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

TL;DR

Under the assumption of the Riemann Hypothesis, the paper shows that the Cesàro means of the periodized zeta function at the nontrivial zeros converge to a fundamental harmonic, revealing a broadband structure of pure tones.

Contribution

The paper demonstrates a novel connection between the zeros of the Riemann zeta function and harmonic structures, using Cesàro means and distributional convergence.

Findings

Cesàro means of $F_{s_n}(a)$ converge to $ ext{exp}(2 extpi i a)$

Reveals broadband structure of pure harmonics

Assumes Riemann Hypothesis for convergence

Abstract

Let $a\in (0,1)$ and let $F_s(a)$ be the periodized zeta function that is defined as $F_s(a) = \sum n^{-s} \exp (2\pi i na)$ for $\Re s >1$, 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 $\exp (2\pi i a)$ 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 $\zeta$.