Sums over Primes II

Rusen Li School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China

                      limanjiashe@163.com

Abstract In this paper, we give explicit asymptotic formulas for some sums over primes involving three types of generalized alternating hyperharmonic numbers. We also consider analogous results for numbers with k prime factors.

1 Introduction and preliminaries

The prime numbers (see the sequence \underline{A000040} in the OEIS [16]) play an essential role in number theory. Let π(x) denote the number of primes up to x. Gauss and Legendre proposed independently that the ratio π(x)/\frac{x}{log x} approaches 1 as x approaches ∞. With the help of analytic tools, Hadamard [5] and de la Vallée Poussin [17] independently and almost simultaneously proved the prime number theorem, i.e.,

π(x) ∼ \frac{x}{log x}.

Let p_{n} be the n-th prime number, and let α be a non-negative integer. It is natural to consider asymptotic formulas for more general sums of type ∑_{p_{n}≤x} p_{n}^{α}. We restate the prime number theorem as

π(x) = ∑_{p_{n}≤x} p_{n}^{0} ∼ \frac{x}{log x}.

An exercise in Granville’s book [4] states that ∑_{p≤x} p ∼ \frac{x^{2}}{2 log x}. In fact, Šalát and Znám [15] proved more general sums ∑_{p_{n}≤x} p_{n}^{α} ∼ \frac{x^{1+α}}{(1+α) log x}. Later, Jakimczuk [7, 8] extended this kind of summation to numbers with k prime factors and functions of slow increase. Gerard and Washington [3] also gave accurate estimates for ∑_{p_{n}≤x} p_{n}^{α} − \frac{x^{1+α}}{(1+α) log x} by using the prime number theorem with error terms.

We now recall some definitions and notation. Let k ≥ 1, and let n be the product of just k prime factors (p_{i} and p_{j} are allowed to be the same), i.e.,

n = p_{1}p_{2} ⋯ p_{k}.  (1)

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 equation(1) are different, n is squarefree. We write π_{k}(x) for the number of these (squarefree) n ≤ x. Landau [6, 9] proved that

π_{k}(x) ∼ τ_{k}(x) ∼ \frac{x(log log x)^{k−1}}{(k − 1)! log x} (k ≥ 2).

For k = 1, this result reduces to the prime number theorem, if, as usual, we take 0! = 1. Conway and Guy [1] introduced the conception of hyperharmonic numbers as

h_{n}^{(r)} := ∑_{j=1}^{n} h_{j}^{(r−1)} (n, r ∈ ℕ := {1, 2, 3, …}) with h_{n}^{(1)} = H_{n} := ∑_{j=1}^{n} 1/j.

Dil, Mező, and Cenkci [2] introduced the notion of generalized hyperharmonic numbers as

H_{n}^{(p,r)} := ∑_{j=1}^{n} H_{j}^{(p,r−1)} (n, p, r ∈ ℕ),

and studied the Euler sums of hyperharmonic numbers. Ömür and Koparal [14] introduced the generalized hyperharmonic numbers H_{n}^{(p,r)} independently and almost simultaneously from a combinatorial point of view, and defined two n × n matrices A_{n} and B_{n} with a_{i,j} = H_{i}^{(j,r)} and b_{i,j} = H_{i}^{(p,j)}, respectively. Ömür and Koparal 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_{n}^{(p,r)} are linear combinations of n’s power times generalized harmonic numbers.

The author [10] introduced the conception of generalized alternating hyperharmonic numbers H_{n}^{(p,r)}. Define the notion of the generalized alternating hyperharmonic numbers of types I, II, and III, respectively, as

H_{n}^{(p,r,1)} := ∑_{k=1}^{n}(−1)^{k−1}H_{k}^{(p,r−1,1)} (H_{n}^{(p,1,1)} = H_{n}^{(p)}),

H_{n}^{(p,r,2)} := ∑_{k=1}^{n} H_{k}^{(p,r−1,2)} (H_{n}^{(p,1,2)} = \overline{H}_{n}^{(p)} := ∑_{j=1}^{n}(−1)^{j−1}/j^{p}),

H_{n}^{(p,r,3)} := ∑_{k=1}^{n}(−1)^{k−1}H_{k}^{(p,r−1,3)} (H_{n}^{(p,1,3)} = \overline{H}_{n}^{(p)}).

