Restricted Factorial And A Remark On The Reduced Residue Classes

Transcription

1 Applid Mathmatics E-Nots, , c ISSN Availabl fr at mirror sits of am/ Rstrictd Factorial Ad A Rmark O Th Rducd Rsidu Classs Mhdi Hassai Rcivd 26 March 2016 Abstract I this papr w study th rstrictd factorial fuctio ñ!! dfid as th product of positiv itgrs k ot xcdig ad coprim to. As a corollary, w cosidr th asymptotic bhaviour of th ratio A G, whr A ad G dot rspctivly th arithmtic ad gomtric mas of all mmbrs of th last positiv rducd st of rsidus modulo. 1 Itroductio Amog svral qustios cocrig gralizatios of th factorial fuctio i [2], aalogus of Stirlig s approximatio for gralizd factorials is proposd. I th prst papr w dfi th rstrictd factorial fuctio for ach itgr 1 by ñ!! = k, whr k, dots th gratst commo divisor of th itgrs k ad. W study th asymptotic growth of ñ!!, ad aalogu to th wll-kow asymptotic rlatio! = log + O, w obtai log ñ!! = φ log + O. Mor prcisly w prov th followig. THEOREM 1. W hav log ñ!! = φ log + E, whr for 7 th rmaidr trm E satisfis 1 2 log E 1 2 log. Mathmatics Subjct Classificatios: 11A25, 11A07, 11N56. Dpartmt of Mathmatics, Uivrsity of Zaja, Uivrsity Blvd., , Zaja, Ira. 244

2 M. Hassai 245 To obtai th abov xplicit bouds, w d som xplicit bouds cocrig!, as follows. LEMMA 2. For ay itgr 1 w hav! = log2π + R, 1 2 whr 0 R Mawhil, as a immdiat cosquc of Thorm 1 w obtai th followig rsult. COROLLARY 3. As, w hav 1 φ ñ!! = + O log. If w dot th arithmtic ad gomtric mas of th positiv ral umbrs a 1, a 2,..., a, by Aa 1,..., a ad Ga 1,..., a, rspctivly, th th abov corollary givs th asymptotic xpasio of G := Gϱ 1,..., ϱ φ, whr R = {ϱ 1,..., ϱ φ } is th last positiv rducd st of rsidus modulo. By cosidrig w obtai th followig. A := Aϱ 1,..., ϱ φ = 1 φ COROLLARY 4. As, w hav A = log G 2 + O. k = 2, Th ratio 2 appars surprisigly i studyig th ratio of th arithmtic to th gomtric mas of som umbr thortic squcs. For th squc cosistig of positiv itgrs, Stirlig s approximatio for! implis s [5] for mor dtails A1,..., G1,..., = 2 + O. Rgardig to th squc of prim umbrs, i [6] w provd that Ap 1,..., p Gp 1,..., p = O,

3 246 Rstrictd Factorial o Rducd Rsidu Classs whr p dots th th prim umbr. Morovr, i [3] w provd validity of th similar ad mor prcis xpasio Aγ 1,..., γ Gγ 1,..., γ = 1 1 log log 2 1 log 2 2 log 2 + O log 3, whr 0 < γ 1 < γ 2 < γ 3 < dot th coscutiv ordiats of th imagiary parts of o-ral zros of th Rima zta-fuctio, which is dfid by ζs = =1 s for Rs > 1, ad xtdd by aalytic cotiuatio to th complx pla with a simpl pol at s = 1. O th othr had, th apparac of th similar limit valu 2 i th abov rsults is ot trivial ad a global proprty. As a xampl, w cosidr th asymptotic bhaviour of th ratio udr study for th valus of th Eulr fuctio. By usig th asymptotic xpasios for Aφ1,..., φ ad Gφ1,..., φ s [13] for th arithmtic ma, ad [7] for th gomtric ma, w gt Aφ1,..., φ Gφ1,..., φ = 3 π p + O, p p whr th product rus ovr all prims. This givs a limit valu diffrt from 2, for th cas of Eulr fuctio. Mor grally, w obsrv that th limit valu of th ratio udr study could b ay arbitrary ral umbr β 1, as th followig rsult cofirms. PROPOSITION 5. For ach ral umbr β 1 thr xists a ral positiv squc with gral trm a such that Aa 1,..., a lim Ga 1,..., a = β. Rgardig to th cas β = 1, w show th followig. PROPOSITION 6. Assum that a > 0 with a l ad l > 0. Th Aa 1,..., a lim Ga 1,..., a = 1. W obsrv that Propositio 6 is ot tru for l = 0. For istac, if w lt a = 1, th by usig Stirlig s approximatio for!, w obtai Aa 1,..., a Ga 1,..., a = 1 + O1. Fially, w ot that if d dot th umbr of positiv divisors of, th i [4] w provd that for ach fixd itgr m 1 o has Ad1,..., d m Gd1,..., d = B 1 1 log 2 r k 1 + log k + O 1 log m+1, whr B ad th coffi cits r k ar computabl costats. This provids a umbr thortic xampl for wh th ratio A G tds to ifiity.

