Tuesday, May 31, 2005

Whither Are The Zeros Of Zeta Of S?

First, the following needs to be got off one's chest.

Two hydrogen atoms meet in a bar. "I think I lost my electron." "Are you sure?" "Yes, I'm positive." Whew...

Desultory flipping through of Karl Sabbagh's book, The Riemann Hypothesis - The Greatest Unsolved Problem In Mathematics. Like other mathematics books for laypeople (such as Zero - The Biography Of A Dangerous Idea and Fermat's Last Enigma), this one is a judicious mix of history, anecdotes and some relatively accesible mathematics.

The Riemann Hypothesis states (more or less) that
that the nontrivial Riemann zeta function zeros, i.e., the values of s other than -2, -4, -6, ... such that ζ(s)=0 (where ζ is the Riemann zeta function) all lie on the "critical line" σ=R[s]=½ (where R denotes the real part of s).
Yes, its really that simple. A child could tackle this.

Why is this important, you ask? Maybe you don't ask. But I tell. In the words of Sabbagh
The Riemann Hypothesis matters because, if it is true, it proves that there is a rule for generating the prime numbers...
If Tom, Li Mu Bai, and Thirunavukkarasu start generating prime numbers, one will have to wonder what the implications for cryptography are. This is only the tip of the iceberg. There are other obscure mathematical reasons (which I fully comprehend, mind you, but which will be so much Greek to you folks) why this is such an important result.

Many great mathematicians have been trying to prove or disprove the Riemann Hypothesis. It features as one of the 23 unsolved (at the time) problems of David Hilbert. Hardy and Ramanujan tried it and couldn't. Hardy was quite a character, but this post is too long already.

Interesting mathematics websites include the MacTutor History of Mathematics archive. Wolfram Research's Mathworld is a positive treasure, they also have similar sections on physics, chemistry and other sciences. If you fancy yourself as a mathematician, you might want to check out IBM Research's Ponder This problem of the month.

Tailpiece
Grafitti on New York City subway wall:

xn + yn = zn

There is no value of n>2 for which the above is true. I have found a truly remarkable proof of this, but my train is coming and I have to go...

6 comments:

Anonymous said...

It is for a post like this that the description of your blog suits me well: Wovon man nicht sprechen kann, darüber muß man schweigen. :)
In reality and hindsight, I think it is just not for this blog, but lots more...
-- Y!

Somus said...

This comment will probably disappear into the ether, since I only just discovered your blog today (for shame!), but still. The bestest history of the Riemann hypothesis was written by one Tom M. Apostol, whose name is no doubt familiar to all, etc etc. You can read it in its magnificent entirety here.

Ludwig said...

Somus, thanks for the link :) Apostol's poem shows up in many places in the book itself. Perhaps it is a fit candidate for a Minstrels post? :PP The blog has been going on, quite desultorily for a few months now. Not a whole lot going on here, just evidence of someone's utter scatterbrainedness...

Somus said...

If you'd like to submit the Apostolic verse as a guest pome on the Minstrels, please feel eminently free to do so. BTW what's your preferred email address these days? m****** at hotmail?

Ludwig said...

Roger that, Somus. A. gozaim. I lurk at m*@gmail.com also.

Somus said...

Hai, wakarimashita.