  In this article we prove the equivalence of certain inequalities for Riesz
means of eigenvalues of the Dirichlet Laplacian with a classical inequality of
Kac. Connections are made via integral transforms including those of Laplace,
Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We
also prove new universal eigenvalue inequalities and monotonicity principles
for Dirichlet Laplacians as well as certain Schr\"odinger operators. At the
heart of these inequalities are calculations of commutators of operators, sum
rules, and monotonic properties of Riesz means. In the course of developing
these inequalities we prove new bounds for the partition function and the
spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about
additional bounds.
