{"paper":{"title":"Lambert series and q-functions near q=1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Blake Wilkerson, Shubho Banerjee","submitted_at":"2016-02-02T18:42:01Z","abstract_excerpt":"We study the Lambert series $\\mathscr{L}_q(s,x) = \\sum_{k=1}^\\infty k^s q^{k x}/(1-q^k)$, for all $s \\in \\mathbb{C}$. We obtain the complete asymptotic expansion of $\\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the q-Pochhammer symbol, and, in closed form, the Jacobi theta functions. Some typical results include $\\Gamma_2(\\frac{1}{4}) \\Gamma_2(\\frac{3}{4}) \\simeq \\frac{2^{13/32} \\pi}{\\log 2}$ and $\\vartheta_4 (0,e^{-1/\\pi}) \\simeq 2 \\pi e^{-\\pi^3\\!/4}$, with relative errors"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01085","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}