{"paper":{"title":"On the distribution of positive and negative values of Hardy's $Z$-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c, Steven M. Gonek","submitted_at":"2016-04-02T15:40:07Z","abstract_excerpt":"We investigate the distribution of positive and negative values of Hardy's function $$ Z(t) := \\zeta(1/2+it){\\chi(1/2+it)}^{-1/2}, \\quad \\zeta(s) = \\chi(s)\\zeta(1-s). $$ In particular we prove that $$ \\mu\\bigl(I_{+}(T,T)\\bigr) \\;\\gg T\\; \\qquad \\hbox{and}\\qquad \\mu\\bigl(I_{-}(T, T)\\bigr) \\; \\gg \\; T, $$ where $\\mu(\\cdot)$ denotes the Lebesgue measure and \\begin{align*}\n  { I}_+(T,H) &\\;=\\; \\bigl\\{T< t\\le T+H\\,:\\, Z(t)>0\\bigr\\},\n  { I}_-(T,H) &\\;=\\; \\bigl\\{T< t\\le T+H\\,:\\, Z(t)<0\\bigr\\}. \\end{align*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00517","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"}