On Certain Sums Extended over Prime Factors

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1 Iteratioal Mathematical Forum, Vol. 9, 014, o. 17, HIKARI Ltd, O Certai Sum Exteded over Prime Factor Rafael Jakimczuk Diviió Matemática, Uiveridad Nacioal de Lujá Bueo Aire, Argetia Copyright c 014 Rafael Jakimczuk. Thi i a ope acce article ditributed uder the Creative Commo Attributio Licee, which permit uretricted ue, ditributio, ad reproductio i ay medium, provided the origial work i properly cited. Abtract Let P () be the greatet prime factor of a poitive iteger. I thi ote we obtai aymptotic formulae for the um P (k) m ( =0, 1,...,m), where m i a arbitrary but fixed poitive iteger. We alo obtai aymptotic formulae for the um p(k) m ( =0, 1,...,m), where p() deote the leat prime factor of. Mathematic Subject Claificatio: 11A99, 11B99 Keyword: Sum over prime factor 1 Itroductio Let P () be the greatet prime factor of a poitive iteger ad let m be a arbitrary but fixed poitive iteger. For completee, let P (1) = 1. Let u coider the m 1 um F m() = P (k) m ( =0, 1,...,m). (1)

2 798 Rafael Jakimczuk The cae m = 1 ad = 0 wa tudied by K. Alladi ad P. Erdő [1]. Thee author obtaied the followig aymptotic formula F 0 1 () = P (k) π 1 log = ζ() log, () where ζ() deote the Riema zeta fuctio. The cae m = 1 ad = 1 wa tudied by J. Kemey [3]. Thi author obtaied the followig aymptotic formula F 1 1 () = P (k) k π 6 log = ζ() log. (3) J. Kemey claim that formula () ca be obtaied from hi formula (3). However, we hall ee that formula (3) ca be obtaied from formula (). The cae = 0 wa tudied by R. Jakimczuk []. Thi author obtaied the followig aymptotic formula Fm 0 () = P (k) m ζ(m 1) m1 m 1 log (m 1). (4) If m = 1 the equatio (4) become equatio (). I thi ote we tudy the more geeral um (1). Let p() be the leat prime factor of a poitive iteger. I thi ote we alo obtai aymptotic formulae for the um fm () = p(k) m ( =0, 1,...,m), where m i a arbitrary but fixed poitive iteger. Mai Reult Lemma.1 Let i=1 a i ad i=1 b i be two erie of poitive term uch that a i b i. If i=1 b i i diverget the we have the followig aymptotic formula a i b i. i=1 Proof. See for example ([4], page 33). Theorem. Let m be a arbitrary but fixed poitive iteger. We have the followig aymptotic formulae F m () = P (k) m i=1 1 m 1 log ( =0, 1,...,m). (5)

3 Sum over prime factor 799 Proof. Let m be a fixed poitive iteger. If = 0 the formula (5) i true (ee formula (4)). Suppoe that formula (5) i true for {0,...,m 1}. We hall prove that formula (5) i alo true for 1. We have F 1 m () = Fm(1) 1 (F m() Fm(1)) 1 3 (F m(3) Fm()) 1 ( (F m() Fm( 1)) = 1 1 ) ( 1 F m(1) 1 Fm() 3) ( ( 1 Fm 4) (3) 1 1 ) Fm ( 1) 1 F m () That i, = 1 1. F m(1) 1.3 F m() F m(3) 1 F m () =1 F m (1) F m() F m() 1 1. F m() 1 Fm(3) ( 1) F m( 1) F m( 1) 1 Fm 1 () =1 F m (1) 1 Fm() 1 Fm(3) 1 Fm() F m() (6) Now, we have (ee (5)) Fm 1 () =f() m 1 log ( ), (7) where f() 1. Therefore Fm () 1 = f() 1m 1log = g() m 1log ( ), (8) where g() 1, ad where h(i) 1. That i, 1 Fm (i) i ζ(m 1) = f(i) i 1 i i 1m 1 ζ(m 1) = h(i) (i ), m 1 1 Fm (i) i 1 i ζ(m 1) m 1 (i ). (9)

4 800 Rafael Jakimczuk Coequetly (9) ad lemma.1 give We have (L Hopital rule) 1 Fm (i) i 1 i ζ(m 1) m 1. (10) x a tm 1 log t lim x x m (m ) dt = lim x ( d dx x m 1 x m (m ) ) =1 ( =0,...,m 1). (11) Note that if =0,...,m the the fuctio xm 1 from a certai 0. Coequetly (ee (11)) i trictly icreaig = O 0 1 ( ) m 1 log 0 x m 1 dx m (m ) log. (1) Sice the left ide of the equality i a um of rectagle of bai 1. O the other had if = m 1 the the fuctio xm 1 i trictly decreaig. Coequetly (ee (11)) = x m 1 dx O (1) m (m ) log. (13) Sice the left ide of the equality i a um of rectagle of bai 1. Equatio (10), (1) ad (13) give 1 Fm(i) i 1 i where k() 1. Equatio (6), (8) ad (14) give Fm 1 () = 1 F m ζ(m 1) (1) k() m 1 = k() m 1(m ) log, (14) m (m ) log m log = ζ(m 1) m ( 1)1 That i, equatio (5). The theorem i proved. If = m the we obtai the followig corollary. g() m 1log m (1)1 log.

5 Sum over prime factor 801 Corollary.3 The followig aymptotic formula hold ( ) Fm m () = m P (k) ζ(m 1) (m 1). (15) k log If m = 1 the equatio (15) become the Kemey formula (3). Note that if k i prime the we have P (k) =k. Coequetly each prime cotribute 1 to the um F m m () (ee (15)). Theorem.4 Let m be a arbitrary but fixed poitive iteger. We have the followig aymptotic formulae f m() = p(k) m 1 m 1 m 1 log ( =0, 1,...,m). Proof. If = 0 the the theorem i true (ee []). Now, the proof i the ame a Theorem.. The theorem i proved. If = m the we obtai the followig corollary. Corollary.5 The followig aymptotic formula hold ( ) m p(k) fm m () = (m 1). (16) k log Note that if k i prime the we have p(k) =k. Coequetly each prime cotribute 1 to the um f m m () (ee (16)). ACKNOWLEDGEMENTS. The author i very grateful to Uiveridad Nacioal de Lujá. Referece [1] K. Alladi ad P. Erdő, O a additive arithmetic fuctio, Pacific Joural of Mathematic, 71 (1977), [] R. Jakimczuk, A ote o um of greatet (leat) prime factor, Iteratioal Joural of Cotemporary Mathematical Sciece, 8 (013), [3] J. Kemey, Larget prime factor, Joural of Pure ad Applied Algebra, 89 (1993), [4] J. Rey Pator, P. Pi Calleja y C. A. Trejo, Aálii Matemático, Kapeluz, Received: April 5, 014

Gaps between Consecutive Perfect Powers

Gaps between Consecutive Perfect Powers Iteratioal Mathematical Forum, Vol. 11, 016, o. 9, 49-437 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1988/imf.016.63 Gaps betwee Cosecutive Perfect Powers Rafael Jakimczuk Divisió Matemática, Uiversia