{"paper":{"title":"On the Sum of Divisors of Mixed Powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinjiang Li, Min Zhang","submitted_at":"2016-09-24T12:50:48Z","abstract_excerpt":"Let $d(n)$ denote the Dirichlet divisor function. Define \\begin{equation*}\n  \\mathcal{S}_{k}(x)=\\sum_{\\substack{1\\leqslant n_1,n_2,n_3 \\leqslant x^{1/2} \\\\ 1\\leqslant n_4\\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k),\n  \\qquad 3\\leqslant k\\in \\mathbb{N}. \\end{equation*} In this paper, we establish an asymptotic formula of $\\mathcal{S}_k(x)$ and prove that \\begin{equation*}\n  \\mathcal{S}_k(x)=C_1(k)x^{3/2+1/k}\\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-\\delta_k+\\varepsilon}), \\end{equation*} where $C_1(k),\\,C_2(k)$ are two constants depending only on $k,$ with $\\delta_3=\\frac{19}{60},\\,\\delta_4=\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07610","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"}