4 M. Hassai Sums Ovr Rducd Rsidu Systms To approximat log G w d to comput rstrictd summatios ruig ovr th lmts of R. W follow th sam mthod as i [1] to obtai th followig. PROPOSITION 7. Assum that f is a arbitrary arithmtic fuctio. Th, w hav fk = µd fdq. 3 d 1 q d PROOF. Th rsult is valid for = 1. W assum that > 1, ad w us th kow idtity d m µd = [ 1 m ] to writ 1 fk = By takig k = dq, w gt 1 d k,d [ fk µdfk = 1 k, ] 1 = fk 1 dq< d = 1 q< d d k, µdfdq µd = 1 µdfdq = µd d d d k,d 1 q< d µdfk. fdq. Now, w ot that if q = d, th fdq = f, ad sic > 1, w imply that d µdf = f[ 1 ] = 0. Thus, w obtai 3, ad th proof is complt. 3 Proofs PROOF OF LEMMA 2. W apply Eulr Maclauri summatio formula s [12] with fk = log k to writ whr! = T, T = 1 B 2 {x} 2x 2 B 2 {x} dx 2x 2 dx, ad B 2 {x} is th Broulli fuctio of ordr 2. Also, {x} dots th fractioal part of th ral x. Thus, w obtai with! = C I, C = B 2 {x} 1 2x 2 dx,

5 248 Rstrictd Factorial o Rducd Rsidu Classs ad Sic I 1 whr I = B 2 {x} 2x 2 dx. as, w gt C = lim! = log lim D, D =! 1. 2 W apply Wallis product formula for π s [14] for a lmtary proof, to gt D 2 = lim D D D 2 2 = lim 2k k = 2π Thus, w obtai D = 2π, ad cosqutly C = log D = log 2π. Also, w hav I This complts th proof. B 2 {x} 2x 2 dx 1 12 PROOF OF THEOREM 1. By usig 3 w hav log ñ!! = log k = µd logdq = d d 1 q d dx x 2 = µd d log d + log!. d W apply th kow rlatio d µd log d = Λ, whr Λ is th Magoldt fuctio, to obtai log ñ!! = φ log + E, with E = 1 2 Λ + d µdr, d ad R is dfid i 1. W hav 0 Λ. Also, by usig th triagl iquality, ad cosidrig th bouds 2, w obtai d µdr d d 1 d R d 6 = σ 6 < 1 log, 2 d whr for dducig th last boud w us th iquality σ < 2.59 log, which is valid for 7 s [8]. Hc, for ach 7 w gt 1 2 log E 1 2 log.

6 M. Hassai 249 This complts th proof. PROOF OF COROLLARY 3. Thorm 1 implis that ñ!! 1 φ = E φ. For ay 1 w hav φ. Also, th iquality φ > is valid for 3 s [10]. Thus, w gt from which w obtai This complts th proof. E φ E φ γ log , log log = log, log = 1 + O. PROOF OF PROPOSITION 5. For ach ral umbr η 0, w st a = η. It is kow [9] that Aa 1,..., a lim Ga 1,..., a = η := lη, η + 1 d η say. W ot that dη lη = lη η+1. Hc lη is strictly icrasig for η 0. Also l0 = 1 ad lim η lη =. Thus, for ay ral umbr β 1 thr xists a ral umbr η 0 such that lη = β, as dsird. This complts th proof. PROOF OF PROPOSITION 6. For th squc a addrssd i th statmt of thorm, it is kow that Aa 1,..., a l s [11], pag 80. Also, sic log a log l as, w obtai log Ga 1,..., a = Alog a 1,..., log a log l, ad cosqutly, Ga 1,..., a l. This cocluds th proof. Ackowldgmt. Th author was supportd by a grat of th Uivrsity of Zaja, rsarch projct umbr Rfrcs [1] T. M. Apostol, Broulli s powr-sum formulas rvisitd, Math. Gaz., , [2] M. Bhargava, Th factorial fuctio ad gralizatios, Amr. Math. Mothly, ,

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