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