{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:3UBRRLB4SZZIBZLEFWNRVXM6PG","short_pith_number":"pith:3UBRRLB4","canonical_record":{"source":{"id":"1402.2746","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-12T06:56:53Z","cross_cats_sorted":[],"title_canon_sha256":"1b92c9985490a57237e4d35a6cfaf571b9d1400bbb94d1c20e71d9bbc1a7d332","abstract_canon_sha256":"05b1cd6c11a5c87d7b1b0cee60064d0efb0fb37e84b0ac2f229e2646d9f25e0f"},"schema_version":"1.0"},"canonical_sha256":"dd0318ac3c967280e5642d9b1add9e79a19b2bd112e23c0400fdcb54aa65b55d","source":{"kind":"arxiv","id":"1402.2746","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.2746","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"arxiv_version","alias_value":"1402.2746v2","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.2746","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"pith_short_12","alias_value":"3UBRRLB4SZZI","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3UBRRLB4SZZIBZLE","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3UBRRLB4","created_at":"2026-05-18T12:28:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:3UBRRLB4SZZIBZLEFWNRVXM6PG","target":"record","payload":{"canonical_record":{"source":{"id":"1402.2746","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-12T06:56:53Z","cross_cats_sorted":[],"title_canon_sha256":"1b92c9985490a57237e4d35a6cfaf571b9d1400bbb94d1c20e71d9bbc1a7d332","abstract_canon_sha256":"05b1cd6c11a5c87d7b1b0cee60064d0efb0fb37e84b0ac2f229e2646d9f25e0f"},"schema_version":"1.0"},"canonical_sha256":"dd0318ac3c967280e5642d9b1add9e79a19b2bd112e23c0400fdcb54aa65b55d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:38.290997Z","signature_b64":"JG4RAtBsMhqWmtBFg5F/YZgGyo61olCe0Fiieo0BunvWnqcwAoQ9zNUzpuk3MZg12S4Hnj5vf/WHVa5jLTPaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd0318ac3c967280e5642d9b1add9e79a19b2bd112e23c0400fdcb54aa65b55d","last_reissued_at":"2026-05-18T01:09:38.290626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:38.290626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.2746","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":"2026-05-18T01:09:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PrwpVel9a7PZlDp48IxoF7+yMd7vY5GGhMKOLFxEDddGNS2iHQpmZWA/uLR4libvMxs6JU1TjBUiLWNuqmZuBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:13:43.524619Z"},"content_sha256":"7b1968aec94250e0e7403465d0923389c849f66852ed0043150abb86729b1cd7","schema_version":"1.0","event_id":"sha256:7b1968aec94250e0e7403465d0923389c849f66852ed0043150abb86729b1cd7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:3UBRRLB4SZZIBZLEFWNRVXM6PG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Moments and oscillations of exponential sums related to cusp forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Esa V. Vesalainen","submitted_at":"2014-02-12T06:56:53Z","abstract_excerpt":"We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the $k$-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.\n  We also consider the mean square of very short sums, proving that on average short linear sums wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2746","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":"2026-05-18T01:09:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7cXa7r40OH9zbAvV5bIVZG2OlVjdt4twz+IG23QyQ31yyn7Gzs5IhIFJXk3HA+fGWAiFbpQbpknLvBa5tcDIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:13:43.525287Z"},"content_sha256":"ca69460a76d0304f1354ed4dc10396c92593f6c54b070a235d7a5ff8168bcd09","schema_version":"1.0","event_id":"sha256:ca69460a76d0304f1354ed4dc10396c92593f6c54b070a235d7a5ff8168bcd09"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/bundle.json","state_url":"https://pith.science/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/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":"2026-06-07T14:13:43Z","links":{"resolver":"https://pith.science/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG","bundle":"https://pith.science/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/bundle.json","state":"https://pith.science/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3UBRRLB4SZZIBZLEFWNRVXM6PG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3UBRRLB4SZZIBZLEFWNRVXM6PG","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":"05b1cd6c11a5c87d7b1b0cee60064d0efb0fb37e84b0ac2f229e2646d9f25e0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-12T06:56:53Z","title_canon_sha256":"1b92c9985490a57237e4d35a6cfaf571b9d1400bbb94d1c20e71d9bbc1a7d332"},"schema_version":"1.0","source":{"id":"1402.2746","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.2746","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"arxiv_version","alias_value":"1402.2746v2","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.2746","created_at":"2026-05-18T01:09:38Z"},{"alias_kind":"pith_short_12","alias_value":"3UBRRLB4SZZI","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3UBRRLB4SZZIBZLE","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3UBRRLB4","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:ca69460a76d0304f1354ed4dc10396c92593f6c54b070a235d7a5ff8168bcd09","target":"graph","created_at":"2026-05-18T01:09:38Z","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 consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the $k$-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.\n  We also consider the mean square of very short sums, proving that on average short linear sums wit","authors_text":"Esa V. Vesalainen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-12T06:56:53Z","title":"Moments and oscillations of exponential sums related to cusp forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2746","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":"sha256:7b1968aec94250e0e7403465d0923389c849f66852ed0043150abb86729b1cd7","target":"record","created_at":"2026-05-18T01:09:38Z","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":"05b1cd6c11a5c87d7b1b0cee60064d0efb0fb37e84b0ac2f229e2646d9f25e0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-12T06:56:53Z","title_canon_sha256":"1b92c9985490a57237e4d35a6cfaf571b9d1400bbb94d1c20e71d9bbc1a7d332"},"schema_version":"1.0","source":{"id":"1402.2746","kind":"arxiv","version":2}},"canonical_sha256":"dd0318ac3c967280e5642d9b1add9e79a19b2bd112e23c0400fdcb54aa65b55d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd0318ac3c967280e5642d9b1add9e79a19b2bd112e23c0400fdcb54aa65b55d","first_computed_at":"2026-05-18T01:09:38.290626Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:38.290626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JG4RAtBsMhqWmtBFg5F/YZgGyo61olCe0Fiieo0BunvWnqcwAoQ9zNUzpuk3MZg12S4Hnj5vf/WHVa5jLTPaAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:38.290997Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.2746","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7b1968aec94250e0e7403465d0923389c849f66852ed0043150abb86729b1cd7","sha256:ca69460a76d0304f1354ed4dc10396c92593f6c54b070a235d7a5ff8168bcd09"],"state_sha256":"69a1d6eea3f4aeacd21169b99c56ebbd30aa140bcd17b6c5e7c9a6538c77bc45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hvEs5js044AMzvt0PzP1CgauLsN20nPaf9HXaR8P2CsOWWosocs5H40G6P2tLMAV+RF8st2bIXpHtvNgjsqMBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T14:13:43.528695Z","bundle_sha256":"835000a2ca2abf64f45a1f11e29bd021c036b6169c13d438223c140649cd7efd"}}