  We show that if $Y$ is a compact Riemannian manifold with improved $L^q$
eigenfunction estimates then, at least for large enough exponents, one always
obtains improved $L^q$ bounds on the product manifold $X\times Y$ if $X$ is
another compact manifold. Similarly, improved Weyl remainder term bounds on the
spectral counting function of $Y$ lead to corresponding improvements on
$X\times Y$. The latter results partly generalize recent ones of Iosevich and
Wyman [14] involving products of spheres. Also, if $Y$ is a product of five or
more spheres, we are able to obtain optimal $L^q(Y)$ and $L^q(X\times Y)$
eigenfunction and spectral cluster estimates for large $q$, which partly
addresses a conjecture from [14] and is related to (and is partly based on)
classical bounds for the number of integer lattice point on $\lambda \cdot
S^{n-1}$ for $n\ge5$.
