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We study the power moments and sign changes of $\\Delta(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $\\Delta(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constan"},"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":"1908.05598","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-14T07:18:08Z","cross_cats_sorted":[],"title_canon_sha256":"07f0b800f269c518b4610a97f5af0a11afcd29d0c3cd54962208ff152ee989d8","abstract_canon_sha256":"c999f04e186d8eb8e3d42cbd3a39fd2a62cf5c98317434a70a435e6850d0f473"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:56:54.153814Z","signature_b64":"DZZHGA4oOXe7tG2fVxzWKnBIi8NyIvxE6qbSI2SwEczH/2qjLfnIo1t498of2Xw0AUQWFpV2WPIwonLFZfuUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e028d11f4e8b2d4f0185b6c707312159f100c683173a4f1509a58a1cb0d64fdc","last_reissued_at":"2026-07-04T23:56:54.153347Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:56:54.153347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the divisor problem with congruence conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lirui Jia, Tianxin Cai, Wenguang Zhai","submitted_at":"2019-08-14T07:18:08Z","abstract_excerpt":"Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\\equiv r_i\\pmod{q_i}$ ($i=1,2$) and $\\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\\geq (q_1q_2)^{1+\\varepsilon}, 1\\leq r_i\\leq q_i$, and $(r_i, q_i)=1$ ($i=1, 2$). We study the power moments and sign changes of $\\Delta(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $\\Delta(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\\sqrt{T}]$ for any large $T$. 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