  We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T}
|\zeta(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all
large T. This is sharp up to the value of the implicit constant.
  Our proof builds on well known work of Soundararajan, who showed, assuming
the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt
\ll_{k,\epsilon} T log^{k^{2}+\epsilon} T for any fixed k \geq 0 and \epsilon >
0. Whereas Soundararajan bounded \log|\zeta(1/2+it)| by a single Dirichlet
polynomial, and investigated how often it attains large values, we bound
\log|\zeta(1/2+it)| by a sum of many Dirichlet polynomials and investigate the
joint behaviour of all of them. We also work directly with moments throughout,
rather than passing through estimates for large values.
