On the error term of the fourth moment of the Riemann zeta-function

We examine the size of $E_2(T)$, the error term in the asymptotic formula for ${\int }_0^T\mathrm{\mid }\zeta (1\mathrm{/}2+it){\mathrm{\mid }}^{4}dt$ where $\zeta (s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for $E_2(T)$ and ${\int }_0^T{E}_2(t{)}^{2}dt$. As a consequence, we obtain small logarithmic improvements for $k$th moments where $8\le k\le 12$. In particular, we make a modest improvement on the 12th power moment for $\zeta (s)$.