{"paper":{"title":"Riemann's zeta function and the broadband structure of pure harmonics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Artur Sowa","submitted_at":"2016-03-10T20:55:09Z","abstract_excerpt":"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 br"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03667","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}