Improved Berezin-Li-Yau inequality and Kröger inequality and consequences

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kröger inequality, in ${\mathbb{R}}^n$, $n\ge 2$. The improvement on Kröger’s inequality resolves an open question raised by Weidl from 2006. The improvements allow us to show that, for any open bounded domains, there are infinite many Dirichlet eigenvalues satisfying Pólya’s conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying Pólya’s conjecture if $n\ge 5$ and the Neumann spectrum is discrete.