{"paper":{"title":"On the distribution of values of the argument of the Riemann zeta-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c, Maxim Korolev","submitted_at":"2018-08-31T14:25:48Z","abstract_excerpt":"Let $S(t) \\;:=\\; \\frac{\\displaystyle 1}{\\displaystyle \\pi}\\arg \\zeta(\\frac{1}{2} + it)$. We prove that, for $T^{\\,27/82+\\varepsilon} \\le H \\le T$, we have $$ {\\rm mes}\\Bigl\\{t\\in [T, T+H]\\;:\\; S(t)>0\\Bigr\\} = \\frac{H}{2} + O\\left(\\frac{H\\log_3T}{\\varepsilon\\sqrt{\\log_2T}}\\right), $$ where the $O$-constant is absolute. A similar formula holds for the measure of the set with $S(t)<0$, where $\\log_kT = \\log(\\log_{k-1}T)$. This result is derived from an asymptotic formula for the distribution of values of $S(t)$, which is uniform in the relevant parameters, and this is of crucial importance. This "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10768","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"}