{"paper":{"title":"On certain integrals involving the Dirichlet divisor problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aleksandar Ivi\\'c, Wenguang Zhai","submitted_at":"2017-11-27T09:15:39Z","abstract_excerpt":"We prove that $$ \\int_1^X\\Delta(x)\\Delta_3(x)\\,dx \\ll X^{13/9}\\log^{10/3}X, \\quad \\int_1^X\\Delta(x)\\Delta_4(x)\\,dx \\ll_\\varepsilon X^{25/16+\\varepsilon}, $$ where $\\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\\zeta^k(s)$ ($\\Delta_2(x) \\equiv \\Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\\Delta_k(x)$. We also obtain the analogues of the above bounds when $\\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\\zeta(1/2+it)|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.09589","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"}