{"paper":{"title":"Higher moments of the error term in the 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":"2009-04-15T09:38:10Z","abstract_excerpt":"It is proved that, if $k\\ge2$ is a fixed integer and $1 \\ll H \\le X/2$, then $$ \\int_{X-H}^{X+H}\\Delta^4_k(x)\\d x \\ll_\\epsilon X^\\epsilon\\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\\Bigr), $$ where $\\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\\\"\\i}--type formula for $\\Delta_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $\\Delta_2(x)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.2271","kind":"arxiv","version":3},"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"}