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.

中文翻译：

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Picard 流形的 Bykovskii 型定理

摘要

我们将 Bykovskii 的结果推广到高斯整数，并在 Picard 流形上证明了短间隔内素数测地线定理的渐近公式。以前的工作表明，余数分别以 $O(X^{13/8+\epsilon })$ 和 $O(X^{3/2+\theta +\epsilon })$ 为界，其中 $\theta$是在 $\mathbb{Q}(i)$ 上的二次 Dirichlet $L$-函数的次凸指数。通过将算术方法与谱指数和和平滑显式公式的估计相结合，我们获得了这两个指数的改进。此外，通过假设 $L$-函数的两个标准猜想，我们证明可以将指数降低到障碍 $3/2$ 以下并有条件地得到 $O(X^{34/23+\epsilon })$。