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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*}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.00517","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-02T15:40:07Z","cross_cats_sorted":[],"title_canon_sha256":"5a16863fdafbc4017d36cf3cd9297f6790ac2adba1e92dec6b66d93cfe8d570d","abstract_canon_sha256":"a70e54349a6e3f78c6b0460b0788c8877d0186c5806552f79c0a7e8cec0fe470"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:49.373447Z","signature_b64":"FhLI/7LbIs6DfF95AhjjQ3WiRK0wBYtpsbGQ9FvA2OpsaOxopQTJOM3XaV+ehGb7Hu0umGn7P3DCppQFBE+GBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cce941494ac2d227277315e387084e73750d9c6194554637ec16d040bf19854a","last_reissued_at":"2026-05-18T01:17:49.372730Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:49.372730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.00517","created_at":"2026-05-18T01:17:49.372835+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.00517v1","created_at":"2026-05-18T01:17:49.372835+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00517","created_at":"2026-05-18T01:17:49.372835+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZTUUCSKKYLJC","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZTUUCSKKYLJCOJ3T","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZTUUCSKK","created_at":"2026-05-18T12:30:55.937587+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON","json":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON.json","graph_json":"https://pith.science/api/pith-number/ZTUUCSKKYLJCOJ3TCXRYOCCOON/graph.json","events_json":"https://pith.science/api/pith-number/ZTUUCSKKYLJCOJ3TCXRYOCCOON/events.json","paper":"https://pith.science/paper/ZTUUCSKK"},"agent_actions":{"view_html":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON","download_json":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON.json","view_paper":"https://pith.science/paper/ZTUUCSKK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.00517&json=true","fetch_graph":"https://pith.science/api/pith-number/ZTUUCSKKYLJCOJ3TCXRYOCCOON/graph.json","fetch_events":"https://pith.science/api/pith-number/ZTUUCSKKYLJCOJ3TCXRYOCCOON/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON/action/storage_attestation","attest_author":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON/action/author_attestation","sign_citation":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON/action/citation_signature","submit_replication":"https://pith.science/pith/ZTUUCSKKYLJCOJ3TCXRYOCCOON/action/replication_record"}},"created_at":"2026-05-18T01:17:49.372835+00:00","updated_at":"2026-05-18T01:17:49.372835+00:00"}