On the Function ω(n)

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1 International Mathematical Forum, Vol. 3, 08, no. 3, 07 - HIKARI Ltd, On the Function ω(n Rafael Jakimczuk Diviión Matemática, Univeridad Nacional de Luján Bueno Aire, Argentina Copyright c 08 Rafael Jakimczuk. Thi article i ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Abtract Let ω(n be the number of ditinct prime in the prime factorization of n. In thi article we prove that the number n uch that ω(n i even have denity /. We alo prove another theorem on quadratfrei number. Mathematic Subject Claification: A99, B99 Keyword: Prime factorization, arithmetical function Ω(n and ω(n, Merten function, prime number theorem. Introduction and Main Reult In thi article (ee [, Chapter XXII] ω(n denote the number of ditinct prime in the prime factorization of n and Ω(n denote the total number of prime in the prime factorization of n. That i, if the prime factorization of n i q r, where the q i (i =,..., ( are the different prime and the r i (i =,..., are the multiplicitie or exponent, we have ω(n = and Ω(n = r + + r. A quadratfrei number or quarefree number i a product of ditinct prime, that i, a number uch that it prime factorization i of the form q q where the q i (i =,..., are the ditinct prime in the prime factorization. Let Q(x be the number of quarefree not exceeding x. It i well-known (ee [, Chapter XVIII]thee number have poitive denity. That i, π Q(x = x + o(x ( π

2 08 Rafael Jakimczuk Let Q p (x be the number of quarefree not exceeding x with an even number of prime factor and let Q i (x be the number of quarefree not exceeding x with a odd number of prime factor. We have Q(x = Q p (x + Q i (x = x + o(x ( π On the other hand, a it i well-known, the prime number theorem i equivalent to M(x = n x µ(n = o(x = Q p (x Q i (x (3 where M(x i the well-known Merten function and µ(n i the well-known Möbiu function. Equation ( and (3 give Q i (x = x + o(x (4 π Q p (x = x + o(x (5 π Thee two equation are alo another equivalent etablihment of the prime number theorem. Let Ω p (x be the number of poitive integer n not exceeding x uch that Ω(n i even and Ω i (x the number of poitive integer n not exceeding x uch that Ω(n i odd. The following theorem i well-known (it i a conequence of the prime number theorem. For ake of completene we give a imple proof. The author do not know if thi imple proof i well-known. Theorem. The following two aymptotic formulae hold Ω i (x = x + o(x ( Ω p (x = x + o(x (7 Proof. Let u conider the poitive integer not exceeding x uch that n i their greatet quare factor. Since Ω(n i even, we have (ee equation (5 N ( x N ( x Ω p (x = Q p + F (x = n= n n= π n + o(x + F (x = π x N n= n + o(x + F (x = x π x n>n n + o(x + F (x (8

3 On the function ω(n 09 = π Since, a it i well-known n= n that N<n ɛ. Therefore we have n 0 F (x N<n Equation (8 and (9 give Ω p (x x ɛ + π ɛ 3ɛ. Let ɛ > 0, we hall chooe N uch x ɛx (9 n That i, equation (7. Since ɛ can be arbitrarily mall. Equation ( can be proved in the ame way or by difference. The theorem i proved. Corollary. The following aymptotic formula hold. Ω p (x Ω i (x = n x( Ω(n = o(x (0 It i well known that equation ( and (7 or the equivalent equation (0 imply the prime number theorem (ee, for example, Theorem.7 in thi paper for a imple proof. Therefore equation ( and (7 or the equivalent equation (0 are equivalent to the prime number theorem, a it i well-known. Let ω p (x be the number of poitive integer n not exceeding x uch that ω(n i even and ω i (x the number of poitive integer n not exceeding x uch that ω(n i odd. A a conequence of the prime number theorem we hall prove a imilar theorem a Theorem.. However the proof i not o imple a the proof of Theorem.. The ret of the paper i dedicated to the proof of a imilar theorem a Theorem. for the function ω(n. We hall need the following well-known theorem. Theorem.3 (Incluion-excluion principlelet S be a et of N ditinct element, and let S,..., S r be arbitrary ubet of S containing N,..., N r element, repectively. For i < j <... < l r, let S ij...l be the interection of S i, S j,..., S l and let N ij...l be the number of element of S ij...l. Then the number K of element of S not in any of S,..., S r i K = N N i + N ij N ijk ( r N...r i r i<j r i<j<k r Proof. See, for example, [3] (page 84 or [] (page 33. Let q,..., q be ditinct prime. Let P q q (x be the number of poitive integer n not exceeding x uch that Ω(n i even and uch that in their prime factorization appear the prime q,..., q with odd multiplicity. On the other hand, let I q q (x be the number of poitive integer n not exceeding x uch that Ω(n i odd and uch that in their prime factorization appear the prime q,..., q with odd multiplicity. We have the following theorem.

