Riemann's zeta function and the broadband structure of pure harmonics
| Autor: | Sowa, Artur |
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| Rok vydání: | 2016 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\zeta$. Comment: 9pages, 1 figure |
| Databáze: | arXiv |
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