Asymptotic Formulas Composite Numbers III

Let k ≥ 1 and h ≥ 1 arbitrary but fixed positive integers. Let us consider the numbers such that in their prime factorization there are k primes with exponent h and the remainder of the primes have exponente greater than h. Let Pk,h(x) be the number of these numbers not exceeding x. We prove the formula Pk,h(x) ∼ Ah+1 hx 1/h(log log x)k−1 (k − 1)! log x , where Ah+1 is a constant defined in this article. Let k ≥ 1, h ≥ 1 and t ≥ 1 arbitrary but fixed positive integers. Let us consider the numbers such that in their prime factorization there are k primes with exponent h and the t primes remaining have exponent greater than h. Let Ak,h,t(x) be the number of these numbers not exceeding x. We prove the formula Ak,h,t(x) ∼ At,h+1 hx 1/h(log log x)k−1 (k − 1)! log x , where At,h+1 is a constant defined in this article. Let Et,h(x) be the number of h-ful numbers with exactly t distinct prime factors in their prime factorization. We prove the asymptotic formula Et,h(x) ∼ hx 1/h(log log x)t−1 (t − 1)! log x .

where A h+1 is a constant defined in this article.
Let k ≥ 1, h ≥ 1 and t ≥ 1 arbitrary but fixed positive integers.Let us consider the numbers such that in their prime factorization there are k primes with exponent h and the t primes remaining have exponent greater than h.Let A k,h,t (x) be the number of these numbers not exceeding x.We prove the formula where A t,h+1 is a constant defined in this article.Let E t,h (x) be the number of h-ful numbers with exactly t distinct prime factors in their prime factorization.We prove the asymptotic formula E t,h (x) ∼ hx 1/h (log log x) t−1 (t − 1)! log x .
In particular if h = 1 then we obtain the following well-known Landau's Theorem E t,1 (x) ∼ x(log log x) t−1 (t − 1)! log x , where E t,1 (x) is the number of numbers not exceeding x with exactly t distinct prime factors in their prime factorization.

Introduction, Notation and Lemmas
Let n be a number such that its prime factorization if of the form where a i ≥ h + 1 (i = 1, 2, . . ., t), (h ≥ 1) is fixed and p 1 , p 2 , . . ., p t (t ≥ 1) are the different primes in the factorization.Note that the a i (i = 1, 2, . . ., t) and t are variable.These number are well known, they are called (h + 1)-ful numbers.
There exist various studies on the distribution of these numbers using not elementary methods (see [1]).
Let C n be the sequence of (h + 1)-ful numbers and let C h+1 (x) be the number of (h + 1)-ful numbers that do not exceed x.It is well known (see [2] for an elementary proof) that where b h+1 and c h+1 are positive constants.Note that C n depends of h + 1.
For sake of simplicity we use this notation.
In this article C denotes a (h + 1)-ful number.
From (1) we can obtain without difficulty the following lemma.
Lemma 1. 1 The following series are convergent.That is, we have Let us consider the sequence P n of the numbers whose prime factorization is of the form where The number of these numbers not exceeding x we shall denote P k,h (x) In this article we prove the asymptotic formula Let us consider the sequence E n of the (h + 1)-ful numbers with t different prime factors, where t ≥ 1 is a fixed positive integer.Note that the sequence E n depends of t and h + 1.For sake of simplicity we use this notation.
We shall denote these numbers in the compact form E.
The number of these numbers not exceeding x we shall denote E t,h+1 (x).
Let us consider the sequence A n of the numbers whose prime factorization is of the form ) is fixed and p 1 , p 2 , . . ., p t+k are the different primes in the factorization.Note that the sequence A n depend of k, h and t.For sake of simplicity we use this notation.
We shall denote these numbers in the compact form Ep The number of these numbers not exceeding x we shall denote A k,h,t (x).Since in this case the E numbers are (h + 1)-ful numbers, Lemma 1.1 imply that the following series are convergent, that is In this article we prove the asymptotic formula On the other hand (2) imply that from a certain value of x we have Let π(x) be the number of primes not exceeding x.We shall need the prime number Theorem which we shall use as a lemma.

