{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5OW7GE5N7CKNGZ7VFNNJSBFNG4","short_pith_number":"pith:5OW7GE5N","canonical_record":{"source":{"id":"1610.03465","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:58:01Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"cf7c3b3b6b395467f6908e7a2095c569fb9be21dc2943bbe4d690ac515e13dfc","abstract_canon_sha256":"ae6f2dc91119efa58ba076a6815f4c98f3bd5ca1712aea77157b0d84c8266387"},"schema_version":"1.0"},"canonical_sha256":"ebadf313adf894d367f52b5a9904ad373235be3333e3a0b6fbdc634ec7bb775f","source":{"kind":"arxiv","id":"1610.03465","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.03465","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"arxiv_version","alias_value":"1610.03465v3","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03465","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"pith_short_12","alias_value":"5OW7GE5N7CKN","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5OW7GE5N7CKNGZ7V","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5OW7GE5N","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5OW7GE5N7CKNGZ7VFNNJSBFNG4","target":"record","payload":{"canonical_record":{"source":{"id":"1610.03465","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:58:01Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"cf7c3b3b6b395467f6908e7a2095c569fb9be21dc2943bbe4d690ac515e13dfc","abstract_canon_sha256":"ae6f2dc91119efa58ba076a6815f4c98f3bd5ca1712aea77157b0d84c8266387"},"schema_version":"1.0"},"canonical_sha256":"ebadf313adf894d367f52b5a9904ad373235be3333e3a0b6fbdc634ec7bb775f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:26.681053Z","signature_b64":"ZJu1MEBKm+IDymACAb+53rNkvksswHlF5QACgOomUoOlmyYN5X3TEKyUrnjOfZiICobXmBtrFSQ1RvxhM1G+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebadf313adf894d367f52b5a9904ad373235be3333e3a0b6fbdc634ec7bb775f","last_reissued_at":"2026-05-17T23:59:26.680442Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:26.680442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.03465","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZIOn02QTg6f2SoQR7LLRwtEGTbZZO2aXLZnvobfLUWA8xU83A1XQ/lSbhaLncH/Nzi0bf9iaQsiX7vsK4/v+DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:18:01.372843Z"},"content_sha256":"1de7b1b6c0cd78dc3f6711ab34a18d7e5329cc6a079afd52c50dca027515a93a","schema_version":"1.0","event_id":"sha256:1de7b1b6c0cd78dc3f6711ab34a18d7e5329cc6a079afd52c50dca027515a93a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5OW7GE5N7CKNGZ7VFNNJSBFNG4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Moments of L-functions and Liouville-Green method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Dmitry Frolenkov, Olga Balkanova","submitted_at":"2016-10-11T18:58:01Z","abstract_excerpt":"We show that the percentage of primitive forms of level one and weight $4k\\rightarrow \\infty$ for which the associated $L$-function at the central point is no less than $(\\log{k})^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03465","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:59:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yhDtkp5nJDsSRfTTyJmnOtXz3hWID4skVSKxF5lni42cnhMDEEmMd9CKfnNf/N/EdmrsXLDBwSHg7JKgtVKjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:18:01.373187Z"},"content_sha256":"60e190f8d23c3482a64bb8eee28db3890f394859a1739266fd0a36b4510aa714","schema_version":"1.0","event_id":"sha256:60e190f8d23c3482a64bb8eee28db3890f394859a1739266fd0a36b4510aa714"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/bundle.json","state_url":"https://pith.science/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T23:18:01Z","links":{"resolver":"https://pith.science/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4","bundle":"https://pith.science/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/bundle.json","state":"https://pith.science/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5OW7GE5N7CKNGZ7VFNNJSBFNG4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5OW7GE5N7CKNGZ7VFNNJSBFNG4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ae6f2dc91119efa58ba076a6815f4c98f3bd5ca1712aea77157b0d84c8266387","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:58:01Z","title_canon_sha256":"cf7c3b3b6b395467f6908e7a2095c569fb9be21dc2943bbe4d690ac515e13dfc"},"schema_version":"1.0","source":{"id":"1610.03465","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.03465","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"arxiv_version","alias_value":"1610.03465v3","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03465","created_at":"2026-05-17T23:59:26Z"},{"alias_kind":"pith_short_12","alias_value":"5OW7GE5N7CKN","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5OW7GE5N7CKNGZ7V","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5OW7GE5N","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:60e190f8d23c3482a64bb8eee28db3890f394859a1739266fd0a36b4510aa714","target":"graph","created_at":"2026-05-17T23:59:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that the percentage of primitive forms of level one and weight $4k\\rightarrow \\infty$ for which the associated $L$-function at the central point is no less than $(\\log{k})^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method.","authors_text":"Dmitry Frolenkov, Olga Balkanova","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:58:01Z","title":"Moments of L-functions and Liouville-Green method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03465","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1de7b1b6c0cd78dc3f6711ab34a18d7e5329cc6a079afd52c50dca027515a93a","target":"record","created_at":"2026-05-17T23:59:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ae6f2dc91119efa58ba076a6815f4c98f3bd5ca1712aea77157b0d84c8266387","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-11T18:58:01Z","title_canon_sha256":"cf7c3b3b6b395467f6908e7a2095c569fb9be21dc2943bbe4d690ac515e13dfc"},"schema_version":"1.0","source":{"id":"1610.03465","kind":"arxiv","version":3}},"canonical_sha256":"ebadf313adf894d367f52b5a9904ad373235be3333e3a0b6fbdc634ec7bb775f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebadf313adf894d367f52b5a9904ad373235be3333e3a0b6fbdc634ec7bb775f","first_computed_at":"2026-05-17T23:59:26.680442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:26.680442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZJu1MEBKm+IDymACAb+53rNkvksswHlF5QACgOomUoOlmyYN5X3TEKyUrnjOfZiICobXmBtrFSQ1RvxhM1G+Aw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:26.681053Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.03465","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1de7b1b6c0cd78dc3f6711ab34a18d7e5329cc6a079afd52c50dca027515a93a","sha256:60e190f8d23c3482a64bb8eee28db3890f394859a1739266fd0a36b4510aa714"],"state_sha256":"9caaa6b7668a588e16d8277b6cd27e0f5681ca3dbcd60bb9591d8116c0f200d5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ID4tlMtzbl1N7UjtFC6YYUSaP9KBckLFLia9xv5wP/BV7IzkSpHnbC82AgkRPo8i/JxESGYl2f9MwM61kM7oBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T23:18:01.375514Z","bundle_sha256":"95cca239662e5e47c8a71fd4ec9c8db31ed1a38c4642594e2b407aded1bedccb"}}