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Let $1\\le \\ell_1 \\le \\ell_2$ be two integers, $\\Lambda$ be the von Mangoldt function and % \\(r_{\\ell_1,\\ell_2}(n) = \\sum_{m_1^{\\ell_1} + m_2^{\\ell_2}= n} \\Lambda(m_1) \\Lambda(m_2) \\) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \\geq 2$ be an integer. We prove that the Ces\\`aro average of weight $k > 1$ of $r_{\\ell_1,\\ell_2}$ over the interval $[1, N]$ has a development as"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1806.04930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-13T10:27:27Z","cross_cats_sorted":[],"title_canon_sha256":"8eaca79cb5fe3e20b309f3ce25251d44cef68b6f2d271c5e745265b0b6db082a","abstract_canon_sha256":"98ad264e6d99b1ac8d1fe5325825f583a2b939112c1da41c0f6d240c710f0ab1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:08.984358Z","signature_b64":"BXfnJOOcuJHeKTyphjy8mFZTFIcf0zTPUZwMUBCvv0iK3/oMEVHodPxgXvLLaCuu/VPQMiRNCAkafWCJDYsoBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"559a7501d9c16fa1391847419a3a3f79d58b5796d98bb8e0a25186b81d1606f3","last_reissued_at":"2026-05-17T23:40:08.983879Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:08.983879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Ces\\`aro average for an additive problem with prime powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alessandro Languasco, Alessandro Zaccagnini","submitted_at":"2018-06-13T10:27:27Z","abstract_excerpt":"In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\\le \\ell_1 \\le \\ell_2$ be two integers, $\\Lambda$ be the von Mangoldt function and % \\(r_{\\ell_1,\\ell_2}(n) = \\sum_{m_1^{\\ell_1} + m_2^{\\ell_2}= n} \\Lambda(m_1) \\Lambda(m_2) \\) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \\geq 2$ be an integer. We prove that the Ces\\`aro average of weight $k > 1$ of $r_{\\ell_1,\\ell_2}$ over the interval $[1, N]$ has a development as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04930","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.04930","created_at":"2026-05-17T23:40:08.983950+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.04930v2","created_at":"2026-05-17T23:40:08.983950+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.04930","created_at":"2026-05-17T23:40:08.983950+00:00"},{"alias_kind":"pith_short_12","alias_value":"KWNHKAOZYFX2","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"KWNHKAOZYFX2COIY","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"KWNHKAOZ","created_at":"2026-05-18T12:32:33.847187+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH","json":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH.json","graph_json":"https://pith.science/api/pith-number/KWNHKAOZYFX2COIYI5AZUOR7PH/graph.json","events_json":"https://pith.science/api/pith-number/KWNHKAOZYFX2COIYI5AZUOR7PH/events.json","paper":"https://pith.science/paper/KWNHKAOZ"},"agent_actions":{"view_html":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH","download_json":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH.json","view_paper":"https://pith.science/paper/KWNHKAOZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.04930&json=true","fetch_graph":"https://pith.science/api/pith-number/KWNHKAOZYFX2COIYI5AZUOR7PH/graph.json","fetch_events":"https://pith.science/api/pith-number/KWNHKAOZYFX2COIYI5AZUOR7PH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH/action/storage_attestation","attest_author":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH/action/author_attestation","sign_citation":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH/action/citation_signature","submit_replication":"https://pith.science/pith/KWNHKAOZYFX2COIYI5AZUOR7PH/action/replication_record"}},"created_at":"2026-05-17T23:40:08.983950+00:00","updated_at":"2026-05-17T23:40:08.983950+00:00"}