{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NQ7JQRGK6ICYPWSR6G5VVP6VA5","short_pith_number":"pith:NQ7JQRGK","canonical_record":{"source":{"id":"1109.4242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-20T09:03:33Z","cross_cats_sorted":[],"title_canon_sha256":"f5ed84d3473c10239a69f693ffc9415a7cafb5bb9a33f5915a20e85f90f6457f","abstract_canon_sha256":"322557f3ed827808770275a0770ba06a891aae61d029213d12ef21a778e61d14"},"schema_version":"1.0"},"canonical_sha256":"6c3e9844caf20587da51f1bb5abfd5077f06e33a6995473919acb0838d0416f7","source":{"kind":"arxiv","id":"1109.4242","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4242","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4242v1","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4242","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"NQ7JQRGK6ICY","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NQ7JQRGK6ICYPWSR","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NQ7JQRGK","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NQ7JQRGK6ICYPWSR6G5VVP6VA5","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-20T09:03:33Z","cross_cats_sorted":[],"title_canon_sha256":"f5ed84d3473c10239a69f693ffc9415a7cafb5bb9a33f5915a20e85f90f6457f","abstract_canon_sha256":"322557f3ed827808770275a0770ba06a891aae61d029213d12ef21a778e61d14"},"schema_version":"1.0"},"canonical_sha256":"6c3e9844caf20587da51f1bb5abfd5077f06e33a6995473919acb0838d0416f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:43.794571Z","signature_b64":"l+JAHyQ8niu5c60UGnzMlOhdH235j98zrlnKRuqzHubHbcA2PTvbvWnRILXluxeXT3bdbk87hTFTDSDSdXWqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c3e9844caf20587da51f1bb5abfd5077f06e33a6995473919acb0838d0416f7","last_reissued_at":"2026-05-18T04:12:43.793785Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:43.793785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4242","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":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5j7L+7QBCDsrqbaVh0SS8f74OekQFgQxEehkFQwOdIOIFpQQqhpGz3rGtX/z21rg8qpHryEw+TPhQjyL5fj8CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:16:50.533081Z"},"content_sha256":"e026b469242fe1f6371dab87ae08e01e447ecf3763e6aebbd70075eb143f93ea","schema_version":"1.0","event_id":"sha256:e026b469242fe1f6371dab87ae08e01e447ecf3763e6aebbd70075eb143f93ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NQ7JQRGK6ICYPWSR6G5VVP6VA5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A modified M\\\"obius $\\mu$-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J\\\"orn Steuding, L\\'aszl\\'o T\\'oth, Rasa Steuding","submitted_at":"2011-09-20T09:03:33Z","abstract_excerpt":"We investigate a modified M\\\"obius $\\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss relations with Riemann's hypothesis."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4242","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":"2026-05-18T04:12:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9KiqNq7lSCBUXZqX2fZ5tqzed48oI0heRPtr8oOqDERsFkL9WgWMoeJsarkbI6x/x5YrXCBU3yrG2brd4IaDAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:16:50.533430Z"},"content_sha256":"f184062ed8dce7ed963943fb603868585543f83e571423a3b541dadaa45d754e","schema_version":"1.0","event_id":"sha256:f184062ed8dce7ed963943fb603868585543f83e571423a3b541dadaa45d754e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/bundle.json","state_url":"https://pith.science/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/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":"2026-06-09T15:16:50Z","links":{"resolver":"https://pith.science/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5","bundle":"https://pith.science/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/bundle.json","state":"https://pith.science/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NQ7JQRGK6ICYPWSR6G5VVP6VA5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NQ7JQRGK6ICYPWSR6G5VVP6VA5","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":"322557f3ed827808770275a0770ba06a891aae61d029213d12ef21a778e61d14","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-20T09:03:33Z","title_canon_sha256":"f5ed84d3473c10239a69f693ffc9415a7cafb5bb9a33f5915a20e85f90f6457f"},"schema_version":"1.0","source":{"id":"1109.4242","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4242","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4242v1","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4242","created_at":"2026-05-18T04:12:43Z"},{"alias_kind":"pith_short_12","alias_value":"NQ7JQRGK6ICY","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NQ7JQRGK6ICYPWSR","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NQ7JQRGK","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:f184062ed8dce7ed963943fb603868585543f83e571423a3b541dadaa45d754e","target":"graph","created_at":"2026-05-18T04:12:43Z","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 investigate a modified M\\\"obius $\\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss relations with Riemann's hypothesis.","authors_text":"J\\\"orn Steuding, L\\'aszl\\'o T\\'oth, Rasa Steuding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-20T09:03:33Z","title":"A modified M\\\"obius $\\mu$-function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4242","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:e026b469242fe1f6371dab87ae08e01e447ecf3763e6aebbd70075eb143f93ea","target":"record","created_at":"2026-05-18T04:12:43Z","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":"322557f3ed827808770275a0770ba06a891aae61d029213d12ef21a778e61d14","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-20T09:03:33Z","title_canon_sha256":"f5ed84d3473c10239a69f693ffc9415a7cafb5bb9a33f5915a20e85f90f6457f"},"schema_version":"1.0","source":{"id":"1109.4242","kind":"arxiv","version":1}},"canonical_sha256":"6c3e9844caf20587da51f1bb5abfd5077f06e33a6995473919acb0838d0416f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c3e9844caf20587da51f1bb5abfd5077f06e33a6995473919acb0838d0416f7","first_computed_at":"2026-05-18T04:12:43.793785Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:43.793785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l+JAHyQ8niu5c60UGnzMlOhdH235j98zrlnKRuqzHubHbcA2PTvbvWnRILXluxeXT3bdbk87hTFTDSDSdXWqAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:43.794571Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4242","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e026b469242fe1f6371dab87ae08e01e447ecf3763e6aebbd70075eb143f93ea","sha256:f184062ed8dce7ed963943fb603868585543f83e571423a3b541dadaa45d754e"],"state_sha256":"94b15ae8e7f3ea3c1d6de5104317a29999c02be9557f74b5990fc5580871574a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PpwEKz3aiv9b40q22ItGvnQYOHdjKWwPY45MImldrw4hEHyg2NNHQnb3LiBws7fcrhBgkhlZlRG7i8HaBSWuAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T15:16:50.535277Z","bundle_sha256":"8c39340f7e2c50cf09f96c1e658bdf0babc3d840e90f5e56805c4c680191ac07"}}