两个渐近公式的注记

两个渐近公式的注记

0非织造材料rewellbrowell.非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,非织造材料,也非其它织物上的hybridmethod参数a的自然列表，如所见，用anypk-1代替循环，并在a的初始位置标记。自由排列的数量。对于继承的数量。一种方法可以添加能量免费。Fromthesetofnaturalnumbers(except0and1)-takeoffallmultiplesof2k+1(i.e.2k+1,2k+2,…).-takeoffallmultiplesof3k+1.-takeoffallmultiplesof5k+1.…,andsoon(takeoffallmultiplesofallk+1-powerprimes).Inreference,Smarandacheaskedustostudythepropertiesofthek+1-powerfreenumbers.Thesecondauthorstudiedthehybridmeanvalueinvolvingthek+1-powerfreenumbers,andobtainedseveralasymptoticformulae.Forothertypesofhybridmeanvalueinvolvingsomenewarithmeticfunctions,seeandthereferencestherein.Nowwedefinetwonewnumber-theoreticfunctionsU(n)andV(n)asfollowing:U(1)=1,U(n)=∏p|nϕV(1)=1,V(n)=V(pα11)…V(pαrr)=(pα1-1)…(pαr-1),wherenisanynaturalnumberwiththeformn=pα11…pαrr.Obviouslytheyarebothmultiplicativefunctions.Inthispaper,weshallusetheanalyticmethodtostudythedistributivepropertiesof∑n∈An≤xU(n)and∑n∈An≤xV(n),andobtaintwointerestingasymptoticformula.Thatis,wehavethefollowingtwotheorems:Theorem1Letkbeanaturalnumber,x≥1arealnumber.Thenwehavetheasymptoticformulathat∑n∈An≤xU(n)=3x2π2∏p(1+p2k-2-1p2k+1+p2k-p2k-1-p2k-2)+Ο(x32+ε),where∏pdenotestheproductofallprimesandεanyfixedpositivenumber.Theorem2Letkbeanaturalnumber,x≥1arealnumber.Thenwehave∑n∈An≤xV(n)=x22∏p(1-1pk+1-p2k+1+p2k-p-1p2k+3+p2k+1)+Ο(x32+ε).1日因子反应主要参数为Inthissection,weshallcompletetheproofoftheorems.FirstweproveTheorem1,letf(s)=1+∑n∈An≤xU(n)ns,thenfromtheEulerproductformulaandthedefinitionofU(n)wehavef(s)=∏p(1+U(p)ps+U(p2)p2s+⋯+U(pk)pks)=∏p(1+1ps-1+1p2s-1+⋯+1pks-1)=∏p(1+1ps-1+p(k-1)s-1p2s-1(p(k-1)s-p(k-2)s))=ζ(s-1)ζ(2(s-1))∏p(1+p(k-1)s-1(p2s-1+ps)(p(k-1)s-p(k-2)s))whereζ(s)isRiemannzetafunction.Obviously,wehavetheinequalitythat|U(n)|≤n,|∞∑n=1U(n)nσ|<1σ-2,whereσ>2istherealpartofs.SobyPerronformulawehave∑n≤xU(n)ns0=12iπ∫b+iΤb-iΤf(s+s0)xssds+Ο(xbB(b+σ0)Τ)+Ο(x1-σ0Η(2x)min(1,logxΤ)+Ο(x-σ0Η(Ν)min(1,x∥x∥),whereNisthenearestintegertox,∥x∥=|x-Ν|.Τakings0=0,b=3,Τ=x32,Η(x)=x,B(σ)=1σ-2,wehave∑n≤xU(n)=12iπ∫3+iΤ3-iΤζ(s-1)ζ(2(s-1))R(s)xssds+Ο(x32+ε),whereR(s)=∏p(1+p2k-2-1p2k+1+p2k-p2k-1-p2k-2).Toestimatethemainterm12iπ∫3+iΤ3-iΤζ(s-1)ζ(2(s-1))R(s)xssds,wemovetheintegrallinefroms=3±iTtos=32±iΤ.Thistime,thefunctionf(s)=ζ(s-1)xsζ(2(s-1))sR(s)hasasimplepolepointats=2withresiduex22ζ(2)R(2).Sowehave12iπ(∫3-iΤ3+iΤ+∫32+iΤ32+iΤ+∫32+iΤ32-iΤ+∫32-iΤ3-iΤ)ζ(s-1)xsζ(2(s-1))sR(s)ds=x22ζ(2)∏p(1+p2k-2-1p2k+1+p2k-p2k-1-p2k-2).Wecaneasilygettheestimates|12πi(∫3-iΤ3+iΤ+∫32-iΤ3-iΤ)ζ(s-1)xsζ(2(s-1))sR(s)ds|≪∫323|ζ(σ-1+iΤ)ζ(2(σ-1+iΤ))R(s)x3Τ|dσ≪x3Τ=x32and|12πi∫32+iΤ32-iΤζ(s-1)xsζ(2(s-2))sR(s)ds|≪∫0Τ|ζ(1/2+it)ζ(1+2it)x32t|dt≪x32+ε.Notethatζ(2)=π26,fromtheabovewehave∑n∈An≤xU(n)=3x2π2∏p(1+p2k-2-1p2k+1+p2k-1-p2k-2)+Ο(x32+ε).ThiscompletestheproofofTheorem1.NowwecometoproveTheorem2.Letg(s)=1+∑n∈An≤xV(n)ns.FromtheEulerproductformulaandthedefinitionofV(n),wealsohaveg(s)=∏p(1+V(p)ps+V(p2)p2s+⋯+V(pk)pks)=∏p(1+p-1ps+p2-1p2s+⋯+pk-1pks)=∏p(1-1p(k+1)(s-1)1-1ps-1-1-1pksps-1)=ζ(s-1)∏p(1-1p(k+1)(s-1)-(pks-1)(ps-1+1)(pks-p(k-1)s)p2s-1).ByPerronformula,andthemeth