Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-Form L -Functions

Source abstract

Abstract In this paper, we prove the following asymptotic formula for the spectral cubic moment of central $L$-values: $$\begin{align*}& \sum_{t_f \leqslant T} \frac{2 L \big( \tfrac 1 2, f \big)^3} {L(1, \textrm{Sym}^2 f)} + \frac{2} {\pi} \int_{ 0}^{T} \frac{\left| \zeta \big(\tfrac 1 2 + it \big) \right|^{6}} { | \zeta (1 + 2 it ) |^2 } \textrm{d} t = T^2 P_3 (\log T) + O (T^{1+{\varepsilon}}), \end{align*}$$where $f$ ranges in an orthonormal basis of (even) Hecke–Maass cusp forms and $P_3$ is a certain polynomial of degree $3$. It improves on the error term $O (T^{8/7+{\varepsilon} })$ in a paper of Ivić and hence confirms his strong conjecture for the cubic moment. This is the 1st time that the (strong) moment conjecture is fully proven in a cubic case. Moreover, we establish the short-interval variant of the above asymptotic formula on intervals of length as short as $T^{{\varepsilon} }$.