You need to know: Complex number, absolute value and real part of complex number z, function of complex variable, meromorphic function, analytic continuation, big O notation. Also, see this previous theorem description for the concept of primitive Dirichlet character modulo integer .
Background: For a prime , denote the set of all primitive Dirichlet characters modulo q, and let be the number of such characters. For , and complex number with , let . By analytic continuation, function can be extended to a meromorphic function on the whole complex plane, and it is called a Dirichlet L-function.
The Theorem: On 22nd September 2006, Matthew Young submitted to the Annals of Mathematics a paper in which he proved that for any prime and any , , where are some explicitly computable absolute constants.
Short context: Dirichlet L-functions are generalisations of famous Riemann zeta function (which corresponds to the case ), and estimating their moments, especially at point (which is called the central point), is an important problem in number theory with many applications. The Theorem computes the fourth moment of Dirichlet L-functions at for prime moduli q, with error term decreasing as a power of q. Ealier, similar formula was derived (by Heath-Brown in 1979) only for the Riemann zeta function.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 
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