{"paper":{"title":"The second moment of $S_n(t)$ on the Riemann hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andr\\'es Chirre, Emily Quesada-Herrera","submitted_at":"2020-06-15T16:01:46Z","abstract_excerpt":"Let $S(t) = \\tfrac{1}{\\pi} \\arg \\zeta \\big({1/2} + it \\big)$ be the argument of the Riemann zeta-function at the point $\\tfrac12 + it$. For $n \\geq 1$ and $t>0$ define its antiderivatives as \\begin{equation*} S_n(t) = \\int_0^t S_{n-1}(\\tau) \\hspace{0.08cm} \\rm d\\tau + \\delta_n, \\end{equation*} where $\\delta_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that $$ \\int_{0}^{T}|S_n(t)|^2 \\hspace{0.06cm} \\rm dt = O(T), $$ for $n\\geq 1$. In 1946, Selberg unconditionally established the explicit asymptotic formulas for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.08503","kind":"arxiv","version":2},"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/2006.08503/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"}