This paper studies the first moment of symmetricsquare L-functions at the critical point in the weight aspect. Asymptotics with the best known error term O(k −1/2 ) were obtained independently by Fomenko in 2005 and by Sun in 2013. We prove that there is an extra main term of size k −1/2 in the asymptotic formula and show that the remainder term decays exponentially in k. The twisted first moment was evaluated asymptotically by Ng Ming Ho with the error bounded by lk −1/2+ǫ . We improve the error bound to l 5/6+ǫ k −1/2+ǫ unconditionally and to l 1/2+ǫ k −1/2 under the Lindelöf hypothesis for quadratic Dirichlet L-functions.

We prove a Weyl-type subconvex bound for cube-free level Hecke characters over totally real number fields. Our proof relies on an explicit inversion to Motohashi's formula. Schwartz functions of various kinds and the invariance of the relevant Motohashi's distributions discovered in a previous paper play central roles.

For Γ a cofinite Kleinian group acting on H 3 , we study the Prime Geodesic Theorem on M = Γ\H 3 , which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on M . Let E Γ (X) be the error in the counting of prime geodesics with length at most log X. For the Picard manifold, Γ = PSL(2, Z[i]), we improve the classical bound of Sarnak, E Γ (X) = O(X 5/3+ ), to E Γ (X) = O(X 13/8+ ). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg L-function attached to Maass-Hecke cusp forms. We also investigate the second moment of E Γ (X) for a general cofinite group Γ, and show that it is bounded by O(X 16/5+ ).

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