The study of moments may be traced back to the early twentieth century when the second and fourth moments of $\zeta(\frac{1}{2}+it)$ were established asymptotically by Hardy-Littlewood and Ingham, respectively. There has been folklore that sixth and higher moments are beyond current techniques. Nonetheless, a few years ago, Ng proved an asymptotic formula for the sixth moment under a conjecture of ternary additive divisor sums.

In this talk, I will explain how the Riemann hypothesis (RH) and a conjecture of quaternary additive divisor sums imply the conjectural asymptotic for the eighth moment of the Riemann zeta function. This is joint work with Nathan Ng and Quanli Shen, and the key new idea is to use sharp bounds for shifted moments of the zeta function on the critical line. If time allows, we shall also discuss how to adapt Harper's celebrated work on sharp bounds for moments of the zeta to prove the required sharp bounds for shifted moments under RH.