{"paper":{"title":"Transcendental sums related to the zeros of zeta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"M. Ram Murty, Purusottam Rath, Sanoli Gun","submitted_at":"2018-07-30T07:37:53Z","abstract_excerpt":"While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \\sum_{\\rho } R(\\rho) x^{\\rho}, $$ where the sum is over the non-trivial zeros $\\rho$ of $\\zeta(s)$, $R(x) \\in \\overline{\\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \\sum_{\\rho } \\frac{x^{\\rho}}{\\rho} $$ has infinitely many zeros in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11201","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"}