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 $|\zeta (\frac {1}{2}+{\mathrm i}t)|^{2r}$ (with 0