Abstract:
Consider the family of automorphic $L$-functions associated with primitive cusp forms of level one, ordered by weight $k$. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted $L$-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.
