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Y. Colin de Verdière 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 \(O(\mu ^{2/3-1/495})\).

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