4 0 Rafael Jakimczuk Theorem.4 The following aymptotic formulae hold. P q q (x = x + o(x ( q i + I q q (x = x + o(x ( q i + Proof. The number of poitive integer n not exceeding x uch that Ω(n i even and relatively prime with q q will be (incluion excluion principle Ω p (x ( x Ω i + ( x Ω p = i q i i<j q i q j x x i q i x + + o(x = ( qi x + o(x (3 q i q j i<j Analogouly, the number of poitive integer n not exceeding x uch that Ω(n i odd and relatively prime with q q will be ( qi x + o(x (4 Let u conider the number whoe prime factorization i of the form q r where r i (i =,..., i odd. We have ( q i qr q r = ( ( + ( q i q q 3 q q 3 = (5 q i + Let ɛ > 0. We hall chooe the number A uch that q r ɛ ( q r qr >A qr A Therefore we have (ee (3, (4, (5 and ( P q q (x = ( ( qi = x = ( qi q i + x x qr A q r ( qi x q r + o(x + F (x qr >A q r + o(x + F (x + o(x + F (x(7

5 On the function ω(n where (ee ( 0 F (x qr >A Equation (, (7 and (8 give P q q (x x q i + x q r ɛx (8 ( qi ɛ + ɛ + ɛ 3ɛ That i (, ince ɛ can be arbitrarily mall. The proof of equation ( i the ame. The theorem i proved. We hall need the following two well-known theorem. Theorem.5 (The econd Möbiu inverion formula Let f(x and g(x be function defined for x. If g(x = ( x f (x n x n then f(x = ( x µ(ng n x n (x where µ(n i the Möbiu function. Proof. See, for example, [, Chapter XVI, Theorem 8]. Theorem. The following formula hold n= µ(n n = π Proof. See, for example, [, Chapter XVII, Theorem 87 and page 45]. Let (MP q q (x be the number of quarefree n not exceeding x multiple of q q uch that Ω(n = ω(n i even. On the other hand, let (MI q q (x be the number of quarefree n not exceeding x multiple of q q uch that Ω(n = ω(n i odd. We have the following theorem. Theorem.7 The following aymptotic formulae hold. (MP q q (x = (MI q q (x = π π x + o(x (9 q i + x + o(x (0 q i +

6 Rafael Jakimczuk Proof. We have (ee Theorem.4 and [, Chapter XVIII, Theorem 333]. P q q (y = cy + o(y = cy + f(y y = ( (y (MP q q d y d where for ake of implicity we put c =. Beide lim q i + x f(x = 0 and f(x < M. By Theorem.5 and Theorem. we have (MP q q (y = ( ( ( y (y y µ(d c + f = y d y d d c d d y + y ( (y µ(d f = d y d d π cy + O(y + o(y = π q i + y + o(y µ(d d If we put y = x then we obtain equation (9. Equation (0 can be proved in the ame way. Note that y c d>y µ(d d = O(y and y ( (y µ(d f = y ( (y µ(d f d y d d d d d y ( + y (y µ(d f = o(y y<d y d d Since for all ɛ > 0 we have ( (y µ(d f d y d d (( y d f d d y (( y + f y<d y d d ɛπ + Mo( 3ɛ The theorem i proved. Let (RP P q q (x be the number of quarefree n not exceeding x, relatively prime to q q and uch that Ω(n = ω(n i even. Let (RP I q q (x be the number of quarefree n not exceeding x, relatively prime to q q and uch that Ω(n = ω(n i odd. The following theorem hold.

