ON THE n-th ELEMENT OF A SET OF POSITIVE INTEGERS

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1 Aales Uiv. Sci. Budapes., Sec. Comp ) ON THE -TH ELEMENT OF A SET OF POSITIVE INTEGERS Jea-Marie De Koick ad Vice Ouelle Québec, Caada) Commuicaed by Imre Káai Received July 8, 05; acceped Augus 5, 05) Absrac. Give a se A of posiive iegers ad is couig fucio Ax) := #{ x : A}, we examie he size of he -h eleme of A usig he size of Ax).. Iroducio ad oaio Deermiig he size of he -h eleme of a se of posiive iegers usig he kow size of he couig fucio of ha se is a classical problem i aalyic umber heory. For example, leig πx) sad for he umber of prime umbers p x, by usig he Prime Number Theorem i he form πx) x/ log x as x, oe ca easily show ha he -h prime umber p saisfies p = + o)) log ). I fac, i 90, by usig he logarihmic iegral fucio, Cipolla [3] improved his esimae by showig ha here exiss a uique sequece of polyomials Key words ad phrases: Prime umbers, powerful umbers, slowly oscillaig fucios. 00 Mahemaics Subjec Classificaio: N37, N64, K65, N36. Research suppored i par by a gra from NSERC.

2 54 J.-M. De Koick ad V. Ouelle Q j ) j wih raioal coefficies such ha, for ay give posiive ieger m,.) m ) p = log j Q j log + log + ) log j j= ). Here ad i wha follows, we wrie log x for max, log log x). ) + o log m Aoher example is give by he search of a esimae for a, he -h composie umber. Bojaricev [] ad Shiu [] showed ha, for ay give posiive ieger m, a = + β.) log + β log + + β )) m log m + o log m ), where he β i are compuable cosas. Fially, recall ha we say ha a umber is a powerful umber or a squarefull umber) if p implies ha p. Le deoe he -h powerful umber. I 98, Ivić ad Shiu [7] showed ha.3) = ) ζ3) + O 5/3) ). ζ3/) Here, we examie he problem of esimaig he size of he -h eleme of a give se A of posiive iegers usig he size of Ax) := #{ x : A}, ofe called he couig fucio of A. We will do so i wo paricular cases. The firs oe is whe Ax) = b x λ + b x λ + Rx), where Rx) = ox λ3 ), for some real cosas b > 0 ad b, wih > λ > λ > λ 3 > 0, from which we will he deduce a improveme of he esimae.3). The secod case is whe Ax) = x )) + O where ϕ is a icreasig fucio which eds o + as x ad L is a differeiable Lx) ϕx) icreasig slowly oscillaig fucio. Recall ha a fucio L : [M, + ) R coiuous o [M, + ), where M is a posiive real umber, is said o be a slowly oscillaig fucio if for each posiive real umber c > 0, Lcx).4) lim x Lx) =. This class of fucios was iroduced by Karamaa [8] i 930. His paper, alog wih [9] as well as he book of Seea [], provide some ieresig

3 O he -h eleme of a se of posiive iegers 55 properies of slowly oscillaig fucios. I paricular, i is possible o show ha a differeiable fucio L is slowly oscillaig if ad oly if.5) xl x) Lx) = o) x ) ad, i fac, ha L is slowly oscillaig if ad oly if here exiss x 0 > 0 such ha { x }.6) Lx) = Cx) exp d, where lim x Cx) = C, for a cerai cosa C 0, ad 0 as. We shall deoe by L he se of icreasig ad differeiable slowly oscillaig fucios. From here o, he leer c, wih or wihou subscrip, sads for a absolue posiive cosa, bu o ecessarily he same a each occurrece, while he leer p, wih or wihou subscrip, will always deoe a prime umber. x 0. Mai resuls Theorem. Give a sequece of posiive iegers a < a <, le A = {a, a,...} wih couig fucio Ax) saisfyig.) Ax) = b x λ + b x λ + Rx), where Rx) = ox λ3 ) x ) ad where b > 0 ad b are real cosas, wih > λ > λ > λ 3 > 0 which saisfy 3λ 3 < λ 3 λ. λ λ λ The.) a = λ b λ λ b b λ + λ λ + λ + λ + λ +o ) λ 3 + λ. λ ) b b λ + λ λ + λ +

