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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

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

It's important because this question is connected to prime numbers... and the deep and mysterious links between

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

This is a very weird physical system, especially if you study it using quantum mechanics. It really doesn't seem to make sense. However,

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,

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

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 haszeta(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

View 66 previous comments

- I don't think so!Jun 9, 2013
- Does it follow that 2+4+6+8+.. = -2/12? so 1+3+5+..=1/12, so it would follow that 2+5+8+11+14+17=0... but 3+6+9+12+..=-3/12, so 1+1+1+1+..=-3/12. I guess I'm breaking some rules.Jun 12, 2013
- +Tom Lowe - You can easily get into contradictions working with divergent series! But you'd enjoy seeing the manipulations Euler did to get his result in the first place. They're near the start of this file:

http://math.ucr.edu/home/baez/numbers/24.pdf

He had to have 'exquisite taste' to get useful results and no trash, given that he didn't know the modern theory for dealing with these series.Jun 12, 2013 - It would be interesting to see more posts on pulling finite numbers out of these divergent series. And pulling physical sense/examples out of that would be even more interesting.Jun 12, 2013
- I may write more about this someday, but if I don't, you can learn a lot about this by googling these buzzwords: "zeta function regularization", "Abel summation", and "Borel summation". These are methods, widely and successfully used in physics, for extracting finite answers out of divergent sums and integrals.Jun 15, 2013
- Thanks, +John Baez I will take a look at that.

On a tangential angle (with tongue in cheek), it strikes me that this is like Dr Who. He can travel in All Time and All Space, which pretty much just means he almost never leaves Britain.Jun 15, 2013