7 On the function ω(n 3 Theorem.8 The following aymptotic formulae hold. (RP P q q (x = (RP I q q (x = π π q i x + o(x ( q i + q i x + o(x ( q i + Proof. By the incluion-excluion principle, Theorem.7 and (5 we have (RP P q q (x = π x + o(x ( i π q i + x + o(x ( + i<j π (q i + (q j + x + o(x = ( π x + o(x q i + That i, equation (. Equation ( can be proved in the ame way. The theorem i proved. A number i powerful or quareful if all the ditinct prime in it prime factorization have multiplicity (or exponent greater than. That i, the number q r i quareful if r i (i =,..., (. Theorem.9 We have qr π q q (q + (q + q r where the um run on all quareful number q r. Proof. We have = π q q π (q + (q + qr q r = ( + p π p p + p + p p + p + 3 π = + p π p p + p p π = π π = p π The theorem i proved. p Now, we can prove our main theorem.

8 4 Rafael Jakimczuk Theorem.0 The following aymptotic formulae hold. ω p (x = x + o(x (3 ω i (x = x + o(x (4 Proof. In the proof of thi theorem q r denote a quareful number. We have (ee Theorem.8, Theorem.9 and (5 + ω p (x = π x + o(x ( q q π (q + (q + qr A = x + o(x + F (x ( x π qr >A q q (q + (q + x q r q r + o(x + F (x (5 It i well-known that the erie of the reciprocal of the quareful number converge and that the quareful number have zero denity, that i, the number of quareful number not exceeding x i o(x. Let ɛ > 0. Conequently we chooe A uch that and hence and qr >A ( π qr >A q r q q (q + (q + 0 F (x qr >A ɛ x q r q r Finally, equation (5, ( and (7 give ω p (x x ɛ + ɛ + ɛ 3ɛ ɛ ( ɛx (7 That i, equation (3, ince ɛ can be arbitrarily mall. The proof of equation (4 i the ame or by difference. The theorem i proved. In the following theorem we prove that the prime number theorem i equivalent to a propoition on a number of quarefree of poitive denity arbitrarily mall.

9 On the function ω(n 5 Theorem. Let p n be the n-th prime. Let u conider the firt n prime. The prime number theorem i equivalent to the following two formulae. (RP P p p n (x = (RP I p p n (x = n π n π p i p i + x + o(x = p i p i + x + o(x = n π n π + p i x + o(x (8 + p i x + o(x (9 Proof. Equation (8 and (9 are a particular cae of Theorem.8 and Theorem.8 i a conequence of the prime number theorem (ee the proof. Reciprocally, uppoe that equation (8 and (9 hold, then Q p (x = n p n ( i π p i + x + o(x + n p i x + o(x π p i + p i ( n p i x + + o(x + = π p i + p i p j π x + o(x i<j n That i, equation (5. In the ame way we obtain (4 or by difference. On the other hand, we have lim n ( + p i =. The theorem i proved. Theorem. Let n be a quarefree relatively prime to p p n. Then, the prime number theorem i equivalent to the following etablihment µ(n = o(x (30 Compare with equation (3. n x Proof. The prime number theorem implie (30 ince µ(n = (RP P p p n (x (RP P p p n (x = o(x (3 n x (ee Theorem.. Reciprocally, uppoe that (30 hold, without the ue of the prime number theorem can be proved (ee [] that (RP P p p n (x + (RP P p p n (x = π n p i x + o(x (3 p i + From (3 and (3 we obtain (8 and (9. Note that the n have poitive denity arbitrarily mall if n i large (Theorem.. The theorem i proved. Acknowledgement. The author i very grateful to Univeridad Nacional de Luján.

10 Rafael Jakimczuk Reference [] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Number, Oxford, 90. [] R. Jakimczuk, On the ditribution of certain ubet of quadratfrei number, International Mathematical Forum, (07, no. 4, [3] W. J. LeVeque, Topic in Number Theory, Volume, Addion-Weley, 958. Received: January, 08; Publihed: February, 08

A Note on the Distribution of Numbers with a Minimum Fixed Prime Factor with Exponent Greater than 1

International Mathematical Forum, Vol. 7, 2012, no. 13, 609-614 A Note on the Distribution of Numbers with a Minimum Fixed Prime Factor with Exponent Greater than 1 Rafael Jakimczuk División Matemática,