; TeX output 2003.07.09:1115 Kb&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ:9|&html: html:.MUVgcolor push Black color popZVg5PSfile=logo129.eps llx=0 lly=0 urx=99 ury=16 rwi=28807Ѝccolor push Black color popcN q cmbx12A mZultidimensionalversionofaresultof ȮDaZvenp ort-Erd}Uqos-s獑=4XQ ff cmr12O-Y4eat/Chan,GeumlanChoi,andAlexandruZaharescu ˕Department/ofMathematics UUniversity/ofIllinois ?1409/W4estGreenStreet Urbana,/IL61801 dUSA *html:color push cmyk 0 1 0 0odCchan@math.uiuc.edu html: color pop x,html:color push cmyk 0 1 0 0g-choi1@math.uiuc.edu html: color pop -html:color push cmyk 0 1 0 0zaharesc@math.uiuc.edu html: color pop5x _color push Black color pop _N cmbx12Abstract鍑 XQ cmr12DarvenpSortandErd} osshorwedthatthedistributionofvXaluesofsumsoftheform g cmmi12S!2 cmmi8he(x)UR= x|{Y cmr8+h &u cmex10X ҍm=x+1#qō-$zm-$z[ z G 0p8qCt;%whereQpisaprimeand Fu m z ğꍑpf "istheLegendresymrbSol,isnormalash;py#!", cmsy10!1Qsuchthatw33log h33 z KRꍍOJlogp !UR0.8WVeprorveasimilarresultforsumsoftheform#7dShe(x1;:::ʜ;xnP)UR= ` xqAa cmr61*+hdOX ҍzq1*=xq1+1'?ex "; cmmi6n7+h Ao+X ҍ7z n7=x n+1^Sqōh[z1j+UN+znh[[ z @D /}p rq :;Bhtml: html: 32 A1. ɚ2- cmcsc10Introduction GivrenaprimenumbSerp,anintegerxandapSositiveintegerh,weconsiderthesum She(x)UR= x+h X ҍm=x+1#qō-$zm-$z[ z G 0p8qCt; :9color push Black 营3o cmr91G color pop *Kb&:9color push Blackhtml:color push gray 0 color pop html:2G color pop3ڍ:9whereWhereandinwhatfollorws Fu~m~ z ğꍑpmu denotestheLegendresymbSol.TheexpectedvXalueof:9sucrhacasumisp ae z Ê :h$.Ifpismuchlargerthanh,|itisaverydicultproblemtoshowthatthere :9isQ/anrycancellationinanindividualsumShe(x)asabSove.lvTheclassicalinequalityofP olya-:9Vinogradorv(see[html:color push cmyk 0 1 0 08 html: color pop],[html:color push cmyk 0 1 0 010 html: color pop])showsthatShe(x)=OS(p z 孟zp孹log#p),andassumingtheGeneralized:9RiemannHypSothesis,%MonrtgomeryandVVaughan[html:color push cmyk 0 1 0 07 html: color pop]provedthatShe(x)=OS(p z 孟zp孹log#log4@?p).:9The@ resultsofBurgess[html:color push cmyk 0 1 0 02 html: color pop]prorvidecancellationinShe(x)forsmallervXaluesofh,gassmallZ:9asp21=4 @.OnedoSesexpecttoharvecancellationinShe(x)forh>p2塀,forxed>0andp :9large.This!wrouldimplythewell-knownhypSothesisofVinogradovthatthesmallestpSositive:9quadraticwnonresiduemoSdpis<Ep2塀,foranryxed>0andplargeenoughintermsof.:9WVekmenrtionthatAnkeny[html:color push cmyk 0 1 0 01 html: color pop]showedthatassumingtheGeneralizedRiemannHypSothesis,R|:9theusmallestpSositivrequadraticnonresiduemodpisO(log-KW2Mp).NGItismruchueasiertoobtain:9cancellation,evrenŋsquareroSotcancellation,ifonearveragesŋShe(x)orverŋx.,Infact,DarvenpSort:9and%Erd} os[html:color push cmyk 0 1 0 05 html: color pop]enrtirelysolvedtheproblemofthedistributionofvXaluesofShe(x),4\0x