You need to know: Complex number, absolute value and real part of complex number z, function of complex variable, meromorphic function, analytic continuation, integration.
Background: For a 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 Riemann zeta function. The Riemann hypothesis states that if then either for some integer or . For , function is called a 2k-th moment of .
The Theorem: On 4th December 2006, Kannan Soundararajan submitted to the Annals of Mathematics a paper in which he, assuming the Riemann hypothesis, proved that for every and there is a constant such that inequality holds for all .
Short context: Riemann zeta function is one of the most studied functions in mathematics, and the Riemann hypothesis (RH) is one of the most important open problems. One line of study related to is understanding the growth of moments . It was known that (assuming RH) for some and the Theorem provides an almost (up to ) matching upper bound. In a later work, Harper (again assuming RH) improved the upper bound to , matching with the lower bound up to the constant factor.
Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.12 of this book for an accessible description of the Theorem.
Theorems of the 21st century 
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