Improvement of P\'{o}lya's conjecture for balls and cylinders

P\'{o}lya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on P\'{o}lya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of P\'{o}lya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann P\'{o}lya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of P\'{o}lya's conjecture for disks and balls; we obtain improvement of P\'{o}lya's conjecture for cylinders and confirm the Neumann P\'{o}lya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.