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

TL;DR

This paper investigates the error term in the fourth moment of the Riemann zeta-function, improving bounds and logarithmic factors, which leads to refined estimates for higher moments up to the 12th power.

Contribution

The paper provides improved bounds on the error term of the fourth moment of the zeta-function and small logarithmic enhancements for moments between 8th and 12th powers.

Findings

Enhanced bounds for the error term $E_2(T)$ with better powers of $ ext{log} T$.

Refined estimates for the 8th to 12th moments of $ ext{zeta}(s)$.

Modest improvement on the 12th power moment of the 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 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 $8\leq k\leq 12$. In particular, we make a modest improvement on the 12th power moment for $\zeta(s)$.