Riemann's zeta function and the broadband structure of pure harmonics

Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a)=\sum n^{-s}\exp (2\pi ina)$ for $\mathrm{\Re }s>1$, and extended to the complex plane via analytic continuation. Let $s_n = \sigma_n + it_n, , t_n >0 $, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Ces`{a}ro means of the sequence $F_{s_n}(a)$ converge to the first harmonic $\exp (2\pi ia)$ in the sense of periodic distributions. This reveals a natural broadband structure of the pure tone. The proof involves Fujii’s refinement of the classical Landau theorem related to the uniform distribution modulo one of the nontrivial zeros of $\zeta$.