{"paper":{"title":"On some mean value results for the zeta-function and a divisor problem II","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":"2015-02-02T09:04:55Z","abstract_excerpt":"Let $d(n)$ be the number of divisors of $n$, let $\\gamma$ denote Euler's constant and $$ \\Delta(x) := \\sum_{n\\le x}d(n) - x(\\log x + 2\\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\\zeta(s)$ denote the Riemann zeta-function. It is shown that $$ \\int_0^T\\Delta(t)|\\zeta(1/2+it)|^2\\,dt \\ll T(\\log T)^{4}. $$ Further, if $2\\le k\\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \\int_1^{T}\\Delta^{k}(t)|\\zeta(1/2+it)|^2\\,dt=c_1(k)T^{1+\\frac k4}\\log T+ c_2(k)T^{1+\\frac k4}+O_\\varepsilon(T^{1+\\frac k4-\\eta_k+\\varepsilon}), $$ where $c_1(k)$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00406","kind":"arxiv","version":2},"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"}