The mathematician Euler found a way to add up the numbers from 1 to infinity... and he got the answer -1/12.   Later this kind of calculation was made illegal.  So we don't teach Euler's tricks to students anymore: they're too dangerous.

But the mathematician Riemann figured out how to extract some sense from Euler's calculation!  He invented a function now called the Riemann zeta function.  This function has

zeta(2)  =  1/1²  +  1/2²  +  1/3²  +  1/4² + ...

and

zeta(3)  =  1/1³  +  1/2³  +  1/3³  +  1/4³  +  ...

and so on, but

zeta(-1)  =  -1/12

and the proof of this is a cleaned-up version of Euler's original calculation.  In case you're wondering,

zeta(-2)  =  0

and in fact the zeta function is zero for all negative even numbers.  But it's also zero at other places!  For example, it's zero at about

1/2 + 14.1347 i

and about

1/2 + 21.0220 i

and about

1/2 + 25.0109 i

where that "i" is the square root of minus 1. 

Riemann guessed that all these other places where the zeta function is zero were 1/2 plus some real number times i.  He only checked this for the first three places... but by now people have checked it for the first 10,000,000,000,000 places.   So it seems to be true, but nobody can prove it.

Proving his guess, called the Riemann Hypothesis, is one the most important challenges in math.  If you succeed in proving it, you'll win a million dollar prize!  But be careful: if you disprove it, an angry mob of mathematicians will burn down your house.

It's important because this question is connected to prime numbers... and the deep and mysterious links between geometry, number theory and physics.  If I didn't have more urgent things to do, I could easily enjoy the rest of my life thinking about nothing else.

For example: what's the deal with these numbers

14.1347,  21.0220,  25.0109,  ...

and so on?  I'm just writing them approximately: there's no simple formula for them.  But around 1912, Pólya guessed that there was some interesting physical system described by quantum mechanics that has these numbers as its allowed energy levels.   If true, finding and understanding this system might help us prove the Riemann Hypothesis.

In 1999, Michael Berry and Jon Keating suggested that this physical system could be the "upside-down version of the quantum harmonic oscillator".  The usual version involves a spring that pulls back with a force proportional to how far it's stretched.  But their "upside-down" version has a spring that pushes with a force proportional to how far it's stretched!

This is a very weird physical system, especially if you study it using quantum mechanics.  It really doesn't seem to make sense.  However, all approaches to the Riemann Hypothesis involve mind-boggling ideas that push us beyond the limits of what we can understand so far.  And that's why it's important: it's telling us there's something big going on, that will blow our minds when we figure it out.

If you're deep into quantum physics or number theory, I really recommend their paper:

• Michael Berry and Jon P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Review 41 (1999), 236–266, http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry307.pdf.

For something a bit easier, I recommend this:

• Daniel Schumayer and David A. W. Hutchinson, Physics of the Riemann hypothesis, http://arxiv.org/abs/1101.3116

I gave a talk explaining Euler's crazy calculation, and how it's related to string theory.  You can see it here:

http://math.ucr.edu/home/baez/numbers/index.html#24

#Riemann #number_theory  
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