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For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\\mathcal B_{\\ell,D}(x,N) := \\sum_{\\substack{n\\in\\mathbb N\\\\ n\\le N}} b(n)n^{-\\beta_m} \\cos\\left( 2\\pi m\\left(\\frac{nx}{D}\\right)^{1/m} +\\frac{\\pi\\ell}{4} \\right),$$ where $D\\ge1$ is the conductor, $\\beta_m:=\\frac{m+1}{2m}$, a"},"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":"2607.16695","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-18T08:06:23Z","cross_cats_sorted":[],"title_canon_sha256":"c40bbc811e1d59c64221502e2314313127accde95d355f7b2591e23a8569a5f2","abstract_canon_sha256":"8b3e48d16c5d7dccc016f2645ee95c4b794a22c19126f19afe4a79077dc073e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T01:20:22.364813Z","signature_b64":"PSLbLMZicb3AiX1Wl2TyoS2VS75oGDUGRQJdxxi+TGetu3AnfZ18PhY4PN1mfT61JYprs9QT/T+9WYJWFFhuAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"788ba318589aff82888cadc0f85d391649a55d06483087c881099fc3ea064faf","last_reissued_at":"2026-07-21T01:20:22.363911Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T01:20:22.363911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counterexamples of Friedlander--Iwaniec dual sums conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Khai-Hoan Nguyen-Dang","submitted_at":"2026-07-18T08:06:23Z","abstract_excerpt":"Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\\sum_{n\\ge1}a(n)n^{-s}, \\qquad B(s)=\\sum_{n\\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \\emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\\mathcal B_{\\ell,D}(x,N) := \\sum_{\\substack{n\\in\\mathbb N\\\\ n\\le N}} b(n)n^{-\\beta_m} \\cos\\left( 2\\pi m\\left(\\frac{nx}{D}\\right)^{1/m} +\\frac{\\pi\\ell}{4} \\right),$$ where $D\\ge1$ is the conductor, $\\beta_m:=\\frac{m+1}{2m}$, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16695","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16695/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.16695","created_at":"2026-07-21T01:20:22.364378+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.16695v1","created_at":"2026-07-21T01:20:22.364378+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16695","created_at":"2026-07-21T01:20:22.364378+00:00"},{"alias_kind":"pith_short_12","alias_value":"PCF2GGCYTL7Y","created_at":"2026-07-21T01:20:22.364378+00:00"},{"alias_kind":"pith_short_16","alias_value":"PCF2GGCYTL7YFCEM","created_at":"2026-07-21T01:20:22.364378+00:00"},{"alias_kind":"pith_short_8","alias_value":"PCF2GGCY","created_at":"2026-07-21T01:20:22.364378+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/PCF2GGCYTL7YFCEMVXAPQXJZCZ","json":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ.json","graph_json":"https://pith.science/api/pith-number/PCF2GGCYTL7YFCEMVXAPQXJZCZ/graph.json","events_json":"https://pith.science/api/pith-number/PCF2GGCYTL7YFCEMVXAPQXJZCZ/events.json","paper":"https://pith.science/paper/PCF2GGCY"},"agent_actions":{"view_html":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ","download_json":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ.json","view_paper":"https://pith.science/paper/PCF2GGCY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.16695&json=true","fetch_graph":"https://pith.science/api/pith-number/PCF2GGCYTL7YFCEMVXAPQXJZCZ/graph.json","fetch_events":"https://pith.science/api/pith-number/PCF2GGCYTL7YFCEMVXAPQXJZCZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ/action/storage_attestation","attest_author":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ/action/author_attestation","sign_citation":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ/action/citation_signature","submit_replication":"https://pith.science/pith/PCF2GGCYTL7YFCEMVXAPQXJZCZ/action/replication_record"}},"created_at":"2026-07-21T01:20:22.364378+00:00","updated_at":"2026-07-21T01:20:22.364378+00:00"}