{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:13:31Z","timestamp":1776795211715,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"278","license":[{"start":{"date-parts":[[2012,8,25]],"date-time":"2012-08-25T00:00:00Z","timestamp":1345852800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We first report on computations made using the GP\/PARI package that show that the error term\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Delta (x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the divisor problem is\n \n \n \n \n =<\/mml:mo>\n \n M<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n +<\/mml:mo>\n \n O<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n 0.35<\/mml:mn>\n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =\\mathscr {M}(x,4)+ O^*(0.35\\, x^{1\/4}\\log x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n when\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ranges\n \n \n \n \n [<\/mml:mo>\n 1<\/mml:mn>\n \n 081<\/mml:mn>\n \n 080<\/mml:mn>\n ,<\/mml:mo>\n \n 10<\/mml:mn>\n \n 10<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n [1\\,081\\,080, 10^{10}]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n M<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n 4<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathscr {M}(x,4)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u0394\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n 0.397<\/mml:mn>\n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:mrow>\n |\\Delta (x)|\\le 0.397\\, {x^{1\/2}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n when\n \n \n \n \n x<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 5<\/mml:mn>\n \n 560<\/mml:mn>\n <\/mml:mrow>\n x\\ge 5\\,560<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and that\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u0394\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n 0.764<\/mml:mn>\n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n |\\Delta (x)|\\le 0.764\\, {x^{1\/3}\\log x}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n when\n \n \n \n \n x<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 9<\/mml:mn>\n \n 995<\/mml:mn>\n <\/mml:mrow>\n x\\ge 9\\,995<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\Delta (x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2011-02535-4","type":"journal-article","created":{"date-parts":[[2011,8,25]],"date-time":"2011-08-25T08:24:22Z","timestamp":1314260662000},"page":"1025-1051","source":"Crossref","is-referenced-by-count":10,"title":["Explicit upper bounds for the remainder term in the divisor problem"],"prefix":"10.1090","volume":"81","author":[{"given":"D.","family":"Berkane","sequence":"first","affiliation":[]},{"given":"O.","family":"Bordell\u00e8s","sequence":"additional","affiliation":[]},{"given":"O.","family":"Ramar\u00e9","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,8,25]]},"reference":[{"issue":"3","key":"1","first-page":"Article 38, 15","article-title":"Explicit upper bounds for the average order of \ud835\udc51_{\ud835\udc5b}(\ud835\udc5a) and application to class number","volume":"3","author":"Bordell\u00e8s, Olivier","year":"2002","journal-title":"JIPAM. 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