The 4.36-th moment of the Riemann zeta-function

Conditionally on the Riemann Hypothesis we obtain bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta-function for all positive real k < 2.181. This provides for the first time an upper bound of the correct order of magnitude for some k > 2; the case of k = 2 corresponds to a classical result of Ingham. We prove our result by establishing a connection between moments with k > 2 and the so-called "twisted fourth moment". This allows us to appeal to a recent result of Hughes and Young. Furthermore we obtain a point-wise bound for |zeta(1/2 + it)|^{2r} (with 0 < r < 1) that can be regarded as a multiplicative analogue of Selberg's bound for S(T). We also establish asymptotic formulae for moments (k < 2.181) slightly off the half-line.