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More on Riemann Hypothesis
Anonymous Coward writes "The NYTimes has a little story on a recent conference at New York University's Courant Institute where mathematicians gathered to discuss potential attacks on the Riemann hypothesis. The Clay Mathematics Institute had announced an award of a million dollars for a proof (or refutation) of the Riemann hypothesis during the millenial celebrations. That million dollars won't be worth much if it takes as long as that Last Theorem by Fermat to solve. There were some interesting observations such as the statistical distribution of the zeros looked just like calculations on the energy levels of large atoms." We did a related story on hard math problems two years ago.
Wrong. Primes do not always plot along one of the axes. Zeroes to the function are always (well, that's the hypothesis) of the form 1/2 + bi. This means they lie on a line parallel to the imaginary axis.
Fault loves the past, worry loves the future, but content enjoys the present.
Apparently there's a distributed computing project called ZetaGrid which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.
Riemann Hypothesis
Riemann Zeta Function
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.
We can neither love nor pity nor forgive. If you make a slip in handling us you die!
"that God -- with whom he waged a very personal war -- would not let Hardy die with such glory."
That has to be the funniest things I've read, today.
Is it me or does it seem that all "hard" mathematicians are either at war with God or trying to "refute"/"prove"/divide/discover/humiliate him/her/it/Taco?
Get your Unix fortune now!
He wrote a function called the "zeta function."
The function had already been discussed by Euler.
For some reason, primes always plot along one of the axes. No one can figure out why.
Actually, that's easy. Primes (at least over the integers) are real numbers, and the zeta function maps real numbers greater than one to real numbers, which is evident from the definition as a Dirichlet series.
Quite a few proofs in analytical number theory rely on the fact that in certain areas on the right side of the line {z : Re z = 1/2} contain no zeroes of the zeta function. So far, mathematicians have tried to carefully choose these areas in order to get good results (so that you can still use them efficiently, but you can also prove that no zeroes lie in it). If we knew that no such zeroes exist at all (the Riemann Hypothesis), we could avoid all these rather technical details and theorems would improve considerably as well (for example, the error term in the Prime Number Theorem).
Even if you are able to get into a cell it can be extremely difficult to stay in and keep your sanity. Many people who do get in just sort of drift off from society and are all but lost. Those few that make it often end up working alone, late at night in the back of dimly lit coffee houses.
There is simply no way to stop someone who is willing to make such sacrifices.
If you can't explain something in ordinary words to a layman, then you really don't understand it.
:)
... on the horizontal axis.
:)
I'm about halfway through writing up my PhD thesis on some applications of homological algebra to knot theory and low-dimensional geometric topology (provisional title liber rerum dementiae, but it'll probably end up being called something more mathematically appropriate).
In principle, yes, I could explain the details of my research to a suitably motivated layman. But I suspect it would take rather a long time.
You see, and this really isn't meant to sound arrogant, supercilious, or dismissive, university-level mathematics is pretty damned difficult, and the details of most cutting-edge research really doesn't make sense until you've spent several years learning the background (the mindset, the language, the fundamental concepts).
My current area of research is essentially the applications of homological algebra to knot theory and low-dimensional geometric topology. To explain this to a non-mathematician, I'd first have to teach them a lot of background stuff (group theory, a bit of stuff about rings and modules, point set topology, basic algebraic topology (the fundamental group, (co)homology theory), some geometric topology (basic course in knot theory, some stuff about 3-manifolds), a bit of category theory, and some homological algebra (broad overview of the (co)homology theory of groups and algebras)).
It's taken me nearly nine years (3-year BA, 1-year MSc specialising in topology and knot theory, plus nearly five years doing a (part-time) PhD) to get to this point myself. If I were a bit cleverer (or didn't have a `proper' job as well) I might have been able to shave a couple of years off that.
My friend Steve has a physics degree. I managed, in ten minutes one evening, with much handwaving, to give him some idea of what my thesis is all about. It helped that he knew what a group was already though. But for me to explain it fully to him would probably necessitate him doing at least one mathematics degree first. And that's not really something I'd wish on one of my friends
Now this really isn't meant in an arrogant way, and I hope you won't read it like that, but Euclid was right: There is no royal road to geometry.
I can have a go at explaining the Riemann hypothesis, though. To fully understand what it's about and why it's so damned difficult you'll need to do an advanced course in complex analysis (which isn't my field either).
A complex number is a sort of two-dimensional number, which you can regard as a point in a plane (the `complex plane' or `argand diagram'). You add them together coordinate-wise, and you multiply them together in a weird manner which involves something which behaves like a `square root of -1' (engineers also like to think of it as a sort of 90-degree phase-shift operator, I'm told).
There's a particular function (`Riemann's zeta function') defined on the complex plane (it takes one complex number as input and returns one complex number). For some complex numbers (`the zeros of the function'), the value of this function is zero.
The `trivial' zeros occur at the points -2, -4, -6,
The `non-trivial' zeros (that is, all the other points for which zeta is zero) all seem to occur on the line parallel to the vertical axis that intersects the horizontal axis at +0.5. Indeed, nobody's ever found one which doesn't.
The Riemann Hypothesis is that *all* the non-trivial zeros lie on this line. It's known to be true for the first (large number which temporarily escapes me), but it turns out to be phenomenally difficult to prove that it's true in every case.
Now that's the basic idea, but it doesn't (and I can't - it's not my field) explain *why* it's so difficult that some of the greatest minds (Hardy, Littlewood, Ramanujan, etc) of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.
- nicholas (we don't just sit around doing big sums, you know