The generalised P\'{o}lya conjecture for the Dirichlet eigenvalues

In this paper, we prove the Generalized P\'{o}lya conjecture for the Dirichlet eigenvalues. In other words, we show that $\lambda_k(\alpha) \ge \frac{(2\pi)^{\alpha} k^{\alpha/n}}{\big(\omega_n \cdot {vol}(\Omega)\big)^{\alpha/n}}, \quad\, {for}\;\; k=1,2,3,....$ where $\lambda_k(\alpha)$ is the $k$-th Dirichlet eigenvalue for the fractional Laplacian $(-\Delta)^{\alpha/2}$ with $\alpha\in (0,2]$ in a bounded domain $\Omega\subset {\Bbb R}^n$.