An Improved Remainder Estimate in the Weyl Formula for the Planar Disk

摘要

Y. Colin de Verdiere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is no more than O(mu 2/3). In this paper, we study this problem from spectral geometry by using analysis and number theory. More precisely, by combining with the method of exponential sum estimation, we give a sharper remainder term estimate O(mu 2/3-1/495).