Lemma 1.2 The following formula holds
Let us consider the numbers whose prime factorization is of the form where k ≥ 2 is fixed and p 1 , p 2 , . . ., p k are different primes.Let B k (x) be the number of these numbers not exceeding x.We have the following theorem (Landau's Theorem) which we shall use as a lemma (see [1]).
Lemma 1. 3 The following asymptotic formula holds We shall also need the following two lemmas whose proofs are simple.
Lemma 1.5 The function (c > 1) is increasing from a certain value of x.Note that f (x) depends of c.
In this article we also prove the asymptotic formula (see ( 4)) In particular if h = 1 then we obtain the following well-known Landau's Theorem where E t,1 (x) is the number of numbers not exceeding x with exactly t distinct prime factors in their prime factorization.

Main Lemmas
The method of proof in the following Lemma 2.1 is similar to the method used in [4].For sake of completeness we give the proof.Note that the meaning of E is different here.
The method of proof in the following Lemma 2.2 is similar to the method used in [5].For sake of completeness we give the proof.Note that the meaning of E is different here.
Lemma 2.1 and Lemma 2.2 can be united in the following lemma.

Lemma 2.3 Let > 0.
There exists x such that if x ≥ x then we have the following inequality Lemma 2.4 Let > 0. There exists x such that if x ≥ x then we have the following inequality

Main Results
Theorem 3.1 We have the following asymptotic formula Proof.In the sums (see ( 6) and (33)) are generated undesirable numbers.The number of these undesirable numbers not exceeding x is F k (x) (k ≥ 1).Let us consider the number (we take h = 1) This number is undesirable when some primes p i appear in the prime factorization of the (73) In particular If k > t we have Equations ( 73), ( 74), ( 75), ( 67) and ( 4) give That is The theorem is proved.
If h = 1 then we obtain as corollary of Theorem 3.3 the following well-known Landau's result.
1, 2, . .., t) are variable, (h ≥ 1) is fixed, (t ≥ 1) is variable, (k ≥ 1) is fixed and p 1 , p 2 , . .., p t+k are the different primes in the factorization.Note that the sequence P n depends of k and h.For sake of simplicity we use this notation.We shall denote these numbers in the compact form Cp h

)
Proof.The proof is the same as Lemma 2.1 and Lemma 2.2.In the proofs of Lemma 2.1 and Lemma 2.2 we replace A k,h,t (x) by P k,h (x), E by C, E i by C i , E n by C n , A t,h+1 by A h+1 , E n+1 by C n+1 and E t,h+1 (x) by C h+1 (x).The lemma is proved.
Clearly we can withdraw of (71) two primes with exponent greater than 2. In contrary case we do not obtain a 2 − f ul number.The number of possible ways is then bounded by 8 2 .Therefore if k ≤ t we have 2 − f ul number with t different prime factors E.
Equations (68) and (78) give (77), since is arbitrarily small.The theorem is proved.Let us consider the h-ful numbers with exactly t distinct primes in their prime factorization.If h = 1 we obtain the numbers with exactly t distinct primes in their prime factorization.The number of these numbers not exceeding x is (see the introduction) E t,h (x).where t ≥ 1 and h ≥ 1 are fixed and p 1 , p 2 , ..., p t are different primes.Let B t,h (x) be the number of these numbers not exceeding x.We have the following asymptotic formula The proof of this formula is an immediate consequence of Lemma 1.2 and Lemma 1.3.We have (see (80), (67) and (4))E t,h (x) = B t,h (x) + A t−1,h,1 (x) + A t−2,h,2 (x) + • • • + A 1,h,t−1 (x) + E t,h+1 (x) 3heorem 3.3The following asymptotic formula holdsE t,h (x) ∼ hx 1/h (log log x) t−1 (t − 1)! log x .(79)Proof.Let us consider the numbers whose prime factorization is of the form