4 56 J.-M. De Koick ad V. Ouelle Theorem. Give a sequece of posiive iegers a < a <, le A = {a, a,...} wih couig fucio Ax) saisfyig.3) Ax) = x )) + O Lx) ϕx) x ), where ϕ is a icreasig fucio which eds o + as x ad where L L wih correspodig fucio defied implicily by.6). Moreover, assume ha is a decreasig fucio ad ha ).4) Cx) = C + O x ), ψx) where ψx) is a icreasig fucio which eds o + as x. The, a = C a { a } )) ) L) exp d + O.5) C) ϕ) ) ad.6) a = L) / δ)) + O ϕ) + )) ψ) ), where δ is some fucio saisfyig ηa ) < δ) < η) for all iegers x 0. Moreover, if here exiss a posiive cosa c such ha.7) η) x 0 d c ), he.8) a = e c + o)) L) ). 3. Proof of Theorem To prove Theorem, we use a approach already used by Copil ad Paaiopol [4] o esimae he size of he -h o powerful umber. Firs, observe ha i follows from.) ha = A a ) = b a λ + b a λ + R a ),

5 O he -h eleme of a se of posiive iegers 57 so ha a λ = a λ b b R a ) ) hereby implyig ha 3.) a = /λ b /λ a I paricular, sice boh expressios b λ have a = /λ + o)) ad b /λ a λ b R a ) /λ ). Ra) ad goes o 0 as, we 3.) a = /λ b /λ + O ) λ + λ. Moreover, for ay α > 0, y) α = αy + α α ) y + O y 3) as y 0. Thus 3.3) a λ b R a ) /λ ) = = b a λ λ + ) b a λ λ + O λ R a ) ). Subsiuig his esimae i 3.) yields 3.4) a = /λ b /λ λ b b λ + λ λ + λ + O ) λ + λ. Usig his esimae ad he fac ha R a ) = o ) ) a λ3 = o λ 3/λ, we ca replace he RHS of 3.3) by 3.5) λ b b λ/λ λ λ + λ + λ λ ) b b λ/λ λ λ + o ) λ 3 λ, which subsiued back i 3.4) yields.).

6 58 J.-M. De Koick ad V. Ouelle 4. Proof of Theorem I follows from esimae.3) ha 4.) = Aa ) = a La ) ad herefore ha 4.) a = La ) + O )) ϕa ) )) + O ϕ) x 0 ) ). Sice L is a slowly oscillaig fucio, we have a ) 4.3) L a ) = C a ) exp d = C a ) L) exp C) Combiig 4.) ad 4.3) proves.5). a ) d. Now, le α = α) be he uique posiive ieger saisfyig α < log < La ) α La ), so ha α = = log L a ) + ɛ), where 0 log log ɛ) <. O he oe had, sice is decreasig ad posiive, we have 4.4) a d O he oher had, α d + d + + d < α < η) log + η) log + + η α ) log ) log L a ) η)α log = η) log + ɛ). log a 4.5) d α d + d + + d > α > η) log + η4) log + + η α ) log > ) log L > η α a ) )α ) log η a ) log + ɛ). log I follows from 4.3), 4.4) ad 4.5) ha here exiss a fucio δ saisfyig ηa ) < δ) < η) for all iegers x 0 such ha a d = δ) log L a ).

