{"paper":{"title":"Large 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":"Kamalakshya Mahatab","submitted_at":"2018-07-23T12:12:27Z","abstract_excerpt":"Let $Z(t):=\\zeta\\left(\\frac{1}{2}+it\\right)\\chi^{-\\frac{1}{2}}\\left(\\frac{1}{2}+it\\right)$ be Hardy's function, where the Riemann zeta function $\\zeta(s)$ has the functional equation $\\zeta(s)=\\chi(s)\\zeta(1-s)$. We prove that for any $\\epsilon>0$, \\begin{align*} &\\quad\\max_{T^{3/4}\\leq t\\leq T} Z(t) \\gg \\exp\\left(\\left(\\frac{1}{2}-\\epsilon\\right)\\sqrt{\\frac{\\log T\\log\\log\\log T}{\\log\\log T}}\\right)\\\\ \\text{ and }& \\max_{T^{3/4}\\leq t\\leq T}- Z(t) \\gg \\exp\\left(\\left(\\frac{1}{2}-\\epsilon\\right)\\sqrt{\\frac{\\log T\\log\\log\\log T}{\\log\\log T}}\\right). \\end{align*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08554","kind":"arxiv","version":3},"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"}