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First, we start with three different double integrals that have been previously used in the literature to show $S(2)=\\pi^2/8,$ which implies Euler's identity $\\zeta(2)=\\pi^2/6.$ Then, we generalize each integral in order to find the considered sums. The $k$ dimensional analogue of the first integral is the density function of the quotient of $k$ independent, nonnegat"},"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":"1710.03637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-10T14:56:53Z","cross_cats_sorted":[],"title_canon_sha256":"368ab3f211934972d65fca7428a54781ab330cb5cc89f01b7094a1f7738c79e4","abstract_canon_sha256":"a9fecf8727409c16d578b0cd92ec49dcf8a9d1e1d0cbb0e361bed3de39ccd9b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:41.418911Z","signature_b64":"Jsfyp8HNBOYY6W93/3uvt7YFaeef0nj1gfSPaIz4JPGqQxojld9owZqkFPHmlPSebvXQrQn2xO8PfV9O/PX2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b22d43f64ca9e30d1d8ccd654f9a3c3bb97e29c47f466ab85bc0181cb95bb52","last_reissued_at":"2026-05-18T00:00:41.418515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:41.418515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluation of Harmonic Sums with Integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniele Ritelli, Vivek Kaushik","submitted_at":"2017-10-10T14:56:53Z","abstract_excerpt":"We consider the sums $S(k)=\\sum_{n=0}^{\\infty}\\frac{(-1)^{nk}}{(2n+1)^k}$ and $\\zeta(2k)=\\sum_{n=1}^{\\infty}\\frac{1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show $S(2)=\\pi^2/8,$ which implies Euler's identity $\\zeta(2)=\\pi^2/6.$ Then, we generalize each integral in order to find the considered sums. 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