Let ℕ_{0} denote the set of nonnegative integers. If p ∈ ℕ_{0}, then H_{n}^{(−p)} and \overline{H}_{n}^{(−p)} are the sum ∑_{j=1}^{n} j^{p} and ∑_{j=1}^{n}(−1)^{j−1}j^{p}, respectively. The author [10] proved that Euler sums of the generalized alternating hyperharmonic numbers of types I, II, and III are linear combinations of classical (alternating) Euler sums.

Let f(n) denote an arithmetical function. It is interesting to consider asymptotic formulas for sums over primes of type ∑_{p_{n}≤x} p_{n}^{α}f(n)^{m}. The author [11] gave explicit asymptotic formulas for sums over primes involving generalized hyperharmonic numbers of type ∑_{p_{n}≤x} p_{n}^{α}(H_{n}^{(p,r)})^{m}. The author [11] also considered analogous results for numbers with k prime factors.

The motivation of this paper arises from an exercise in Granville’s book [4] and the author’s recent work [10] on generalized alternating hyperharmonic numbers. This paper is a continuation of the previous paper of the author with the same title [11]. In this paper, we derive explicit asymptotic formulas for some sums over primes involving three types of generalized alternating hyperharmonic numbers. We also consider analogous results for numbers with k prime factors.

2 Some notation and lemmas

We now recall some notation and lemmas.

Lemma 1 ([13]). For all n ∈ ℕ and a fixed order r ≥ 1, we have

h_{n}^{(r)} ∼ \frac{1}{(r − 1)!}n^{r−1} log(n).

Lemma 2 ([11]). For r, n, p ∈ ℕ with p ≥ 2, we have

H_{n}^{(p,r)} ∼ \frac{1}{(r − 1)!}n^{r−1}ζ(p),

where ζ(p) := ∑_{n=1}^{∞} n^{−p} is the Riemann zeta function.

Lemma 3 ([12]). For r, n, p ∈ ℕ, we have

H_{n}^{(p,r,2)} = ∑_{m=0}^{r−1} ∑_{j=0}^{r−1−m} a(r, m, j)n^{j}\overline{H}_{n}^{(p−m)}.

The coefficients a(r, m, j) satisfy the following recurrence formulas:

a(r + 1, r, 0) = − ∑_{m=0}^{r−1} a(r, m, r − m − 1)\frac{1}{r − m},

a(r + 1, m, ℓ) = ∑_{j=ℓ−1}^{r−1−m} \frac{a(r, m, j)}{j + 1}\binom{j + 1}{j − ℓ + 1}B_{j−ℓ+1}

(0 ≤ m ≤ r − 1, 1 ≤ ℓ ≤ r − m),

a(r + 1, m, 0) = − ∑_{y=0}^{m} ∑_{j=max{0,m−y−1}}^{r−1−y} a(r, y, j)D(r, m, j, y) (0 ≤ m ≤ r − 1),

where

D(r, m, j, y) = ∑_{ℓ=max\{0,m−y−1\}}^{j} \frac{1}{j + 1}\binom{j + 1}{j − ℓ}B_{j−ℓ}\binom{ℓ + 1}{m − y}(−1)^{1+ℓ−m+y}.

The Bernoulli numbers B_{n} satisfy the following recurrence formula

∑_{j=0}^{k} \binom{k + 1}{j}B_{j} = k + 1 (k ≥ 0).

The initial value is a(1, 0, 0) = 1. Definition 4. For m, j ∈ ℕ_{0}, define the quantities c(m, j), d(m, j), c_{1}(m, j), and d_{1}(m, j) as

c(m, j) = \frac{1}{m + 1}\binom{m + 1}{m + 1 − j}B_{m+1−j},

d(m, j) = \frac{1}{m + 1} ∑_{k=j−1}^{m} \binom{m + 1}{m − k}B_{m−k}\binom{1 + k}{j}(−1)^{1+k−j},

c_{1}(m, j) = \frac{1}{2(m + 1)} ∑_{k=0}^{m−j} \binom{m + 1}{k}B_{k}2^{k}\binom{m + 1 − k}{j}(−1)^{m−k−j},

d_{1}(m, j) = ∑_{k=j}^{m} \binom{k}{j}(−1)^{k−j}c_{1}(m, k).

