The Nyman–Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in space of square integrable functions on (0,?), involving dilations fractional part function by factors ? k ?(0,1), k?1. Randomizing generates new structures and criteria. One them a sufficient condition for RH that splits into

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagn...