On the Distribution of Square-full Numbers

oINTRODUCTIONLetxbeasuffcientlylargepositivenumber,Q(x)bethenumberofsquare-fullnumbersnotexceedingx.Anaturalnumberniscalledtobesquare-full,ifpInalwaysimpliesthatp'ln,herepdenotesaprimenumber.From[l]weknowthatwhereA(x)istheerrorterm.Letsbeasuff-cientlysmallgivenpositivenumber.Theproblemoffinding-anassmallaspossiblepositivenumberosuchthatA(x)=O(x'+')byassumingtheRiemannhypothesistobetenablehasbeenstudiedinpapers[2],[3],[4],[51and[6],inwhichthesepermissiblevaluesofohavebeenproved-l3/8l,ll…     

On An Average Sum

0~urnAccoldingto[I],evecallthefollowingsumasBarbansum:whereQ$x,A(n)isthevonMangoldtfunction,9(q)istheEulerfUnction,xisasufficiendylargePOsitivenumber.actAbeagivenPOsitiveconstant.TheasylnptoticformulaofS(Q,x)whenx(logx)--'$Q$xiswell--known.Recendy,in[2]itisshownthatS(Q,suitabl}'smallpositiveconstant.Intheworkof[2]thefollowingaveragesumplaysanimPOrtantrole:wheree($)=e-cp(Zrris)forarealE,p(q)istheM0biusfUnction.Infactthemethodof[2]showsthatifxexp(M)>Qxlogx.Then,bym…     

Remarks on the lower bounds for sums of the BDH type

Theso calledBDHsum(hereBDHistheabbre viationof“Barban Davenport Halberstam”)isdefinedasfollows:S(Q,x)=∑q≤Q∑1≤a≤q(a,q)=1φ(x;q,a)-xφ(q)2,wherexisalargepositivenumber,x(logx)-A≤Q≤x,Aisagivenpositiveconstant,φ(q)istheEulerfunc tion,φ(x;q,a)=∑n≤     

Positive integers possessing a weak order

For a positive integerm,we call a positive integertwhich possesses a weak order(modm),if there doesnot exist a positive integer nsuch thattn 1≡ t(modm). Letg(m)be the number of positive integerstforwhich1≤ t≤ mandtpossess a weak order(modm).Forx≥3,letG(x) =∑m≤xg(m).In 1981, V.S. Joshi[1]first proved the following as-ymptotic formula:G(x) =αx2 O(x(logx)3), (1)whereα=2ζ(12)∑n∞ =1C(nn2), C(n) =∏p npp 1.In 1986, Yang Zhaohua[2]improved (1) toG(x) =αx2 O(x(logx)2). (2) The key ne…