Mean square in the prime geodesic theorem

Abstract

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.

The prime geodesic theorem in square mean

“…This estimate improves on the result of Cherubini-Guerreiro [ChGu,Th. 1.4], where the right hand side was A 5/4+ε , and in fact our analysis is based on theirs.…”

“…The last bound improves on the display before [ChGu,Prop. 4.5] in that we have T 3+ε in place of T 4+ε .…”

“…Specifically, on the right hand side of (8), the square mean of the first j-sum can be estimated via Kuznetsov's formula and the Hardy-Littlewood-Pólya inequality (cf. [ChGu,Lemma 4.2]), while the square mean of the second j-sum can be estimated in terms of the spectral second moment of symmetric square L-functions (cf. [LuSa,(33)]).…”

“…Following the proof of [ChGu,Prop. 4.5], which is based on [LuSa,, we see that (9) can be deduced from the following smoothened variant, itself a strengthening of [ChGu,Lemma 4.4]:…”

“…We abbreviate L j (τ ) := L(1/2 + iτ, sym 2 u j ), and we average over A X 2A. Applying [ChGu,Lemma 4.2] 1 for the contribution of the n-sum on the right hand side, we obtain…”