{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:TZ33RPD5EYHV65VH2H5ZJ6ZUK7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6c66b8a12b8354a3ac0993de19ff4100cfd50e9904355389cbba658c5ae600e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-25T19:22:04Z","title_canon_sha256":"0040cc46be44332c4cde3aea62384bf3f08f7cff8c988788a3689882778df0e6"},"schema_version":"1.0","source":{"id":"1709.08686","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.08686","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"arxiv_version","alias_value":"1709.08686v2","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.08686","created_at":"2026-05-18T00:33:58Z"},{"alias_kind":"pith_short_12","alias_value":"TZ33RPD5EYHV","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"TZ33RPD5EYHV65VH","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"TZ33RPD5","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:5e6947d5ae27bfaf70f5d6215db01c4ab4ce873bd67844b832e371969327fb17","target":"graph","created_at":"2026-05-18T00:33:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $I(n):=\\int_0^1 [x^n+(1-x)^n]^\\frac1n dx.$ In this paper, we show that $I(n)= \\sum_0^\\infty \\frac{I_i}{n^i},n\\rightarrow \\infty$ and we compute $I_i, i =0..5$, obtained by polylog functions and Euler sums. As a corollary, we obtain explicit expressions for some integrals involving functions $ u^i, exp(-u), (1 +exp(-u))^j , ln(1 + exp(-u))^k$ . As another asymptotic result, let $S_0(z):=\\frac{Li_m(1)}{Li_m(1)-Li_m(z)}$, where $Li_m(z)$ is the polylog function. We provide the asymptotic behaviour of $S_n,n\\rightarrow \\infty$ where $S_n:=[z^n]S_0(z)$. This paper fits within the framework of a","authors_text":"Guy Louchard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-25T19:22:04Z","title":"Two applications of polylog functions and Euler sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08686","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b78a4e412efbd06a5146f7739348f4e758fe5b170483acbccdc46d5fe554b3fd","target":"record","created_at":"2026-05-18T00:33:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6c66b8a12b8354a3ac0993de19ff4100cfd50e9904355389cbba658c5ae600e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-25T19:22:04Z","title_canon_sha256":"0040cc46be44332c4cde3aea62384bf3f08f7cff8c988788a3689882778df0e6"},"schema_version":"1.0","source":{"id":"1709.08686","kind":"arxiv","version":2}},"canonical_sha256":"9e77b8bc7d260f5f76a7d1fb94fb3457effb3bb8cc47271fe75a9663c9faccd7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9e77b8bc7d260f5f76a7d1fb94fb3457effb3bb8cc47271fe75a9663c9faccd7","first_computed_at":"2026-05-18T00:33:58.304070Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:58.304070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jNyYRg1kTX9L5NrmHo8u0A4jYk1Y/jWIs/ePnQex+3YzG4oZWCFEfP47uQ5A9JHNRp2SATpX8LmrdyeNw3jYAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:58.304668Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.08686","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b78a4e412efbd06a5146f7739348f4e758fe5b170483acbccdc46d5fe554b3fd","sha256:5e6947d5ae27bfaf70f5d6215db01c4ab4ce873bd67844b832e371969327fb17"],"state_sha256":"5f18246da35e830a34f81af6cf9a2259091977c3b02b0b163b5abb564f6a228a"}