This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear ...
and even PDE (on periodic domains); it asserts that for any large number n, only a “logarithmically small” set of numbers less than n will actually divide n exactly, even in the worst-case scenario when n issmooth. (The average value of d(n) is much smaller, being about ...