  In this paper, we study the two-point Weyl Law for the Laplace-Beltrami
operator on a smooth, compact Riemannian manifold $M$ with no conjugate points.
That is, we find the asymptotic behavior of the Schwartz kernel,
$E_\lambda(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum
of eigenspaces with eigenvalue smaller than $\lambda^2$ as $\lambda \to\infty$.
In the regime where $x,y$ are restricted to a compact neighborhood of the
diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the
remainder of the asymptotic expansion for $E_\lambda$ and its derivatives of
all orders, which generalizes a result of B\'erard, who treated the on-diagonal
case $E_\lambda(x,x)$. When $x,y$ avoid a compact neighborhood of the diagonal,
we obtain this same improvement in an upper bound for $E_\lambda$. Our results
imply that the rescaled covariance kernel of a monochromatic random wave
locally converges in the $C^\infty$ topology to a universal scaling limit at an
inverse logarithmic rate.
