We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_{ \sigma}(z) := \sum_k{ (z -\lambda_k)_+^{ \sigma}}. Here ${ \lambda_k}_{ k=1}^{ \infty}$ are the ordered eigenvalues of the Laplacian on a bounded domain $\Omega \subset \R^d$, and $x_+ := \max(0, x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on $\lambda_k$, on averages such as $\bar{ \lambda_k} := { \frac 1 k}\sum_{ \ell \le k}\lambda_\ell$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k \ge j \frac{ 1+\frac d 2}{ 1+\frac d 4}$, { \bar{ \lambda_{ k}}}/{ \bar{ \lambda_{ j}}} \le 2 (\frac{ 1+\frac d 4}{ 1+\frac d 2})^{ 1+\frac 2 d}({ \frac k j})^{ \frac 2 d}.
Categories: math.SP math-ph math.MP