Improved Berezin-Li-Yau inequality and Kr\"oger inequality and consequences

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger'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 P\'olya's conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying P\'olya's conjecture if $n\ge 5$ and the Neumann spectrum is discrete.