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We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov and Yu on the case $N=1$."},"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":"1812.07863","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-19T10:23:53Z","cross_cats_sorted":[],"title_canon_sha256":"760decde244c38b52dc7e07e9a14f47ca2b04cd2a9e49cd71c8b0fb6fe08a725","abstract_canon_sha256":"2685c99ec34ee5253f10ae82478b1b8e0d22c6c2dc0f79e209f6fef646c707ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:55.697599Z","signature_b64":"8ZO1zGrrcCaMTR80kj4Hg3on1tpWaVxOzPunsEC70VJkio32f2EgwXXrCre/7fvTomNlI51M4Y+16eFYD+JtCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c6fffccd331072058e892980c5dbe9b45a1d532d19ab3187c88a61dac5e7c83","last_reissued_at":"2026-05-17T23:57:55.696747Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:55.696747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean values of divisors of forms $n^2+Nm^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liangyi Zhao, Peng Gao","submitted_at":"2018-12-19T10:23:53Z","abstract_excerpt":"Let $N$ be any fixed positive integer and define \\begin{align*}\n  S_N(x)=\\sum_{m, n \\leq x}d(n^2+Nm^2), \\end{align*}\n  where $d(n)$ is the divisor function. 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