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

Abstract

Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a) = \sum n^{-s} \exp (2πi na)$ for $\Re s >1$, and extended to the complex plane via analytic continuation. Let $s_n = σ_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 Cesàro means of the sequence $F_{s_n} (a)$ converge to the first harmonic $\exp (2πi a)$ 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 $ζ$.