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

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 \(O(\mu ^{2/3-1/495})\).