Sums Over Primes II

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.


Introduction and preliminaries
It is known to all that prime numbers play an essential role in number theory. The infamous problem (known as the Prime Number Theorem) that how many primes there are up to a given point has attracted many excellent mathematicians' interests since the time of Euclid. Let π(x) denote the number of primes up to x. Gauss and Legendre proposed independently that the ratio π(x)/ x log x would approach 1 as x approaches ∞. With the help of analytic tools, Hadamard [6] and de la Vallée Poussin [2] independently and almost simultaneously proved the Prime Number Theorem, i.e., Here, and through out this paper, we use the natural logarithm (to base e). We write A(x) ∼ B(x), that is A(x) is asymptotic to B(x), which is equivalent to lim x→∞ Let p n be the sequence of prime numbers, the Prime Number Theorem can be restated as It is natural to consider asymptotic formulas for more general sums of type pn≤x p α n . An exercise in Granville's book [5] states that p≤x p ∼ x 2 2 log x . This result can be proved with the help of asymptotic formula of p n , which is equivalent to the Prime Number Theorem. In fact, we can prove asymptotic formulas for pn≤x p α n , i.e., pn≤x p α n ∼ x 1+α (1+α) log x , which is first obtained by Salát and Znám [16]. Later, Jakimczuk [8,9] extends this kind of summation to numbers with k prime factors and functions of slow increase. Gerard and Washington [4] also give accurate estimates for pn≤x p α n − x 1+α (1+α) log x by using the Prime Number Theorem with error terms.
The above results remind the author that it would be interesting to obtain asymptotic formulas for sums over primes of types pn≤x p α n f (n) m , where f (n) denotes an arithmetical function. Motivated by an exercise in Granville's book [5] and the author's recent work [12] on generalized hyperharmonic numbers H (p,r) n , the author [13] give explicit asymptotic formulas for sums over primes involving generalized hyperharmonic numbers of type pn≤x p α n (H (p,r) n ) m . Analogous results for numbers with k-prime factors have also be considered by the author [13].
We now recall the definition of numbers with k-prime factors and the hyperharmonic numbers. Let k ≥ 1 and consider a positive integer n which is the product of just k prime factors, i.e., We write τ k (x) for the number of such n ≤ x. If we impose the additional restriction that all the prime divisors p in (3) shall be different, n is squarefree. We write π k (x) for the number of these (squarefree) n ≤ x. It was proved by Landau [7,10] that For k = 1, this result would reduce to the Prime Number Theorem, if, as usual, we take 0! = 1.
The conception of hyperharmonic numbers are first introduced by Conway and Guy in their famous book [1] as From the definition of hyperharmonic numbers we can see that these numbers can be obtained by taking repeated partial sums of harmonic numbers H n . Starting from the classical generalized harmonic numbers H , respectively. They also gave some interesting factorizations and determinant properties of the matrices A n and B n . The author [12] proved that the generalized hyperharmonic numbers H (p,r) n could be written in terms of linear combinations of n's power times generalized harmonic numbers.
The conception of generalized alternating hyperharmonic numbers are introduced by the author [11] as an alternating analogue of the generalized hyperharmonic numbers H (p,r) n . Define the notion of the generalized alternating hyperharmonic numbers of types I, II, and III, respectively, as Let N 0 denote the set of nonnegative integers. If p ∈ N 0 , then H (−p) n and H (−p) n are understood to be the sum n j=1 j p and n j=1 (−1) j−1 j p , respectively. The author [11] proved that Euler sums of the generalized alternating hyperharmonic numbers of types I, II, and III could be expressed in terms of linear combinations of classical (alternating) Euler sums.
The motivation of this paper arises from an exercise in Granville's book [5] and the author's recent work [11] on generalized alternating hyperharmonic numbers of types I, II and III. This paper is a continuation of the previous paper of the author with the same title [13]. In this paper, we will derive explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with k-prime factors will also be considered.

Some notations and lemmata
We now recall some notations and lemmata which will be useful in later sections.

Lemma 3 ([12]
). For r, n, p ∈ N, we have The coefficients a(r, m, j) satisfy the following recurrence relations: and Bernoulli numbers B + n are determined by the recurrence formula or by the generating function The initial value is given by a(1, 0, 0) = 1.  and for k = 0, 1, 2, 3, b 1 (r, m, j, k) satisfy the following recurrence relations: when r is odd, when r is even, ) .
Lemma 5 ( [8,9]). Let ∞ i=1 a i and ∞ i=1 b i be two series of positive terms such that lim n→∞ Lemma 6 ( [7,13]). Let p n,k denote the nth squarefree number with just k prime factors and q n,k denote the nth number with just k prime factors. Then the following asymptotic relations hold: n log(n) (log log(n)) k−1 , p n,k (log log(p n,k )) k−1 ∼ q n,k (log log(q n,k )) k−1 ∼ (k − 1)!n log(n) .
For k = 1, we have p n ∼ n log(n) .
3 Sums over primes involving generalized alternating hyperharmonic numbers of type H (p,r,1) n Now we will provide the asymptotic formula for the generalized alternating hyperharmonic numbers of type H (p,r,1) n . Lemma 8. Let y, p ∈ N with p ≥ 2, the following asymptotic formulas hold: where ζ(s) denotes the well-known alternating zeta function Proof. From Definition 2 and Lemma 4, we have the following identities: when r = odd, when r = even, When r = odd, note that b 1 (r, m, j, 3) = 0 (m + j = 2r−(−1) r −3

4
), we know that the main term of H . From Definition 2 we can obtain the following recursive formulas: when r is odd with r ≥ 3, , when r is even, By using the initial values b 1 (1, 0, 0, 2) = 1 , b 1 (1, 0, 0, 3) = 0 , and the above recursive formulas, we can give the following explicit formulas: For y ∈ N, b 1 (2y + 1, y, 0, 2) = 1 2 y · y! , Thus we get the desired results. Now we will prove our main theorems of this section.
The other eleven asymptotic formulas can be proved in a similar manner.