{"paper":{"title":"Omega Theorems for The Twisted Divisor Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anirban Mukhopadhyay, Kamalakshya Mahatab","submitted_at":"2018-07-26T10:06:52Z","abstract_excerpt":"For a fixed $\\theta\\neq 0$, we define the twisted divisor function $$ \\tau(n, \\theta):=\\sum_{d\\mid n}d^{i\\theta}\\ .$$ In this article we consider the error term $\\Delta(x)$ in the following asymptotic formula $$ \\sum_{n\\leq x}^*|\\tau(n, \\theta)|^2=\\omega_1(\\theta)x\\log x + \\omega_2(\\theta)x\\cos(\\theta\\log x) +\\omega_3(\\theta)x + \\Delta(x),$$ where $\\omega_i(\\theta)$ for $i=1, 2, 3$ are constants depending only on $\\theta$. We obtain $$\\Delta(T)=\\Omega\\left(T^{\\alpha(T)}\\right) \\text{ where } \\alpha(T) =\\frac{3}{8}-\\frac{c}{(\\log T)^{1/8}} \\text{ and } c>0,$$ along with an $\\Omega$-bound for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10047","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"}