Definition 5. Let g(r) := (2r − (−1)^{r} − 3)/4. For r ∈ ℕ, define the boundary values of the quantities b_{1}(r, m, j, k), k = 0, 1, 2, 3 as

• b_{1}(1, 0, 0, 2) = 1, b_{1}(1, 0, 0, 3) = 0;

• b_{1}(r, m, j, 0) = b_{1}(r, m, j, 1) = 0 (r odd);

• b_{1}(r, m, j, 2) = b_{1}(r, m, j, 3) = 0 (r even);

• b_{1}(r, m, j, 3) = 0 (r odd, m + j = g(r)). For k = 0, 1, 2, 3, the quantities b_{1}(r, m, j, k) satisfy the following recurrence formulas:

When r is odd,

• b_{1}(r + 1, m, j, 0) = ∑_{ℓ=m}^{g(r)} b_{1}(r, ℓ, j, 2)c_{1}(ℓ, m) (1 ≤ m ≤ g(r), 0 ≤ j ≤ g(r) − m);

• b_{1}(r + 1, 0, j, 0) = ∑_{ℓ=0}^{g(r)−j} b_{1}(r, ℓ, j, 2)c_{1}(ℓ, 0) (0 ≤ j ≤ g(r));

• b_{1}(r + 1, m, j, 1) = ∑_{ℓ=m−1}^{g(r)−1} b_{1}(r, ℓ, j, 3)c(ℓ, m) (1 ≤ m ≤ g(r), 0 ≤ j ≤ g(r) − m);

• b_{1}(r + 1, 0, j, 1) = ∑_{m=0}^{g(r)} ∑_{\begin{matrix}j_{1}+ℓ=j \\ 0≤j_{1}≤g(r)−m \\ 1≤ℓ≤m \end{matrix}} b_{1}(r, m, j_{1}, 2)d_{1}(m, ℓ) + ∑_{m=0}^{g(r)−j} b_{1}(r, m, j, 2)d_{1}(m, 0) +

b_{1}(r, 0, j, 3) − ∑_{m=0}^{g(r)−1} ∑_{\begin{matrix}j_{1}+ℓ=j \\ 0≤j_{1}≤g(r)−m−1 \\ 1≤ℓ≤m+1 \end{matrix}} b_{1}(r, m, j_{1}, 3)d(m, ℓ) (0 ≤ j ≤ g(r)).

When r is even,

• b_{1}(r+1, m, j, 2) = ∑_{ℓ=m−1}^{g(r)} b_{1}(r, ℓ, j, 0)c(ℓ, m) (1 ≤ m ≤ g(r)+1, 0 ≤ j ≤ g(r)+1−m);

• b_{1}(r + 1, 0, j, 2) = − ∑_{m=0}^{g(r)} ∑_{\begin{matrix}j_{1}+ℓ=j \\ 0≤j_{1}≤g(r)−m \\ 1≤ℓ≤m+1 \end{matrix}} b_{1}(r, m, j_{1}, 0)d(m, ℓ) + ∑_{m=0}^{g(r)−j} b_{1}(r, m, j, 1)d_{1}(m, 0) +

b_{1}(r, 0, j, 0) + ∑_{m=0}^{g(r)} ∑_{\begin{matrix}j_{1}+ℓ=j \\ 0≤j_{1}≤g(r)−m \\ 1≤ℓ≤m \end{matrix}} b_{1}(r, m, j_{1}, 1)d_{1}(m, ℓ) (0 ≤ j ≤ g(r) + 1);

• b_{1}(r + 1, m, j, 3) = ∑_{ℓ=m}^{g(r)} b_{1}(r, ℓ, j, 1)c_{1}(ℓ, m) (1 ≤ m ≤ g(r), 0 ≤ j ≤ g(r) − m);

• b_{1}(r + 1, 0, j, 3) = ∑_{ℓ=0}^{g(r)−j} b_{1}(r, ℓ, j, 1)c_{1}(ℓ, 0) (0 ≤ j ≤ g(r)).

Lemma 6 ([10]). For r, n, p ∈ ℕ, we have

H_{n}^{(p,r,1)} = ∑_{m=0}^{\frac{2r−(−1)^{r}−3}{4}} ∑_{j=0}^{\frac{2r−(−1)^{r}−3}{4} −m}(b_{1}(r, j, m, 0)(−1)^{n−1}H_{n}^{(p−m)} + b_{1}(r, j, m, 1)\overline{H}_{n}^{(p−m)}
+ b_{1}(r, j, m, 2)H_{n}^{(p−m)} + b_{1}(r, j, m, 3)(−1)^{n−1}\overline{H}_{n}^{(p−m)})n^{j},

H_{n}^{(p,r,3)} = ∑_{m=0}^{\frac{2r−(−1)^{r}−3}{4}} ∑_{j=0}^{\frac{2r−(−1)^{r}−3}{4} −m}(b_{1}(r, j, m, 0)(−1)^{n−1}\overline{H}_{n}^{(p−m)} + b_{1}(r, j, m, 1)H_{n}^{(p−m)}

+ b_{1}(r, j, m, 2)\overline{H}_{n}^{(p−m)} + b_{1}(r, j, m, 3)(−1)^{n−1}H_{n}^{(p−m)})n^{j}.

