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Unsolved Mathematical Problems

_AVG_
OK, I'm starting this topic for people to post any unsolved mathematical problem which they think is very important. I'll start with the Riemann Zeta function (if you solve this, \$1,000,000 is the prize money!).

zeta(s) = sum(n^(-s)) from n = 1 to infinity
s can be any complex number except s = 1

The Riemann Hypothesis states that the only non-trivial zeroes of this function have a real part equal to 1/2.
Any negative even integer is a trivial zero. Any other zero is nontrivial.
Nobody has yet been able to prove the hypothesis true or false.

What unsolved problem do you consider very important?
kelseymh
http://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics

All of them are important. None of them are going to be solved here (unless the solver posts a link to their peer-reviewed publication of the proof).

http://en.wikipedia.org/wiki/Hilbert%27s_problems

In particular, Hilbert's sixth, eighth, twelfth and sixteenth problems.
Afaceinthematrix
There are also the Millennium Prize Problems:
http://en.wikipedia.org/wiki/Millennium_Prize_Problems

Given that I have a rather large interest in partial differential equations and that those are what I mostly study, I would really like to see the Navier-Stokes existence and smoothness problem solved.

Many PDEs have a weak solution but we do not know if they have a strong solution.