The 4.36th moment of the Riemann Zeta-function

Conditionally on the Riemann Hypothesis, we obtain bounds of the correct order of magnitude for the 2kth 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 [11]. 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 [10]. Furthermore we obtain a point-wise bound for |ζ(1/2+ it)|^2r(with 0<r<1) that can be regarded as a multiplicative analog of Selberg's bound for S(T) [18]. We also establish asymptotic formulae for moments (k<2.181) slightly off the half-line.