Lemma 7 ([7, 8]). Let ∑_{i=1}^{∞} a_{i} and ∑_{i=1}^{∞} b_{i} be two series of positive terms such that a_{i} ∼ b_{i}. Then if ∑_{i=1}^{∞} b_{i} is divergent, the following result holds:

∑_{i=1}^{n} a_{i} ∼ ∑_{i=1}^{n} b_{i}.

Lemma 8 ([6, 11]). 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 formulas hold:

p_{n,k} ∼ q_{n,k} ∼ (k − 1)!\frac{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). Lemma 9 ([11]). For m, n, k, x ∈ ℕ, we have

∑_{ℓ=1}^{x} ℓ^{m}(log(ℓ))^{n} ∼ \frac{x^{m+1}(log(x))^{n}}{m + 1} ,

∑_{ℓ=1}^{x} \frac{ℓ^{m}(log(ℓ))^{n}}{(log log(ℓ))^{k}} ∼ \frac{x^{m+1}(log(x))^{n}}{(m + 1)(log log(x))^{k}} .

3 Sums over primes involving generalized alternating hyperharmonic numbers of type H_{n}^{(p,r,1)}

Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type H_{n}^{(p,r,1)}. Lemma 10. Let y, p ∈ ℕ with p ≥ 2, the following asymptotic formulas hold:

H_{n}^{(1,2y+1,1)} ∼ \frac{1}{2^{y} · y!}n^{y} log(n), H_{n}^{(p,2y+1,1)} ∼ \frac{1}{2^{y} · y!}n^{y}ζ(p),

H_{n}^{(1,2y,1)} ∼ \frac{1}{2^{y} · (y − 1)!}n^{y−1}(−1)^{n−1} log(n),

H_{2n}^{(p,2y,1)} ∼ −\frac{1}{2 · (y − 1)!}n^{y−1}(ζ(p) − \overline{ζ}(p)),

H_{2n−1}^{(p,2y,1)} ∼ \frac{1}{2 · (y − 1)!}n^{y−1}(ζ(p) + \overline{ζ}(p)),

where \overline{ζ}(s) is the well-known alternating zeta function

\overline{ζ}(s) := ∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n^{s}} = (1 − 2^{1−s})ζ(s) with \overline{ζ}(1) = log 2.

Proof. By applying Definition 5 and Lemma 6, we have the following identities: when r is odd,

H_{n}^{(p,r,1)} = ∑_{m=0}^{g(r)} ∑_{j=0}^{g(r)−m}(b_{1}(r, j, m, 2)H_{n}^{(p−m)} + b_{1}(r, j, m, 3)(−1)^{n−1}\overline{H}_{n}^{(p−m)})n^{j};

when r is even,

H_{n}^{(p,r,1)} = ∑_{m=0}^{g(r)} ∑_{j=0}^{g(r)−m}(b_{1}(r, j, m, 0)(−1)^{n−1}H_{n}^{(p−m)} + b_{1}(r, j, m, 1)\overline{H}_{n}^{(p−m)})n^{j}.

When r is odd, by b_{1}(r, m, j, 3) = 0 (m + j = g(r)), we know that the main term of H_{n}^{(p,r,1)} is b_{1}(r, g(r), 0, 2)H_{n}^{(p)}n^{g(r)}.

When r is even and p = 1, we know that the main term of H_{n}^{(1,r,1)} is

b_{1}(r, g(r), 0, 0)(−1)^{n−1}H_{n}n^{g(r)}.

When r is even and p ≥ 2, we know that the main term of H_{n}^{(p,r,1)} is

(b_{1}(r, g(r), 0, 0)(−1)^{n−1}H_{n}^{(p−m)} + b_{1}(r, g(r), 0, 1)\overline{H}_{n}^{(p)})n^{g(r)}.

