{"paper":{"title":"Bounding $S_n(t)$ on the Riemann hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Andr\\'es Chirre, Emanuel Carneiro","submitted_at":"2017-02-14T07:35:02Z","abstract_excerpt":"Let $S(t) = \\tfrac{1}{\\pi} \\arg \\zeta (\\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\\tfrac12 + it$. For $n \\geq 1$ and $t>0$ define its iterates \\begin{equation*} S_n(t) = \\int_0^t S_{n-1}(\\tau) \\,{\\rm d}\\tau\\, + \\delta_n\\,, \\end{equation*} where $\\delta_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that $S_n(t) = O(\\log t/ (\\log \\log t)^{n+1})$. The order of magnitude of this estimate was never improved up to this date. The best bounds for $S(t)$ and $S_1(t)$ are currently due "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.04099","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1702.04099/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}