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We study a Poincar\\'e series $E(z,s)$ defined for $z$ in $\\mathcal{H}^{25}$, convergent for $\\operatorname{Re}(s) > 25$, invariant under $\\operatorname{Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a \"Leech cusp\" and prove that it can be meromorphically continued to $\\operatorname{Re}(s) > 25/2$. Analytic continuation of Kloosterman sum zeta functi"},"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":"1707.07790","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-25T02:24:40Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"07b5bc5760cf42216ebb0a8905aa1c169b35a4d8f22a1225dd500901c090e88d","abstract_canon_sha256":"cba6116c15141c94280270ebdf71225bd1f26839835665156ab1ab33239df614"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:30.348554Z","signature_b64":"c6OnBQisXtI1F6qtueldgzF9sJ3bMyT6qAXcdeQdl5XWUuedzggWlLtTl44dZHUsYAedbVpsWk2fO6YCeMroCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"330453a7839455cd1103f503f99d92ef6d7fdcc3fb27d514104ced1e10ad18d7","last_reissued_at":"2026-05-18T00:39:30.347857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:30.347857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Poincar\\'e series on hyperbolic space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Tathagata Basak","submitted_at":"2017-07-25T02:24:40Z","abstract_excerpt":"Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\\operatorname{Aut}(L)$ acts on the hyperbolic space $\\mathcal{H}^{25}$. We study a Poincar\\'e series $E(z,s)$ defined for $z$ in $\\mathcal{H}^{25}$, convergent for $\\operatorname{Re}(s) > 25$, invariant under $\\operatorname{Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a \"Leech cusp\" and prove that it can be meromorphically continued to $\\operatorname{Re}(s) > 25/2$. 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