By applying Definition 5, we can obtain the following recursive formulas: When r is odd with r ≥ 3,

b_{1}(r + 1, g(r + 1), 0, 0) = b_{1}(r, g(r), 0, 2)\frac{1}{2},

b_{1}(r + 1, g(r + 1), 0, 1) = b_{1}(r, g(r) − 1, 0, 3)\frac{1}{g(r)}.

When r is even,

b_{1}(r + 1, g(r + 1), 0, 2) = b_{1}(r, g(r), 0, 0)\frac{1}{g(r) + 1},

b_{1}(r + 1, g(r + 1) − 1, 0, 3) = b_{1}(r, g(r) − 1, 0, 1)\frac{1}{2}.

Let y ∈ ℕ. By using the initial values b_{1}(1, 0, 0, 2) = 1 and b_{1}(1, 0, 0, 3) = 0, and the above recursive formulas, we can obtain the following explicit formulas:

b_{1}(2y + 1, y, 0, 2) = \frac{1}{2^{y} · y!},

b_{1}(2y + 1, y − 1, 0, 3) = \frac{1}{2^{y+1} · (y − 1)!},

b_{1}(2y, y − 1, 0, 0) = b_{1}(2y, y − 1, 0, 1) = \frac{1}{2^{y} · (y − 1)!}.

Thus we get the desired results.

Now we state our main theorems of this section. Theorem 11. For α, m, q, y ∈ ℕ with q ≥ 2, we have

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ \frac{x^{α+my+1}(log(x))^{α+m}}{(2^{y} · y!)^{m}(α + my + 1)};

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ \frac{x^{α+my+1}}{(2^{y} · y!)^{m}(α + my + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(q,2y+1,1)})^{m} ∼ \frac{ζ(q)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{ℓ}^{(q,2y+1,1)})^{m} ∼ \frac{ζ(q)^{m}x^{α+my+1}}{(2^{y} · y!)^{m}(α + my + 1)(log(x))^{my+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}((−1)^{ℓ−1}H_{ℓ}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log(x))^{α+m}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)};

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}((−1)^{ℓ−1}H_{ℓ}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log(x))^{α+m}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)};

• ∑_{ℓ≤x} p_{ℓ}^{α}(−H_{2ℓ}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log(x))^{α+m}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)};

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(−H_{2ℓ}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(−H_{2ℓ}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(−H_{2ℓ}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} .

Proof. By using Lemmas 7, 8, 9, and 10, we have

∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ ∑_{ℓ≤x} \frac{ℓ^{α+my}(log(ℓ))^{α+m}}{(2^{y} · y!)^{m}} ∼ \frac{x^{α+my+1}(log(x))^{α+m}}{(2^{y} · y!)^{m}(α + my + 1)}.

We can prove thirteen additional asymptotic formulas in a similar manner. Theorem 12. For α, m, k, q, y ∈ ℕ with q ≥ 2, we have

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ \frac{((k − 1)!)^{α}x^{α+my+1}(log(x))^{α+m}}{(2^{y} · y!)^{m}(α + my + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ \frac{x^{α+my+1}(log log(x))^{(my+1)(k−1)}}{(2^{y} · y!)^{m}((k − 1)!)^{my+1}(α + my + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{q,2y+1,1)})^{m} ∼ \frac{((k − 1)!)^{α}ζ(q)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q,2y+1,1)})^{m} ∼ \frac{ζ(q)^{m}x^{α+my+1}(log log(x))^{(my+1)(k−1)}}{(2^{y} · y!)^{m}((k − 1)!)^{my+1}(α + my + 1)(log(x))^{my+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}((−1)^{ℓ−1}H_{ℓ}^{1,2y,1)})^{m} ∼ \frac{((k − 1)!)^{α}x^{α+m(y−1)+1}(log(x))^{α+m}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}((−1)^{ℓ−1}H_{ℓ}^{(1,2y,1)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2^{y} · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{1,2y,1)})^{m} ∼ ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(−H_{2ℓ}^{(1,2y,1)})^{m}

∼ \frac{((k − 1)!)^{α}x^{α+m(y−1)+1}(log(x))^{α+m}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{(1,2y,1)})^{m} ∼ ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(−H_{2ℓ}^{(1,2y,1)})^{m}

∼ \frac{x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2 · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{(q,2y,1)})^{m} ∼ \frac{((k − 1)!)^{α}(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2 · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(−H_{2ℓ}^{(q,2y,1)})^{m} ∼ \frac{((k − 1)!)^{α}(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(−H_{2ℓ}^{(q,2y,1)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2 · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} .

