Hi,
I know a little bit on the Riemann hypothesis which is a long standing conjecture of Riemann. Basically the conjecture is about the non-trivial (complex) zeroes of the Riemann Zeta function,
Re(s)>1 , which Riemann claims to lie in the critical line that is on the line for which the real part equals 1/2. The conjecture is not yet proved nor it is disproved. But the assertion of the hypothesis is checked for a large number of complex numbers which ensures its validity. However the conjecture if it is solved attempts to answer a large number of mysteries including the distribution of primes. Purely a analytical number theoretic problem, several attempts have been made to solve this conjecture from the point of view of non-commutative geometry. The problem remains a challenge to the entire mathematical community and in particular to the number theorist.
P.S.
I haven't discussed any kind of technicalities pertaining to the zeta function, which has in itself much richer theory, one example is being its applicability in proving the infinitude of primes using Euler product of Dirichlet series, of which the above series is a type. Another important aspect of the zeta function is that the inverse of zeta function evaluated on natural number say n gives the probability that set of n integers selected at random are pairwise coprime.