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

Abstract

We examine the size of $E_2(T)$, the error term in the asymptotic formula for ${\int }_0^T|\zeta (1/2+it){|}^{4}dt$ where $\zeta (s)$ is the Riemann zeta-function. We reduce 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 obtain ${\int }_0^T|\zeta (1/2+it){|}^{12}dt\ll T^2(\log T{)}^{15}$.

arXiv comment14 pages