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We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and $p\\equiv 3\\mod 4$, then there are no odd rational-valued functions $f\\not\\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases there are examples of odd functions $f$ such that $D_k(1,f)=0$.\n  As a consequence, we obtain, for example, that the set of values $L(1,\\chi)^2$, where $\\chi$ ranges over odd characters mod $p$, are linearly independent over $\\math"},"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":"1704.08358","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-04-26T21:35:58Z","cross_cats_sorted":[],"title_canon_sha256":"4d2a418326df4eb1a1e17125cc424782b9f410c8463e36e396d263b3d2c29e80","abstract_canon_sha256":"ec4bcad9d4169a4e78b0301b791785c07b7de814e3428e3be7b357bf26f9fd57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:52.335901Z","signature_b64":"zV1ME5nBRVymrdcX3mIBU7vyXF2mgtgE5i85vedn7+vvJO+4nPpLhEGb2wLOpnWbLCr2rcDSN9KLGEWFo/ZzBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22277691fcca5c89556964bfa57995c425513f80c83e623801c71680570d3b6d","last_reissued_at":"2026-05-18T00:20:52.335402Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:52.335402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the non-vanishing of certain Dirichlet series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bruno Martin, Sandro Bettin","submitted_at":"2017-04-26T21:35:58Z","abstract_excerpt":"Given $k\\in\\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\\sum_{n\\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. 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