Proof. By using Lemmas 7, 8, 9, and 10, we have

∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(1,2y+1,1)})^{m} ∼ ∑_{ℓ≤x} \frac{((k − 1)!)^{α}ℓ^{α+my}(log(ℓ))^{α+m}}{(2^{y} · y!)^{m}(log log(ℓ))^{α(k−1)}}

∼ \frac{((k − 1)!)^{α}x^{α+my+1}(log(x))^{α+m}}{(2^{y} · y!)^{m}(α + my + 1)(log log(x))^{α(k−1)}} .

We can prove eleven additional asymptotic formulas in a similar manner.

4 Sums over primes involving generalized alternating

hyperharmonic numbers of type H_{n}^{(p,r,2)}

Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type H_{n}^{(p,r,2)}.

Lemma 13. For r, n, p ∈ ℕ, we have

H_{n}^{(p,r,2)} ∼ \frac{1}{(r − 1)!}n^{r−1}\overline{ζ}(p).

Proof. By using Lemma 3, we know that the main term of H_{n}^{(p,r,2)} is a(r, 0, r − 1)n^{r−1}\overline{H}_{n}^{(p)}. The author [11] proves that a(r, 0, r − 1) = \frac{1}{(r−1)!} and \overline{H}_{n}^{(p)} ∼ \overline{ζ}(p). Thus we get the desired result.

Now we show our main theorems of this section. Theorem 14. For α, m, q, k, r ∈ ℕ, we have

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(q,r,2)})^{m} ∼ \frac{\overline{ζ}(q)^{m}x^{α+m(r−1)+1}(log(x))^{α}}{((r − 1)!)^{m}(α + m(r − 1) + 1)};

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{ℓ}^{(q,r,2)})^{m} ∼ \frac{\overline{ζ}(q)^{m}x^{α+m(r−1)+1}}{((r − 1)!)^{m}(α + m(r − 1) + 1)(log(x))^{m(r−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q,r,2)})^{m} ∼ \frac{((k − 1)!)^{α}\overline{ζ}(q)^{m}x^{α+m(r−1)+1}(log(x))^{α}}{((r − 1)!)^{m}(α + m(r − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q,r,2)})^{m} ∼ \frac{\overline{ζ}(q)^{m}x^{α+m(r−1)+1}(log log(x))^{(m(r−1)+1)(k−1)}}{((k − 1)!)^{m(r−1)+1}((r − 1)!)^{m}}

× \frac{1}{(α + m(r − 1) + 1)(log(x))^{m(r−1)+1}} .

Proof. By using Lemmas 7, 8, 9, and 13, we have

∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q,r,2)})^{m} ∼ ∑_{ℓ≤x} \frac{((k − 1)!)^{α}\overline{ζ}(q)^{m}ℓ^{α+m(r−1)}(log(ℓ))^{α}}{((r − 1)!)^{m}(log log(ℓ))^{α(k−1)}}

∼ \frac{((k − 1)!)^{α}\overline{ζ}(q)^{m}x^{α+m(r−1)+1}(log(x))^{α}}{((r − 1)!)^{m}(α + m(r − 1) + 1)(log log(x))^{α(k−1)}} .

We can prove three other asymptotic formulas in a similar manner. Theorem 15. For q_{1}, q_{2}, α, β, m, k, s, n, r_{1}, r_{2} ∈ ℕ with q_{1} ≥ 2, we have

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q_{1},r_{1})})^{m}(h_{ℓ}^{(s)})^{n}(H_{ℓ}^{(q_{2},r_{2},2)})^{β} ∼ \frac{((k − 1)!)^{α}ζ(q_{1})^{m}\overline{ζ}(q_{2})^{β}(log(x))^{α+n}}{((r_{1} − 1)!)^{m}((s − 1)!)^{n}((r_{2} − 1)!)^{β}}

× \frac{x^{α+m(r_{1}−1)+n(s−1)+β(r_{2}−1)+1}}{(α + m(r_{1} − 1) + n(s − 1) + β(r_{2} − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q_{1},r_{1})})^{m}(h_{ℓ}^{(s)})^{n}(H_{ℓ}^{(q_{2},r_{2},2)})^{β} ∼ \frac{ζ(q_{1})^{m}\overline{ζ}(q_{2})^{β}}{(α + m(r_{1} − 1) + n(s − 1) + β(r_{2} − 1) + 1)}

× \frac{x^{α+m(r_{1}−1)+n(s−1)+β(r_{2}−1)+1}(log log(x))^{(m(r_{1}−1)+n(s−1)+β(r_{2}−1)+1)(k−1)}}{((r_{1} − 1)!)^{m}((s − 1)!)^{n}((r_{2} − 1)!)^{β}((k − 1)!)^{m(r_{1}−1)+n(s−1)+β(r_{2}−1)+1}}

× \frac{1}{(log(x))^{m(r_{1}−1)+n(s−2)+β(r_{2}−1)+1}} .

