{"paper":{"title":"Moments of Hardy's function over short intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c","submitted_at":"2016-12-06T08:18:34Z","abstract_excerpt":"Let as usual $Z(t) = \\zeta(1/2+it)\\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\\zeta(s) = \\chi(s)\\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\\log T)^{k^2} \\ll_{k,\\alpha} \\sum_{T<\\gamma\\le T+H}\\max_{\\gamma\\le \\tau_\\gamma\\le \\gamma^+} |\\zeta(1/2 + i\\tau_\\gamma)|^{2k} \\ll_{k,\\alpha} H(\\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $\\gamma, \\gamma^+$ denote ordinates of consecutive complex zeros of $\\zeta(s)$ and $T^\\alpha \\le H \\le T$, where $\\alpha$ is a fixed constant such t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01698","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"}