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Bykovskii-Type Theorem for the Picard Manifold.
- Source :
-
IMRN: International Mathematics Research Notices . Feb2022, Vol. 2022 Issue 3, p1893-1921. 29p. - Publication Year :
- 2022
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10737928
- Volume :
- 2022
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- IMRN: International Mathematics Research Notices
- Publication Type :
- Academic Journal
- Accession number :
- 154976226
- Full Text :
- https://doi.org/10.1093/imrn/rnaa128