Proof. By using Lemmas 1, 2, 7, 8, 9, and 13, we have

∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(q_{1},r_{1})})^{m}(h_{ℓ}^{(s)})^{n}(H_{ℓ}^{(q_{2},r_{2},2)})^{β}

∼ ∑_{ℓ≤x} \frac{((k − 1)!)^{α}ζ(q_{1})^{m}\overline{ζ}(q_{2})^{β}ℓ^{α+m(r_{1}−1)+n(s−1)+β(r_{2}−1)}(log(ℓ))^{α+n}}{((r_{1} − 1)!)^{m}((s − 1)!)^{n}((r_{2} − 1)!)^{β}(log log(ℓ))^{α(k−1)}}

∼ \frac{((k − 1)!)^{α}ζ(q_{1})^{m}\overline{ζ}(q_{2})^{β}(log(x))^{α+n}}{((r_{1} − 1)!)^{m}((s − 1)!)^{n}((r_{2} − 1)!)^{β}(log log(x))^{α(k−1)}}

× \frac{x^{α+m(r_{1}−1)+n(s−1)+β(r_{2}−1)+1}}{(α + m(r_{1} − 1) + n(s − 1) + β(r_{2} − 1) + 1)}.

We can prove the other asymptotic formula in a similar manner.

5 Sums over primes involving generalized alternating

hyperharmonic numbers of type H_{n}^{(p,r,3)}

Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type H_{n}^{(p,r,3)}. Lemma 16. Let y, p ∈ ℕ, the following asymptotic formulas hold:

• H_{n}^{(p,2y+1,3)} ∼ \frac{1}{2^{y} · y!}n^{y}\overline{ζ}(p);

• H_{n}^{(1,2y,3)} ∼ \frac{1}{2^{y} · (y − 1)!}n^{y−1} log(n);

• H_{2n}^{(p,2y,3)} ∼ \frac{1}{2 · (y − 1)!}n^{y−1}(ζ(p) − \overline{ζ}(p)) (p ≥ 2);

• H_{2n−1}^{(p,2y,3)} ∼ \frac{1}{2 · (y − 1)!}n^{y−1}(ζ(p) + \overline{ζ}(p)) (p ≥ 2).

Proof. By applying Definition 5 and Lemma 6, we have the following identities: when r is odd,

H_{n}^{(p,r,3)} = ∑_{m=0}^{g(r)} ∑_{j=0}^{g(r)−m}(b_{1}(r, j, m, 2)\overline{H}_{n}^{(p−m)} + b_{1}(r, j, m, 3)(−1)^{n−1}H_{n}^{(p−m)})n^{j};

when r is even,

H_{n}^{(p,r,1)} = ∑_{m=0}^{g(r)} ∑_{j=0}^{g(r)−m}(b_{1}(r, j, m, 0)(−1)^{n−1}\overline{H}_{n}^{(p−m)} + b_{1}(r, j, m, 1)H_{n}^{(p−m)})n^{j}.

When r is odd, by b_{1}(r, m, j, 3) = 0 (m + j = g(r)), we know that the main term of H_{n}^{(p,r,3)} is b_{1}(r, g(r), 0, 2)\overline{H}_{n}^{(p)}n^{g(r)}.

When r is even and p = 1, we know that the main term of H_{n}^{(1,r,3)} is b_{1}(r, g(r), 0, 1)H_{n}n^{g(r)}. When r is even and p ≥ 2, we know that the main term of H_{n}^{(p,r,3)} is

(b_{1}(r, g(r), 0, 0)(−1)^{n−1}\overline{H}_{n}^{(p−m)} + b_{1}(r, g(r), 0, 1)H_{n}^{(p)})n^{g(r)}.

Let y ∈ ℕ. By applying Lemma 10, we have the following explicit formulas:

b_{1}(2y + 1, y, 0, 2) = \frac{1}{2^{y} · y!},

b_{1}(2y + 1, y − 1, 0, 3) = \frac{1}{2^{y+1} · (y − 1)!},

b_{1}(2y, y − 1, 0, 0) = b_{1}(2y, y − 1, 0, 1) = \frac{1}{2^{y} · (y − 1)!}.