7 O he -h eleme of a se of posiive iegers 59 Combiig his resul wih.4), we ge )) 4.6) L a ) = L) / δ)) + O. ψ) Subsiuig 4.6) i 4.) proves.6). Fially,.8) follows easily from.7). Ideed, by 4.6), we have 4.7) L a ) = L) +δ)+oδ )). We have ad L) δ) L) η) = C) η) exp η) x 0 ) d = e c + o) ) ηa) ) ηa) L) δ) L) ηa) = L a ) ηa) L) L) = e c + o)). L a ) L a ) Sice log L a ) log L) = a d + o)) = δ) log L a ) + o)), i follows ha ) ηa) L) = + o) L a ) ad hus 4.8) L) δ) = e c + o). Moreover, usig 4.8), L) δ) L) η)) δ) = e c + o)) δ) = + o). Combiig his las resul wih 4.8) ad 4.7) gives L a ) = L) e c + o)) ). Subsiuig his esimae i 4.) yields a = e c + o)) L).

8 60 J.-M. De Koick ad V. Ouelle 5. Applicaios of Theorem We provide wo applicaios. Firs, we shall prove ha here exiss a posiive cosa C such ha, as, 5.) ) ζ3) = ζ/3) ) 8 ζ3) ζ3/) ζ) ζ3/) + 7 ) ) 0 ζ/3) ζ3) R0 ), 3 ζ) ζ3/) where 5.) R 0 ) 4/3 exp C log ) 3/5 log ) /5). I order o prove 5.), we firs recall he 958 resul of Baema ad Grosswald [] 5.3) P x) := #{ x : powerful} = ζ3/) ζ3) x/ + ζ/3) ζ) x/3 + Rx), where 5.4) Rx) x /6 exp C log x) 3/5 log x) /5), which is he bes kow error erm ad is due o Suryaarayaa ad Siaramachadra Rao [4]. The seig λ = /, λ = /3, λ 3 = /6, b = ζ3/) ζ3) ad b = ζ/3) ζ) i Theorem, keepig rack of he explici error erm give by 5.), esimae 5.) follows. As a secod applicaio, we cosider he geeral case of k-full umbers. Recall ha, give a ieger k, we say ha a posiive ieger is said o be k-full if p implies ha p k. We deoe by P k x) he umber of k-full iegers x ad by,k he -h k-full umber. Ivić ad Shiu [7] obaied ha 5.5) P k x) = γ 0,k x /k + γ,k x /k+) + + γ k,k x /k ) + k x), where he cosas γ i,k are give explicily ad k x) is a suiable error erm.

9 O he -h eleme of a se of posiive iegers 6 Usig 5.5) i he paricular case k = 3 ad Theorem, we ca prove ha here exiss a posiive cosa C ad cosas A 3, A 4 ad A 5 such ha, as,,3 = A 3 ) A 3 ) 5 4 A A3 ) 8 5 A A 3) 9 A 4 ) 5 + R), where R) 9 8 exp C log ) 3/5 log ) /5). Remark. Observe ha explici values for he cosas A i were obaied by Shiu []. Moreover, for k 4, as, oe ca prove ha ) k γ,k 5.6),k = k k +k γ γ 0,k γ 0,k ) kk+) k+,k k k +k k+ γ 0,k ) kk+3) k+ +R k ), k+ where R k ) k +k k+. Remark. Observe ha explici values for he cosas γ i,k are give i Baema ad Grosswald [] ad Erdős ad Szekeres [6]. Moreover, for k > 4, addiioal erms o he righ had side of 5.6) ca be provided. 6. Applicaios of Theorem We provide hree applicaios.. Fix a posiive ieger k ad le A = A k = { N : ω) = k} = {a : N}, where ω) sads for he umber of disic prime facors of. I is well kow ha, as x, Ax) = x )) + O, Lx) log log x where Lx) = k )! log x log x) k see for isace Theorem 0.4 i he book of De Koick ad Luca [5]). I follows from Theorem ha )) k )! log a = log ) k + O ). log log

