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Sums that appear in transformation formulas are generalizations of the Hardy--Berndt sums $s_{j}(d,c),$ $j=1,2,5$. As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented."},"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":"1710.05001","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-10-13T16:49:16Z","cross_cats_sorted":[],"title_canon_sha256":"f971e7430f9502c1fee9069a2afe22c4b07bc6d1a8a881ce78168818c09437fa","abstract_canon_sha256":"f9c1a3d6a3ff99c3af2991675f0cea01c69e078851a3a9a96309bdfb577e4000"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:39.907310Z","signature_b64":"r0yYfvV5LHkbXzzKp5fAFU7oBiUKxXC/a0L6TUh1AYa8KpqEg0ESRcKSAKpPgo/4H+mq477yygxvvlA/gd8KCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dbb8e4138ee537e41ec26428e92caafef3b7d94ba18ca0c66dcbf2bd9384037","last_reissued_at":"2026-05-18T00:04:39.906707Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:39.906707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transformation formulas of a character analogue of $\\log\\theta_{2}(z)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Merve \\c{C}elebi Bozta\\c{s}, M\\\"um\\\"un Can","submitted_at":"2017-10-13T16:49:16Z","abstract_excerpt":"In this paper, transformation formulas for the function \\[ A_{1}\\left(z,s:\\chi\\right)=\\sum\\limits_{n=1}^{\\infty}\\sum\\limits_{m=1}^{\\infty}\\chi\\left(n\\right)\\chi\\left(m\\right)\\left(-1\\right)^{m}n^{s-1}e^{2\\pi imnz/k} \\] are obtained. 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