Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

Abstract

In this paper, we prove the following asymptotic formula for the spectral cubic moment of central $L$-values: $$\sum _{t_f⩽T}\frac{2L(\frac{1}{2},f{)}^3}{L(1,{\mathrm{Sym}}^{2}f)}+\frac{2}{\pi }{\int }_0^T\frac{{\left|\zeta (\frac{1}{2}+it)\right|}^6}{|\zeta (1+2it){|}^2}\mathrm{d}t=T^2{P}_3(\log T)+O(T^{1+\epsilon }),$$ 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 $O(T^{8/7+\epsilon })$ in a paper of Ivić and hence confirms his strong conjecture for the cubic moment. This is the first 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 $T^{\epsilon }$.

20 pages