{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KC72LJQENGFZ2OMY7X2QOY7OKB","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":"895fd91c0ce3b63976a2ffa4afd9e064ff94b00ef43878ac6dffdede53148bc1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-08-27T05:23:25Z","title_canon_sha256":"d75c8cba6ed97498d157c4ab62e474e3554e7b3211d07713a53f92be93a62e84"},"schema_version":"1.0","source":{"id":"1308.5756","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5756","created_at":"2026-05-18T02:44:06Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5756v2","created_at":"2026-05-18T02:44:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5756","created_at":"2026-05-18T02:44:06Z"},{"alias_kind":"pith_short_12","alias_value":"KC72LJQENGFZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KC72LJQENGFZ2OMY","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KC72LJQE","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:691f6b6245a6efdbcdf2747b6e746bd7d80695e179a569177cd1944c519bef74","target":"graph","created_at":"2026-05-18T02:44:06Z","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":"This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, which was shown by him to have all its zeros on the critical line. With a rescaled complex argument, this is denoted here by ${\\cal T}_-(s)$, and is considered together with a counterpart function ${\\cal T}_+(s)$, symmetric rather than antisymmetric about the critical line. We prove that ${\\cal T}_+(s)$ has all its zeros on the critical line, and that the zeros of both functions are all of first order. We establish a link between the zeros of ${\\cal T}_-(s)$ and of ${\\cal T}_+(s)$ with those of t","authors_text":"Christopher G. Poulton, Ross C. McPhedran","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-08-27T05:23:25Z","title":"The Riemann Hypothesis for Symmetrised Combinations of Zeta Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5756","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":"sha256:8a3b958ea14364e3d6c386536ffe0998d2dd6e2882c3a5de4119aaf8a654cbb1","target":"record","created_at":"2026-05-18T02:44:06Z","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":"895fd91c0ce3b63976a2ffa4afd9e064ff94b00ef43878ac6dffdede53148bc1","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-08-27T05:23:25Z","title_canon_sha256":"d75c8cba6ed97498d157c4ab62e474e3554e7b3211d07713a53f92be93a62e84"},"schema_version":"1.0","source":{"id":"1308.5756","kind":"arxiv","version":2}},"canonical_sha256":"50bfa5a604698b9d3998fdf50763ee5051e5c78cd354f50930a5db7da3b0cabb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50bfa5a604698b9d3998fdf50763ee5051e5c78cd354f50930a5db7da3b0cabb","first_computed_at":"2026-05-18T02:44:06.912858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:06.912858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DSRcUOVysWbISWmJT5Fy/tmb1ku3KkyamX6a76xe6O+cdQkl5JU33xQtTvow+KSR2LvTgKtDjhpktYwD6bUiBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:06.913333Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.5756","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a3b958ea14364e3d6c386536ffe0998d2dd6e2882c3a5de4119aaf8a654cbb1","sha256:691f6b6245a6efdbcdf2747b6e746bd7d80695e179a569177cd1944c519bef74"],"state_sha256":"772aa0801c232cd3b48ce4b9f4397d60a11582535fa1ae14603e43f0e6b9a52e"}