ib where a and b are real numbers and i is that "unexistent" square root of -1 some of us meet in highschool some in college. We can represent the complex numbers in the cartesian plane assigning a+ib to the pair (a,b). Given a complex number a+ib, the Real Part (Re) of it is a and the Imaginary Part (Im) is b.

Ok, back to RH, the known properties of the non-trivial zeroes of the zeta function were two:

1. The non-trivial zeroes were all located in the vertical strip with Real Part between 0 and 1. This is, comples numbers a+ib where 0 2. If we had a non-trivial zero and we look for the symmetrical complex number (in the cartesian plane) with respect to the line Re(x)=1/2 (The vertical line of complex numbers whose real part is 1/2), then we would find ANOTHER non-trivial zero.

These people studing the zeta function, however, noticed something else: Most of the non-trivial zeros they knew were actually ON the line Re(x)=1/2.

Riemann Hypothesis claims that ALL the non trivial zeros are on the Re(x)=1/2 line. That's it.

The connection between the RH and the distribution of primes is another story and it should probably be told some other time. (Actually, that's the part that's really hard to explain in elementary terms.)

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TEV DEFINED


  • The Elegant Variation is "Fowler’s (1926, 1965) term for the inept writer’s overstrained efforts at freshness or vividness of expression. Prose guilty of elegant variation calls attention to itself and doesn’t permit its ideas to seem naturally clear. It typically seeks fancy new words for familiar things, and it scrambles for synonyms in order to avoid at all costs repeating a word, even though repetition might be the natural, normal thing to do: The audience had a certain bovine placidity, instead of The audience was as placid as cows. Elegant variation is often the rock, and a stereotype, a cliché, or a tired metaphor the hard place between which inexperienced or foolish writers come to grief. The familiar middle ground in treating these homely topics is almost always the safest. In untrained or unrestrained hands, a thesaurus can be dangerous."

SECOND LOOK

  • The Bookshop by Penelope Fitzgerald

    Bs

    Penelope Fitzgerald's second novel is the tale of Florence Green, a widow who seeks, in the late 1950s, to bring a bookstore to an isolated British town, encountering all manner of obstacles, including incompetent builders, vindictive gentry, small minded bankers, an irritable poltergeist, but, above all, a town that might not, in fact, want a bookshop. Fitzgerald's prose is spare but evocative – there's no wasted effort and her work reminds one of Hemingway's dictum that every word should fight for its right to be on the page. Florence is an engaging creation, stubbornly committed to her plan even as uncertainty regarding the wisdom of the enterprise gnaws at her. But The Bookshop concerns itself, finally, with the astonishing vindictiveness of which provincials are capable, and, as so much English fiction must, it grapples with the inevitabilities of class. It's a dense marvel at 123 pages, a book you won't want to – or be able to – rush through.
  • The Rider by Tim Krabbe

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    Tim Krabbé's superb 1978 memoir-cum-novel is the single best book we've read about cycling, a book that will come closer to bringing you inside a grueling road race than anything else out there. A kilometer-by-kilometer look at just what is required to endure some of the most grueling terrain in the world, Krabbé explains the tactics, the choices and – above all – the grinding, endless, excruciating pain that every cyclist faces and makes it heart-pounding rather than expository or tedious. No writer has better captured both the agony and the determination to ride through the agony. He's an elegant stylist (ably served by Sam Garrett's fine translation) and The Rider manages to be that rarest hybrid – an authentic, accurate book about cycling that's a pleasure to read. "Non-racers," he writes. "The emptiness of those lives shocks me."