Asymptotic Formulas Composite Numbers II

Transcription

1 Iteratoal Matematcal Forum, Vol. 8, 203, o. 34, HIKARI Ltd, ttp://d.do.org/0.2988/mf Asymptotc Formulas Composte Numbers II Rafael Jakmczuk Dvsó Matemátca, Uversdad Nacoal de Lujá Bueos Ares, Argeta Copyrgt c 203 Rafael Jakmczuk. Ts s a ope access artcle dstrbuted uder te Creatve Commos Attrbuto Lcese, wc permts urestrcted use, dstrbuto, ad reproducto ay medum, provded te orgal work s properly cted. Abstract I ts artcle we study te dstrbuto of composte umbers wc ave ter prme factorzato a fed umber of dfferet prme factors, te epoets beg fed. I a prevous artcle [4] we studed a partcular case of ts problem. Let us cosder te sequece A of composte umbers wose prme factorzato s of te form p a pa pa k k. Were a,a 2,...,a k are postve tegers fed ad p,p 2,...,p k are dfferet prmes. Let A) be te umber of tese umbers ot eceedg. I ts artcle we prove A) α β ) γ log Were te umbers α, β, γ are defed ts artcle. Matematcs Subject Classfcato: A99, B99 Keywords: Composte umbers, coutg fuctos, asymptotc formulas Itroducto, otato ad lemmas. Let be a umber suc tat ts prme factorzato f of te form = p a p a p ar r

2 652 R. Jakmczuk were a q 2 =, 2,...,r) ad p,p 2,...,p r r ) are dfferet prmes te factorzato. Tese umber are well kow, tey are called powerful umbers or q-ful umbers. Tere est varous studes o te dstrbuto of tese umbers usg ot elemetary metods see [2]). Let C,q be te sequece of tese umbers ad let C q ) be te umber of tese umbers tat do ot eceed. It s well kow see [3] for a elemetary proof) tat C,q c q q ) C q ) b q q 2) were b q ad c q are costats depedg of q. From ) we ca obta wtout dffculty te followg lemma. Lemma. Te followg seres are coverget q 2) = C,q ) q = log C,q C,q ) q Let us cosder te sequece E of te umbers wose prme factorzato s of te form p a p a p at t Were a a 2... a t 2t ) are postve tegers fed ad te p j j =, 2,...,t) are dfferet prmes. For eample te sequece of te umbers of te form p 9 p5 2 p5 3 p3 4 were p,p 2,p 3,p 4 are dfferet prmes. I ts case a =9,a 2 =5,a 3 =5,a 4 = 3,t=4. We sall deote tese umbers te compact form E. Te umber of tese umbers ot eceedg we sall deote E). Let us cosder te sequece A of te umbers wose prme factorzato s of te form p a p a p at t p t+...p t+k Were a a 2... a t > t ) k ) are postve tegers fed ad te p j j =, 2,...,t+ k) are dfferet prmes. For eample te umbers of te form p 9 p5 2 p5 3 p3 4 p 5p 6 p 7 p 8 were p, p 2, p 3, p 4, p 5, p 6, p 7, p 8 are dfferet prmes. I ts case a =9,a 2 =5,a 3 =5,a 4 = 3,=,t=4,k =4. We sall deote tese umbers te compact form Ep...p k. Were E deotes te umbers of te form p a p a p at t see above), p deotes p t+,..., p k deotes p t+k.

3 Composte umbers II 653 Sce ts case te E umbers are +)-ful umbers, lemma. mply te followg seres are coverget log E = A = E E, = B = E E, 3) O te oter ad 2) mply tat from a certa value of we ave E) + ɛ)b + + ɛ >0) 4) Te umber of tese umbers ot eceedg we sall deote A). I [4] was proved tat f k = te A) A E, log 5) I teorem 2. we sall prove te geeral asymptotc formula Ts formula becomes 5) f k =. A) A E, ) k k )! log Let us cosder te umbers wose prme factorzato s of te form p p 2...p k were k 2 s fed ad p,p 2,...,p k are dfferet prmes. Let B) be te umber of tese umbers ot eceedg. We ave te followg teorem Ladau s teorem) wc we sall use as a lemma see []). Lemma.2 Te followg asymptotc formula olds B) = )k k )! log + f)log log )k k )! log were f) M f 3 ad f) 0. We sall also eed te followg two lemmas wose proofs are smple. Lemma.3 Te oegatve fucto e )k 2 ) f) = )k log s bouded. Tat s, tere est H k > 0 suc tat f) H k.

4 654 R. Jakmczuk Lemma.4 Te fucto c >) f) = ) c s creasg from a certa value of. For sake of completeess we establs te followg remark. Remark. Note tat our problem s trval f all epoets are equal, tat s, f te umbers are of te form p...p k Were ad k. I ts case te asymptotc formula wc correspod to A) s a medate cosequece of te prme umber teorem or te Ladau s teorem. Namely 2 Ma results A) ) k k )! log Teorem 2. Te followg asymptotc formula olds A) A E, ) k k )! log 6) Proof. We sall use matematcal ducto. Te teorem s true for k = see 5) ad [4 ]). Suppose te teorem s true for, 2,...,k k 2). We sall prove t s also true for k. Let P k be te product of te frst k prmes, tat s, P 2 =2.3 =6,P 3 =2.3.5 = 30, etc. We ave Ep p 2...p k p p 2...p k E E p p 2...p k E P k P k E P k 6