Thus we get the desired results.

Now we show our main theorems of this section.

Theorem 17. For α, m, p, q, y ∈ ℕ with q ≥ 2, we have

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(p,2y+1,3)})^{m} ∼ \frac{\overline{ζ}(p)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{ℓ}^{(p,2y+1,3)})^{m} ∼ \frac{\overline{ζ}(p)^{m}x^{α+my+1}}{(2^{y} · y!)^{m}(α + my + 1)(log(x))^{my+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(1,2y,3)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log(x))^{α+m}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)};

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{ℓ}^{(1,2y,3)})^{m} ∼ \frac{x^{α+m(y−1)+1}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{2ℓ−1}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ}^{α}(H_{2ℓ}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)} ;

• ∑_{p_{ℓ}≤x} p_{ℓ}^{α}(H_{2ℓ}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} .

Proof. By using Lemmas 7, 8, 9, and 16, we have

∑_{ℓ≤x} p_{ℓ}^{α}(H_{ℓ}^{(p,2y+1,3)})^{m} ∼ ∑_{ℓ≤x} \frac{\overline{ζ}(p)^{m}ℓ^{α+my}(log(ℓ))^{α}}{(2^{y} · y!)^{m}} ∼ \frac{\overline{ζ}(p)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)} .

We can prove seven other asymptotic formulas in a similar manner. Theorem 18. For α, m, k, p, q, y ∈ ℕ with q ≥ 2, we have

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{p,2y+1,1)})^{m} ∼ \frac{((k − 1)!)^{α}\overline{ζ}(p)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(p,2y+1,1)})^{m} ∼ \frac{\overline{ζ}(p)^{m}x^{α+my+1}(log log(x))^{(my+1)(k−1)}}{(2^{y} · y!)^{m}((k − 1)!)^{my+1}(α + my + 1)(log(x))^{my+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{1,2y,3)})^{m} ∼ \frac{((k − 1)!)^{α}x^{α+m(y−1)+1}(log(x))^{α+m}}{(2^{y} · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(1,2y,3)})^{m} ∼ \frac{x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2^{y} · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−2)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{(q,2y,3)})^{m} ∼ \frac{((k − 1)!)^{α}(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{2ℓ−1}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) + \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2 · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} ;

• ∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{2ℓ}^{(q,2y,3)})^{m} ∼ \frac{((k − 1)!)^{α}(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log(x))^{α}}{(2 · (y − 1)!)^{m}(α + m(y − 1) + 1)(log log(x))^{α(k−1)}} ;

• ∑_{p_{ℓ,k}≤x} p_{ℓ,k}^{α}(H_{2ℓ}^{(q,2y,3)})^{m} ∼ \frac{(ζ(q) − \overline{ζ}(q))^{m}x^{α+m(y−1)+1}(log log(x))^{(m(y−1)+1)(k−1)}}{(2 · (y − 1)!)^{m}((k − 1)!)^{m(y−1)+1}}

× \frac{1}{(α + m(y − 1) + 1)(log(x))^{m(y−1)+1}} .

Proof. By using Lemmas 7, 8, 9, and 16, we have

∑_{ℓ≤x} p_{ℓ,k}^{α}(H_{ℓ}^{(p,2y+1,3)})^{m} ∼ ∑_{ℓ≤x} \frac{((k − 1)!)^{α}\overline{ζ}(p)^{m}ℓ^{α+my}(log(ℓ))^{α}}{(2^{y} · y!)^{m}(log log(ℓ))^{α(k−1)}}

∼ \frac{((k − 1)!)^{α}\overline{ζ}(p)^{m}x^{α+my+1}(log(x))^{α}}{(2^{y} · y!)^{m}(α + my + 1)(log log(x))^{α(k−1)}} .

We can prove seven other asymptotic formulas in a similar manner.

6 Acknowledgment

The author would like to thank the anonymous referee for many helpful comments.

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2020 Mathematics Subject Classification: Primary 11B83; Secondary 11L20, 11N25, 11N37. Keywords: sum over primes, generalized alternating hyperharmonic number, asymptotic formula, number with k prime factors.

(Concerned with sequence \underline{A000040}.)

Received January 7 2024; revised versions received March 11 2024; March 12 2024. Published in Journal of Integer Sequences, March 17 2024.

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