10 6 J.-M. De Koick ad V. Ouelle. Cosider he se A of hose iegers such ha z) z ) = =, where z) = /e log. Usig a compuer, we easily obai he firs elemes of A, so ha we may wrie A = {3, 9, 6, 4, 33, 4, 5, 6, 7, 8, 93,...} = {a : N}. Clearly Ax) zx) for all x. The, sice codiio.7) of Theorem is saisfied wih c = /, we ge from.8) ha a = e + o) ) e log ). 3. Le W = {a, a,...} be he se of hose posiive iegers which ca be wrie as he sum of wo squares. I has bee kow sice Euler ha a posiive ieger ca be represeed as a sum of wo squares if ad oly if each of is prime facors of he form 4k + 3 occurs wih a eve power, so ha W = {, 5, 8, 0, 3, 7, 8, 0, 5, 6, 9, 3, 34, 37, 40, 4, 45, 50,...}. I 908, Ladau [0] showed ha x W x) = B + o)) x ), log x where B = ) p = I 986, Shiu [3] p 3 mod 4) showed ha 6.) W x) = Bx )) + O. log x log x Sice oe ca show ha 6.) Lx) := B { x log x = B exp e log } d, i follows from 6.) ad 6.) ha he correspodig fucios ϕx), ηx), Cx) ad ψx) from he saeme of Theorem are give by ϕx) = log x, ηx) =, Cx) = /B, ψx) =. log x Hece, i follows from.5) ha a = B log a + O )). log Thus, usig he logarihm o his formula, oe ca improve i o a = B log + )) log log 4 log + O ). log

11 O he -h eleme of a se of posiive iegers 63 Refereces [] Baema, P. ad E. Grosswald, O a heorem of Erdős ad Szekeres, Illiois J. Mah., 958), [] Bojaricev, A.E., Asympoic expressios for he h composie umber, Ural. Gos. Uiv. Ma. Zap., 6 987), 43. [3] Cipolla, M., La deermiazioe dell imo umero primo, Red. Acad. Sci. Fis. Ma. Napoli, 8 90), [4] Copil, V. ad L. Paaiopol, Properies of o powerful umbers, J. Iequal. Pure Appl. Mah., 9 008), o., Aricle 9, 8. [5] De Koick, J.-M. ad F. Luca, Aalyic Number Theory: Explorig he Aaomy of Iegers, Graduae Sudies i Mahemaics, Vol. 34, America Mahemaical Sociey, Providece, Rhode Islad, 0. [6] Erdős, P. ad G. Szekeres, Über die Azahl der Abelsche Gruppe gegebeer Ordug ud über ei verwades zahleheoreisches Problem i Germa), Aca Li. Sci. Szeged, 934), [7] Ivić, A. ad P. Shiu, The disribuio of powerful iegers, Illiois J. Mah., 6 98), o. 4, [8] Karamaa, J., Sur u mode de croissace régulière des focios, Mahemaica Cluj), 4 930), [9] Karamaa, J., Sur u mode de croissace régulière. Théorèmes fodameaux, Bull. Soc. Mah. Frace, 6 933), [0] Ladau, E., Über die Eieilug der posiive gaze Zahle i vier Klasse ach der Mideszahl der zu ihrer addiive Zusammesezug erforderliche Quadrae, Arch. Mah. Phys., 3)3 908), 305-3; Colleced works, Vol. 4, Thales Verlag, Esse, 986, [] Seea, E., Regularly Varyig Fucios, Spriger-Verlag, Berli, 976. [] Shiu, P., The disribuio of cube-full umbers, Glasgow Mah. J., 33 99), o. 3, [3] Shiu, P., Couig sums of wo squares: he Meissel Lehmer mehod, Mah. Comp., ), o. 75, [4] Suryaarayaa, D. ad R.R. Siaramachadra, The disribuio of square-full iegers, Arkiv für Maemaik, 973), 95 0.

12 64 J.-M. De Koick ad V. Ouelle J.-M. De Koick ad V. Ouelle Déparme de mahémaiques e de saisique Uiversié Laval Québec Québec GV 0A6 Caada jmdk@ma.ulaval.ca vice.ouelle.7@ulaval.ca