5 Composte umbers II 655 Terefore lemma.2) A) = P k = P k + P k = p...p k E F k ) = B F E k ) = f E E ) k P k P k E k )! log p...p k E E )) k E F k ) k )! log ) F k ) ) k E )) k E k )! log E log E P k log + G k ) F k ) 7) Substtutg = P k E to E ) P k E log E log we obta te sequece Note tat f E E te = E log E log P k E 8) E log E log P k E ad f E s fed te E log E log P k E = E log P k + log E = + log P k E log P k log E E 9) lm E log E log P k E = E 0)

6 656 R. Jakmczuk We ave see 8)) = E log E log P k E j = = E log E log P k E + =j+ E log E log P k E ) were see 9)) =j+ E log E log P k E =j+ + E log P k =j+ log E 2) E Tere ests j suc tat ɛ >0) see 3)) A E, ɛ< log P k If j +, 2), 3) ad 4) gve j = =j+ E <A E, 3) log E E <ɛ 4) 0 =j+ E log E log P k E 2ɛ 5) O te oter ad see 0)) j lm = E log E log P k E = j = E Cosequetly tere ests >j+ suc tat for all we ave j = 3) ad 6) gve E j ɛ = E log E log P k E j = E + ɛ 6) j A E, 2ɛ = E log E log P k E A E, + ɛ 7)

7 Composte umbers II 657 Terefore for all we ave see ), 5) ad 7)) Cosequetly Now, we ave lm = ad = lm A E, 3ɛ = E log E log P k E lm = E log E log P k E E log E log P k E = E log E log P k E + = E log E log P k E + = E log E log P k E + A E, +3ɛ 8) = A E, 9) log P k E E log P k + + = 0 20) 20) ad 2) gve lm = Terefore see 9)) lm E log E log P k E + log P k E = 0 2) E log P k + = E log E log P k E + = 0 22) lm = E log E log P k E lm = E log E log P k E + = A E, 23) = A E, 24)

8 658 R. Jakmczuk Te fucto of E fed, E E ) E log E log s decreasg te terval [ Pk E,Pk E [ +). Terefore f P k E,Pk E + we ave = E log E log P k E + = E Cosequetly 23), 24) ad 26) gve lm P k Te fucto of E fed) E log E log log E log 0 < E ) k = E log E log P k E 25) ) 26) = A E, 27) < s creasg from a certa value of see lemma.4) ad lm Let us cosder te fucto P k E Tere ests 0 suc tat log E log A E, ɛ E ) k = E 0 P k E ) k <A E, + ɛ 28) Now lm 0 P k E log E log = 0 P k E 29)

9 Composte umbers II 659 Terefore tere est > 0 suc tat f we ave see 28) ad 29)) ad f E 0 P k 30) ad 3) gve 0 < 0 P k A E, 2ɛ we ave E 0 P k E log E log 0 < E log E log O te oter ad see 30) ) 0 < 0 P k 0 P k 32) ad 33) gve lm P k 27) ad 34) gve < P k < P k E lm E E E log E log log E log P k log E log E ) k A E, +2ɛ 30) ) k <ɛ 3) A E, +2ɛ)ɛ 32) E ) k 4ɛ 33) E ) k E log E log ) k = 0 34) = A E, 35) Tere ests 0 suc tat lemma.2) f E <ɛ f E 0, tat s f E 0

10 660 R. Jakmczuk Terefore f E M f P k G k ) = f E P k P k ɛ 0 = ɛ E f E E E E E < 0, tat s f k )! log )) k E ) E )) k E k )! log E k )! log ) k k )! log + M 0 0 < P k Now see lemma.3 ad 4)) )) k E 0 k )! log E ) ) + M 0 E E 0 < P k 0 ) k E )) k E log E log < P k E k )! log )) k E ) 36) ) M 0 0 < P k E k )! log )) k E ) M 0H k )! E M 0H k )! + ɛ)b ) ad see 35)) ɛ ) k k )! log 0 E ) k E log E log

11 Composte umbers II 66 ɛ ) k k )! log ) k P k E ) k E log E log ɛ A E, + ɛ) 38) k )! log Cosequetly 36), 37) ad 38) gve G k ) =o ) k 39) log O te oter ad from te ductve ypotess ad 4) f t s ecessary) we ca obta wtout dffculty tat F k ) =o ) k 40) log Fally 7), 35), 39) ad 40) gve 6). Te teorem s proved. Corollary 2.2 Te followg asymptotc formula olds A k )!) log A E, 4) )k ) Proof. From 6) we obta log A) log 42) 42) gves A) 43) Terefore substtutg 42) ad 43) to 6) we fd tat A)) k A) A E, 44) k )! log A) Fally, substtutg = A to 44) we obta A ) k A E, k )! log Tat s 4). Te corollary s proved. ACKNOWLEDGEMENTS. Te autor s very grateful to Uversdad Nacoal de Lujá.

12 662 R. Jakmczuk Refereces [] G. H. Hardy ad E. M. Wrgt, A troducto to te teory of umbers, Oford, 960. [2] A. Ivc, Te Rema zeta-fucto, Dover, [3] R. Jakmczuk, O te dstrbuto of certa composte umbers, Iteratoal Joural of Cotemporary Matematcal Sceces, ), [4] R. Jakmczuk, Asymptotc formulas. Composte umbers, Iteratoal Joural of Cotemporary Matematcal Sceces, 7 202), Receved: August 5, 203