{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:DZOZ7XQADGHI7T4J3VQ3LJY73A","short_pith_number":"pith:DZOZ7XQA","canonical_record":{"source":{"id":"1705.10766","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-30T17:39:17Z","cross_cats_sorted":[],"title_canon_sha256":"8d115508927ef8ebcee3c86c0b6de8c7307a1034066c2fbc008342cc13ced6dc","abstract_canon_sha256":"1bdc487a17c08c0061b99da596b55bab670e909c3354d3fba8477ca22e6dde51"},"schema_version":"1.0"},"canonical_sha256":"1e5d9fde00198e8fcf89dd61b5a71fd82f0efc8c6d97cdbb3c13d5393a13e788","source":{"kind":"arxiv","id":"1705.10766","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.10766","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"arxiv_version","alias_value":"1705.10766v1","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.10766","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"pith_short_12","alias_value":"DZOZ7XQADGHI","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"DZOZ7XQADGHI7T4J","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"DZOZ7XQA","created_at":"2026-05-18T12:31:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:DZOZ7XQADGHI7T4J3VQ3LJY73A","target":"record","payload":{"canonical_record":{"source":{"id":"1705.10766","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-30T17:39:17Z","cross_cats_sorted":[],"title_canon_sha256":"8d115508927ef8ebcee3c86c0b6de8c7307a1034066c2fbc008342cc13ced6dc","abstract_canon_sha256":"1bdc487a17c08c0061b99da596b55bab670e909c3354d3fba8477ca22e6dde51"},"schema_version":"1.0"},"canonical_sha256":"1e5d9fde00198e8fcf89dd61b5a71fd82f0efc8c6d97cdbb3c13d5393a13e788","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:23.562930Z","signature_b64":"TF/dXNb7OFiudNKA/D0bPhzH9wCm4mbEHAp3t5u0/ZM8JhESvhrOQCIZIujA8IAXQdceicQY3tJOOoP7tobuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e5d9fde00198e8fcf89dd61b5a71fd82f0efc8c6d97cdbb3c13d5393a13e788","last_reissued_at":"2026-05-18T00:43:23.562271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:23.562271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.10766","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mx6Nj44OfBFeLVVrS5w+GM3S0DMk33g8JOw9/PCuoCxzvMgP+b0XXL4DgUmkeCqK6HgrwK0+G+dhi/8aZy9ABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:18:51.580018Z"},"content_sha256":"02f90de2eb0645704e36ce4bc77f496f6c3991030284b0b627a1c43e5aef89b4","schema_version":"1.0","event_id":"sha256:02f90de2eb0645704e36ce4bc77f496f6c3991030284b0b627a1c43e5aef89b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:DZOZ7XQADGHI7T4J3VQ3LJY73A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the moments of the gaps between consecutive primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Marek Wolf","submitted_at":"2017-05-30T17:39:17Z","abstract_excerpt":"We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$\\pi(x)$ --- the number of primes up to: $M_k(x)= \\Gamma(k+1)x^k/\\pi^{k-1}(x)+\\mathcal{O}(x)$, We illustrate obtained results by computer data."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.10766","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"esdxLnja6koV73a1Yh5rG4olxj3/kIOxXJRufxW8pozZi63HXQIoGORv5wyEkBqCvRVvq3W00rQGEhDWM7WeCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:18:51.580363Z"},"content_sha256":"29f992b4dddab9fecdb6003a3491add658239334d322c2ad760b42bf5dc4f8dd","schema_version":"1.0","event_id":"sha256:29f992b4dddab9fecdb6003a3491add658239334d322c2ad760b42bf5dc4f8dd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/bundle.json","state_url":"https://pith.science/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/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":"2026-06-09T01:18:51Z","links":{"resolver":"https://pith.science/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A","bundle":"https://pith.science/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/bundle.json","state":"https://pith.science/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DZOZ7XQADGHI7T4J3VQ3LJY73A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:DZOZ7XQADGHI7T4J3VQ3LJY73A","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":"1bdc487a17c08c0061b99da596b55bab670e909c3354d3fba8477ca22e6dde51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-30T17:39:17Z","title_canon_sha256":"8d115508927ef8ebcee3c86c0b6de8c7307a1034066c2fbc008342cc13ced6dc"},"schema_version":"1.0","source":{"id":"1705.10766","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.10766","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"arxiv_version","alias_value":"1705.10766v1","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.10766","created_at":"2026-05-18T00:43:23Z"},{"alias_kind":"pith_short_12","alias_value":"DZOZ7XQADGHI","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"DZOZ7XQADGHI7T4J","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"DZOZ7XQA","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:29f992b4dddab9fecdb6003a3491add658239334d322c2ad760b42bf5dc4f8dd","target":"graph","created_at":"2026-05-18T00:43:23Z","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 derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$\\pi(x)$ --- the number of primes up to: $M_k(x)= \\Gamma(k+1)x^k/\\pi^{k-1}(x)+\\mathcal{O}(x)$, We illustrate obtained results by computer data.","authors_text":"Marek Wolf","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-30T17:39:17Z","title":"On the moments of the gaps between consecutive primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.10766","kind":"arxiv","version":1},"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":"sha256:02f90de2eb0645704e36ce4bc77f496f6c3991030284b0b627a1c43e5aef89b4","target":"record","created_at":"2026-05-18T00:43:23Z","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":"1bdc487a17c08c0061b99da596b55bab670e909c3354d3fba8477ca22e6dde51","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-30T17:39:17Z","title_canon_sha256":"8d115508927ef8ebcee3c86c0b6de8c7307a1034066c2fbc008342cc13ced6dc"},"schema_version":"1.0","source":{"id":"1705.10766","kind":"arxiv","version":1}},"canonical_sha256":"1e5d9fde00198e8fcf89dd61b5a71fd82f0efc8c6d97cdbb3c13d5393a13e788","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1e5d9fde00198e8fcf89dd61b5a71fd82f0efc8c6d97cdbb3c13d5393a13e788","first_computed_at":"2026-05-18T00:43:23.562271Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:23.562271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TF/dXNb7OFiudNKA/D0bPhzH9wCm4mbEHAp3t5u0/ZM8JhESvhrOQCIZIujA8IAXQdceicQY3tJOOoP7tobuDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:23.562930Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.10766","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02f90de2eb0645704e36ce4bc77f496f6c3991030284b0b627a1c43e5aef89b4","sha256:29f992b4dddab9fecdb6003a3491add658239334d322c2ad760b42bf5dc4f8dd"],"state_sha256":"43def52e5483bbc3443120b23fcd8074c6081728b79dc44bf03800c14615becf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Do16oorIHmhAKnRKTL8hcFIQRSOQmuDSaqtt6/DqTVLh33Z3BwVONVIMywI0k96+8ngwHdnPBHXAjJjXGdwaAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T01:18:51.582489Z","bundle_sha256":"1b6a5966f44a23872d16b89501065edb6e652ff38ff5341fa3dad7ae79ce5785"}}