Just ask Cleo

My real name is Cleo, I'm female. I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that. I like math and do my best to be useful at this site, although I realize my answers might be not useful for everyone.

There's a website called Math StackExchange where people ask and answer questions. When hard integrals come up, Cleo often does them - with no explanation! She has a lot of fans now.

The integral here is a good example. When you replace ln³(1+x) by ln²(1+x) or just ln(1+x), the answers were already known. The answers involve the third Riemann zeta value:

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

They also involve the fourth polylogarithm function:

Li₄(x) = x + x²/2⁴ + x³/3⁴ + ...

Cleo found that the integral including ln³(1+x) can be done in a similar way - but it's much more complicated. She didn't explain her answer... but someone checked it with a computer and showed it was right to 1000 decimal places. Then someone gave a proof.

The number

ζ(3) = 1.202056903159594285399738161511449990764986292...

is famous because it was proved to be irrational only after a lot of struggle. Apéry found a proof in 1979. Even now, nobody is sure that the similar numbers ζ(5), ζ(7), ζ(9)... are irrational, though most of us believe it. The numbers ζ(2), ζ(4), ζ(6)... are much easier to handle. Euler figured out formulas for them involving powers of pi, and they're all irrational.

But here's a wonderful bit of progress: in 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational. Sometimes we can only snatch tiny crumbs of knowledge from the math gods, but they're still precious.

For Cleo's posts, go here:

https://math.stackexchange.com/users/97378/cleo

For more on ζ(3), go here:

https://en.wikipedia.org/wiki/Apery's_constant

This number shows up in some physics problems, like computing the magnetic field produced by an electron! And that's just the tip of an iceberg: there are deep connections between Feynman diagrams, the numbers ζ(n), and mysterious mathematical entities glimpsed by Grothendieck, called 'motives'. Very roughly, a motive is what's left of a space if all you care about are the results of integrals over surfaces in this space.

The world record for computing digits of ζ(3) is currently held by Dipanjan Nag: in 2015 he computed 400,000,000,000 digits. But here's something cooler: David Broadhurst, who works on Feynman diagrams and numbers like ζ(n), has shown that there's a linear-time algorithm to compute the nth binary digit of ζ(3):

• David Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), available at https://arxiv.org/abs/math/9803067

He exploits how Riemann zeta values ζ(n) are connected to polylogarithms... it's easy to see that

Liₙ(1) = ζ(n)

but at a deeper level this connection involves motives. For more on polylogarithms, go here:

https://en.wikipedia.org/wiki/Polylogarithm

Thanks to +David Roberts for pointing out Cleo's posts on Math StackExchange!

#bigness