The Moments of the Riemann Zeta-Function Part I: The fourth moment off the critical line

Abstract

In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|\zeta(\sigma+it)|^4g(t) dt \qquad(\tfrac{1}{2} < \sigma < 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function \begin{align*} E_2(T,\sigma) &: =\int_0^T|\zeta(\sigma+it)|^4 dt - {\zeta^4(2\sigma)\over\zeta(4\sigma)}T -{\frac{T}{3-4\sigma}}{\left(\frac{T}{2\pi} \right)}^{2-4\sigma}{\zeta^4(2-2\sigma)\over\zeta(4-4\sigma)}\cr & \quad- T^{2-2\sigma}(a_0(\sigma) + a_1(\sigma)\log T + a_2(\sigma)\log^2T), \end{align*} the error term in the asymptotic formula for the fourth moment of $|\zeta(\sigma+it)|$.