The first and second moments of the Riemann zeta function were established asymptotically by Hardy-Littlewood and Ingham, respectively. It is believed that sixth and higher moments are beyond current techniques. Ng recently proved an asymptotic formula for the sixth moment by assuming a conjecture on additive divisor sums. In this talk, we will discuss about how to compute the eighth moment under a conjecture on additive divisor sums and the Riemann hypothesis. It is a joint work with Nathan Ng and Peng-Jie Wong.