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We explore the covariance of error terms coming from Weyl's conjectureregarding the number of Dirichlet eigenvalues up to size $X$. We also considerthis problem in short intervals, i.e. the error term of the number ofeigenvalues in the window $[X, X+S]$ for some $S(X)$. We look at these errorterms for planar domains where the Dirichlet eigenvalues can be explicitlycalculated. In these cases, the error term is closely related to the error termfrom the classical lattice points counting problem of expanding planar domains.We give a formula for the covariance of such error terms, for general planardomains. We also give a formula for the covariance of error terms in shortintervals, for sufficiently large intervals. Going back to the Dirichleteigenvalue problem, we give results regarding the covariance of the error termsin short intervals of 'generic' rectangles. We also explore a specific example,namely we compute the covariance between the error terms of an equilateraltriangle and various rectangles.

Covariance of Error Terms Related to the Dirichlet Eigenvalue Problem