A Note on Sums of Greatest (Least) Prime Factors

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1 It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, HIKARI Ltd, A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos Aires, Argetia Copyright c 203 Rafael Jakimczuk. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Let a m ( be the m-th power of the least prime factor i the prime factorizatio of. We prove the asymptotic formula a m (i m+ m +log. Let b m ( be the m-th power of the greatest prime factor i the prime factorizatio of. We prove the asymptotic formula b m (i ζ(m + m+ m + log, where ζ(s is the Riema s Zeta Fuctio. Cosequetly b m (i lim ζ(m +. a m (i I particular if m we obtai b (i π2 lim ζ(2 a (i 6. Mathematics Subect Classificatio: A4 Keywords: Sums of greatest (least prime factors

2 424 R. Jakimczuk Itroductio ad Lemmas Let m be a positive iteger ad let b m ( be the m-th power of the greatest prime factor i the prime factorizatio of. For example if 2 the b (2 3 ad b 4 (2 3 4,if 8 the b (8 3 ad b 2 (8 3 2,if 5 the b (5 5 ad b 4 ( I this ote we prove the asymptotic formula b m (i ζ(m + m+ m + log. ( If m is asymptotic formula is well-kow (see either [] or [4]. I the proof of ( we use a similar method of proof already used i the proof of other theorems (see [3]. The followig lemma is a cosequece of the prime umber theorem (see for example [2]. Lemma. Let m be a oegative iteger ad let s m (x be the sum of the m-th powers of the primes ot exceedig x. We have the followig asymptotic formula s m (x p m x m+ h(xxm+ p x (m + log x + log x, (2 where p deotes a positive prime ad h(x 0. Note that h(x depeds of m. Note that if m 0 equatio (2 becomes the Prime Number Theorem. That is, s 0 (x π(x, where π(x is the prime coutig fuctio. Let m be a positive iteger ad let a m ( be the m-th power of the least prime factor i the prime factorizatio of. For example if 2 the a (2 2 ad a 4 (2 2 4,if 8 the a (8 2 ad a 2 (8 2 2,if 5 the a (5 5 ad a 4 ( I this ote we prove the asymptotic formula m+ a m (i (m + log. (3 We also shall eed the followig lemma. Lemma.2 Let m be a positive iteger. We have the followig formula m+ ( + m+ where ζ(s is the Riema s Zeta Fuctio. ζ(m +, m ( + m+

3 Sums of greatest (least prime factors 425 Proof. We have ( m ( ( + m+ (( + m+ m+ ( + m+ ( m+ ( +. m+ ( + m+ Therefore m ( + m+ m+ ( + m+. Now The lemma is proved. ζ(m + lim m+ m+. Note that a cosequece of equatio (2 is the followig iequality x m+ s m (x p m <h p x (m + log x, (4 where h>. This iequality hods for x x 0, where x 0 deped of m. 2 Mai Results Now, we shall prove the metioed results. Namely, formulas ( ad (3. Theorem 2. We have the followig asymptotic formula where m is a arbitrary but fixed positive iteger. a m (i m+ m +log, (5 Proof. Let A(, p be the umber of positive itegers ot exceedig such that their least prime factor is the prime p. Therefore A(, p. We have a m (i 2 p 2 p p m A(, p 2 p k p m A(, p+ p m A(, p, (6 k <p

4 426 R. Jakimczuk where k 2 is a positive iteger. Cosider the first sum i (6. Namely p m A(, p. 2 p k We have the followig trivial iequality A(, p p p. Therefore (see (4 That is where 2 p k p m A(, p h(m + mk m log k log h(m + + λ mk m 2 p k 0 <g( < 2 p k p m p m+ (m + log m+ (m + log 2 p k (λ >0. m+ p m h ( k m m log k p m A(, p g( (m + log, (7 Cosider the secod iequality i (6. Namely h(m + mk m + λ (λ >0. (8 p m A(, p. k <p If is large the k<p. O the other had kp >. Cosequetly the uique multiple of p less tha or equal to such that p is its least prime factor is p. That is, we have A(, p. Therefore (see lemma. m+ (m + log + h( p m A(, p p m p m k <p 2 p 2 p k m+ m+ k (m + log k h k log k k + h( h k k m+ log k log + p k ( m+ log m+ (m + log m+ k m+ (m + log + r k( k m+ log k log m+ log ( k m+ + q k( m+ log m+ (m + log m+ (m + log m+ (m + log,

5 Sums of greatest (least prime factors 427 where h( 0, q k ( 0, p k ( 0 ad r k ( 0. That is p m A(, p k <p where r k ( 0. We have m+ (m + log m+ k m+ m+ (m + log + r k( (m + log, (9 a m (i m+ m +log + f( m+ m +log. (0 Substitutig equatios (7 ad (9 ito (6 we obtai a m (i m+ ( (m + log + k + r k(+g( m+ m+ (m + log. Cosequetly f( k + r k(+g(. ( m+ Let ɛ>0. If we choose k sufficietly large the Therefore we have (see ( k m+ < ɛ 3, r k( < ɛ 3, 0 <g( < ɛ 3. f( <ɛ, if is sufficietly large. Now, ɛ is arbitrarily little. Therefore lim f( 0. (2 Equatios (0 ad (2 give (5. The theorem is proved. Theorem 2.2 We have the followig asymptotic formula b m (i ζ(m + m+ m + log, (3 where m is a arbitrary but fixed positive iteger. Proof. Let B(, p be the umber of positive itegers ot exceedig such that their greatest prime factor is the prime p. Therefore 2 p B(, p.

