Bykovskii-type theorem for the Picard manifold

TL;DR

This paper extends Bykovskii's theorem to Gaussian integers, providing improved bounds for the prime geodesic theorem on the Picard manifold, and explores conditional reductions under standard L-function conjectures.

Contribution

It generalizes Bykovskii's result to Gaussian integers and improves bounds for the prime geodesic theorem using spectral and arithmetic methods.

Findings

Improved bounds for the remainder in the prime geodesic theorem.

Conditional reduction of the exponent below 3/2 assuming L-function conjectures.

Connection between the remainder estimate and the Gauss circle problem.

Abstract

We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\epsilon})$ and $O(X^{3/2+\theta+\epsilon})$, where $\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+\epsilon})$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.