Title: On Weyl's asymptotics and remainder term for the orthogonal and unitary groups,
Abstract: We examine the asymptotics of the spectral counting function of a compact
Riemannian manifold by V.G.~Avakumovic \cite{Avakumovic} and L.~Hörmander
\cite{Hormander-eigen} and show that for the scale of orthogonal and unitary
groups ${\bf SO}(N)$, ${\bf SU}(N)$, ${\bf U}(N)$ and ${\bf Spin}(N)$ it is not
sharp. While for negative sectional curvature improvements are possible and
known, {\it cf.} e.g., J.J.~Duistermaat $\&$ V.~Guillemin \cite{Duist-Guill},
here, we give sharp and contrasting examples in the positive Ricci curvature
case [non-negative for ${\bf U}(N)$]. Furthermore here the improvements are
sharp and quantitative relating to the dimension and {\it rank} of the group.
We discuss the implications of these results on the closely related problem of
closed geodesics and the length spectrum.