On some conjectures and results for the Riemann zeta-function and Hecke series

Abstract

Some results and conjectures on $Z_2(s)={\int }_1^{\infty }|\zeta (1/2+ix){|}^4{x}^{-s}dx(\Re s>1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|\zeta (1/2+it)$ and the error term in the fourth moment of $|\zeta (1/2+it)$ are discussed.