6 428 R. Jakimczuk We have b m (i p m B(, p p m B(, p+ 2 p 2 p k+ + p m B(, p+ + p m B(, p k <p k + 3 <p 2 k+ <p k p m B(, p p m B(, p. (4 2 <p Cosider the first sum i (4. Namely 2 p k+ We have the followig trivial iequality B(, p p m B(, p. p p. As i theorem 2. we obtai p m m+ B(, p g( (m + log, (5 2 p k+ where h(m + 0 <g( < + λ m(k + m (λ >0. (6 Now, cosider the sum (see (4 p m B(, p (, 2,...,k. (7 + <p If is large the k<p. O the other had p ad ( +p>. Cosequetly the multiples of p less tha or equal to such that p is their greatest prime factor are p, 2p,..., p. That is, we have A(, p. Cosequetly (see (7 p m B(, p p m (, 2,...,k. (8 + <p Lemma. gives + <p + <p p m ( m+ (m + log + h ( m+ log ( m+ + (m + log ( +

7 Sums of greatest (least prime factors 429 m+ + h + log + m+ m+ log ( + log m+ log(+ (m + log log + h h m+ log + ( + log m+ log(+ log m+ log m+ ( + + q m+ ( m+ (m + log + p ( m+ log m+ m+ ( + m+ (m + log m+ + r ( (m + log, where h( 0, q ( 0, p ( 0 ad r ( 0. That is p m m+ m+ ( + m+ (m + log + r m+ ( (m + log, (9 + <p where r ( 0. Substitutig (9 ito (8 we obtai + <p p m B(, p m+ ( m ( + m+ m+ (m + log + r ( (, 2,...,k, (20 (m + log where r ( r ( 0. We have ζ(m + m+ b m (i m + log + f( m+ (m + log. (2 Substitutig (5 ad (20 ito (4 we fid that (see lemma.2 k b m (i g(+ k + r m ( + m+ ( ζ(m + m+ m + log + g( k + m ( + m+ m+ (m + log. k+ m+ (m + log r (

8 430 R. Jakimczuk Cosequetly f( g( k+ k+ + m ( + m+ Let ɛ>0. If we choose k sufficietly large the 0 < < ɛ m ( + m+ 3, 0 <g( < ɛ 3. O the other had, sice k r (. (22 r ( 0 (, 2,...,k, if is sufficietly large the we have r ( ɛ < (, 2,...,k. 3k Therefore we have (see (22 f( <ɛ. Now, ɛ is arbitrarily little. Hece lim f( 0. (23 Equatios (2 ad (23 give (3. The theorem is proved. Corollary 2.3 The followig limits hold lim b m (i a m (i I particular if m we obtai b (i lim a (i ζ(m +. ζ(2 π2 6. Proof. It is a immediate cosequece of Theorem 2. ad Theorem 2.2. The corollary is proved. Let c be a composite umber. If we cosider oly composite umbers i Corollary 2.3 the we have the followig corollary. Corollary 2.4 We have the followig limit c b m (c lim c a m (c.

9 Sums of greatest (least prime factors 43 Proof. Let p be a prime umber. We have a m (p b m (p p m. Theorefore (Theorem 2., Theorem 2.2 ad Lemma. c b m (c c a m (c ζ(m+ m+ + o m+ m+ log log o m+ log The corollary is proved. b m (i p b m (p a m (i p a m (p b m (i p p m a m (i p p m ζ(m+ m+ + o( o( Let p k be a prime power. We have a m (p k b m (p k p m. O the other had, if d is ot a prime power the a m (d <b m (d. We have the followig corollary Corollary 2.5 The followig limit holds d b m (d lim d a m (d. Proof. We have p m p m a m (i. p p k Cosequetly (Lemma. ad Theorem 2. p k. p m m+ m +log. (24 Therefore (Theorem 2., Theorem 2.2 ad equatio (24 d b m (d d a m (d ζ(m+ m+ + o m+ m+ log log o m+ log The corollary is proved. b m (i p k b m (p k a m (i p k a m (p k b m (i p k p m a m (i p k p m ζ(m+ m+ + o( o( ACKNOWLEDGEMENTS. The author is very grateful to Uiversidad Nacioal de Luá. Refereces [] K. Alladi ad P. Erdős, O a additive arithmetic fuctio, Pacific Joural of Mathematics, 7 (977,

10 432 R. Jakimczuk [2] R. Jakimczuk, A ote o sums of powers which have a fixed umber of prime factors, Joural of Iequalities i Pure ad Applied Mathematics, 6 (2005, Article 3. [3] R. Jakimczuk, Sums of prime umbers, the zeta fuctio ad the π umber, Iteratioal Mathematical Forum, 3 (2008, [4] J. Kemey, Largest prime factor, Joural of Pure ad Applied Algebra, 89 (993, Received: February 0, 203

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