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$\lambda_k(\alpha)$isthe$k$-th Dirichlet eigenvalue for the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$ with $\alpha \in (0,2]$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$.