{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7G3JLGZ7B3IHRC2HQGBZLDWJBV","short_pith_number":"pith:7G3JLGZ7","canonical_record":{"source":{"id":"1610.07652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-24T21:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"2e841e710b23719a9b4d1dd48fd3e7cfc9be70d8426e3cd7c5d155c4f811684f","abstract_canon_sha256":"1ccdb29280548754f4861dbbd7610c694f3857a1006e69413fea500359a4d910"},"schema_version":"1.0"},"canonical_sha256":"f9b6959b3f0ed0788b478183958ec90d56879dd364868cd196fb6c21c4f71f63","source":{"kind":"arxiv","id":"1610.07652","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07652","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07652v2","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07652","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"pith_short_12","alias_value":"7G3JLGZ7B3IH","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7G3JLGZ7B3IHRC2H","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7G3JLGZ7","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7G3JLGZ7B3IHRC2HQGBZLDWJBV","target":"record","payload":{"canonical_record":{"source":{"id":"1610.07652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-24T21:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"2e841e710b23719a9b4d1dd48fd3e7cfc9be70d8426e3cd7c5d155c4f811684f","abstract_canon_sha256":"1ccdb29280548754f4861dbbd7610c694f3857a1006e69413fea500359a4d910"},"schema_version":"1.0"},"canonical_sha256":"f9b6959b3f0ed0788b478183958ec90d56879dd364868cd196fb6c21c4f71f63","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:08.042082Z","signature_b64":"9M7kRobsaUjgUtiikl4inmSm5KpwHszTlo+1AEtpj2SsRe/U/6uH1La9fS18BzowC+WSNSYWHriKb1HvUxUUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9b6959b3f0ed0788b478183958ec90d56879dd364868cd196fb6c21c4f71f63","last_reissued_at":"2026-05-18T01:01:08.041481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:08.041481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.07652","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qlFZdE7z0imgSNcYMxcQsOQGfQFZydx4Std1i0uPjuNshue9EcN6ZF6oG2tJTXVPAM/grLcss8PNOLCPjnsdDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T06:02:03.172746Z"},"content_sha256":"3706aae560fb768ebae0912729c06c4e04fc3a70bb8869d72e23400a4fd3d344","schema_version":"1.0","event_id":"sha256:3706aae560fb768ebae0912729c06c4e04fc3a70bb8869d72e23400a4fd3d344"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7G3JLGZ7B3IHRC2HQGBZLDWJBV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The first moment of central values of symmetric square $L$-functions in the weight aspect","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shenhui Liu","submitted_at":"2016-10-24T21:05:08Z","abstract_excerpt":"In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\\frac{1}{2},{\\rm{sym}}^2f)$ for $f\\in\\mathcal{H}_k$ as even $k\\rightarrow\\infty$, where $\\mathcal{H}_k$ is an orthogonal basis of weight-$k$ Hecke eigencuspforms for $SL(2,\\mathbb{Z})$. The approach taken in this note allows us to extract two secondary main terms from the error term $O(k^{-\\frac{1}{2}})$ in previous studies. More interestingly, our resul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07652","kind":"arxiv","version":2},"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":"2026-05-18T01:01:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jwhJKWxBxeOVkPY0oXQk9/3dvfsXhzlk5AHyalgM2F62Y2RL2vcBpJ0vKhsUNdrZmU2hIHvhRN9ck6eJG8buCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T06:02:03.173115Z"},"content_sha256":"15a713c34bb006dcff55dc100598c8e307b72fa60412748f90cde56bb5969ebc","schema_version":"1.0","event_id":"sha256:15a713c34bb006dcff55dc100598c8e307b72fa60412748f90cde56bb5969ebc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/bundle.json","state_url":"https://pith.science/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/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":"2026-06-07T06:02:03Z","links":{"resolver":"https://pith.science/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV","bundle":"https://pith.science/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/bundle.json","state":"https://pith.science/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7G3JLGZ7B3IHRC2HQGBZLDWJBV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7G3JLGZ7B3IHRC2HQGBZLDWJBV","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":"1ccdb29280548754f4861dbbd7610c694f3857a1006e69413fea500359a4d910","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-24T21:05:08Z","title_canon_sha256":"2e841e710b23719a9b4d1dd48fd3e7cfc9be70d8426e3cd7c5d155c4f811684f"},"schema_version":"1.0","source":{"id":"1610.07652","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07652","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07652v2","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07652","created_at":"2026-05-18T01:01:08Z"},{"alias_kind":"pith_short_12","alias_value":"7G3JLGZ7B3IH","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7G3JLGZ7B3IHRC2H","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7G3JLGZ7","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:15a713c34bb006dcff55dc100598c8e307b72fa60412748f90cde56bb5969ebc","target":"graph","created_at":"2026-05-18T01:01:08Z","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":"In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\\frac{1}{2},{\\rm{sym}}^2f)$ for $f\\in\\mathcal{H}_k$ as even $k\\rightarrow\\infty$, where $\\mathcal{H}_k$ is an orthogonal basis of weight-$k$ Hecke eigencuspforms for $SL(2,\\mathbb{Z})$. The approach taken in this note allows us to extract two secondary main terms from the error term $O(k^{-\\frac{1}{2}})$ in previous studies. More interestingly, our resul","authors_text":"Shenhui Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-24T21:05:08Z","title":"The first moment of central values of symmetric square $L$-functions in the weight aspect"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07652","kind":"arxiv","version":2},"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":"sha256:3706aae560fb768ebae0912729c06c4e04fc3a70bb8869d72e23400a4fd3d344","target":"record","created_at":"2026-05-18T01:01:08Z","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":"1ccdb29280548754f4861dbbd7610c694f3857a1006e69413fea500359a4d910","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-24T21:05:08Z","title_canon_sha256":"2e841e710b23719a9b4d1dd48fd3e7cfc9be70d8426e3cd7c5d155c4f811684f"},"schema_version":"1.0","source":{"id":"1610.07652","kind":"arxiv","version":2}},"canonical_sha256":"f9b6959b3f0ed0788b478183958ec90d56879dd364868cd196fb6c21c4f71f63","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9b6959b3f0ed0788b478183958ec90d56879dd364868cd196fb6c21c4f71f63","first_computed_at":"2026-05-18T01:01:08.041481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:08.041481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9M7kRobsaUjgUtiikl4inmSm5KpwHszTlo+1AEtpj2SsRe/U/6uH1La9fS18BzowC+WSNSYWHriKb1HvUxUUAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:08.042082Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.07652","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3706aae560fb768ebae0912729c06c4e04fc3a70bb8869d72e23400a4fd3d344","sha256:15a713c34bb006dcff55dc100598c8e307b72fa60412748f90cde56bb5969ebc"],"state_sha256":"5fd4be61f6634601b4caf6a3a0b2a72bb7358edcda4e3129ecf54b1ecd17b578"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YPJ5KnZ0n+Sjb6Wb/uZj9vZ1+etJDG75vY3UCHXh47h5iv1yiWIsPqATM3Fibcg/6VVI6CdrRJc1tw5lcRrpAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T06:02:03.176549Z","bundle_sha256":"945e1872005b62146d6c7a64d53ad4a18480bc7d48eb9746b1583e46eb5d8a7e"}}