Aperiodic diffraction grating based on the relationship between primes and zeros of the Riemann zeta function

The Riemann hypothesis is one of the most famous unsolved problems of modern science. One approach to proving the Riemann hypothesis is based on the assumption that the nontrivial zeros of the Riemann zeta function represent the spectrum of some self-adjoint operator. In this paper, we show that the duality with respect to the Fourier transform between the distribution of nontrivial zeros of the Riemann zeta function along the critical line, on the one hand, and the distribution of logarithms of prime numbers and powers of primes, on the other hand, can be used as a theoretical basis for creating new diffractive optical elements. In particular, we manufactured an aperiodic diffraction grating, the slits of which are ordered in accordance with the distribution of nontrivial zeros of the Riemann zeta function. Atomic force microscopy lithography was used for nanopatterning. The resulting diffraction pattern shows the presence of discrete diffraction maxima at the logarithms of primes and prime powers, which is the direct experimental visualization of the Hilbert–Polya conjecture.