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\begin{center}
\vskip 1cm{\LARGE\bf Sums over Primes II
}
\vskip 1cm
\large
Rusen Li\\
School of Mathematics and Computational Science\\
Xiangtan University\\
Xiangtan 411105\\
China\\
\href{mailto:email}{\tt limanjiashe@163.com} \\
\end{center}

\vskip .2 in
\begin{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.
\end{abstract}

\section{Introduction and preliminaries}
The prime numbers (see the sequence \seqnum{A000040} in the OEIS \cite{Sloane}) play an essential role in number theory. Let $\pi (x)$ denote the number of primes up to $x$. Gauss and Legendre proposed independently that the ratio $\pi (x)/\frac{x}{\log x}$ approaches $1$ as $x$ approaches $\infty$. With the help of analytic tools, Hadamard \cite{Hadamard} and de la Vall\'{e}e Poussin \cite{Poussin} independently and almost simultaneously proved the prime number theorem, i.e.,
\begin{align*}
\pi (x)\sim\frac{x}{\log x}.  
\end{align*}

Let $p_{n}$ be the $n$-th prime number, and let $\alpha$ be a non-negative integer. It is natural to consider asymptotic formulas for more general sums of type
$\sum_{p_{n}\leq x}p_{n}^{\alpha}$. We restate the prime number theorem as
\begin{align*}
\pi (x)=\sum_{p_{n}\leq x}p_{n}^{0}\sim\frac{x}{\log x}.
\end{align*}
An exercise in Granville's book \cite{Granville} states that $\sum_{p\leq x}p\sim\frac{x^2}{2\log x}$. In fact, \v Sal\'at and Zn\'am \cite{Salat} proved more general sums $\sum_{p_{n}\leq x}p_{n}^{\alpha}\sim\frac{x^{1+\alpha}}{(1+\alpha)\log x}$. Later, Jakimczuk \cite{Jakimczuk2005,Jakimczuk2010} extended this kind of summation to numbers with $k$ prime factors and functions of slow increase. Gerard and Washington \cite{Gerard} also gave accurate estimates for $\sum_{p_{n}\leq x}p_{n}^{\alpha}-\frac{x^{1+\alpha}}{(1+\alpha)\log x}$ by using the prime number theorem with error terms.

We now recall some definitions and notation. Let $k\ge1$, and let $n$ be the product of just $k$ prime factors ($p_i$ and $p_j$ are allowed to be the same), i.e.,
\begin{align}\label{p1p2pk}
n=p_{1}p_{2}\cdots p_{k}.
\end{align}
We write $\tau_k (x)$ for the number of such $n \leq x$. If we impose the additional restriction that all the prime divisors $p$ in equation(\ref{p1p2pk}) are different, $n$ is squarefree. We write $\pi_k (x)$ for the number of these (squarefree) $n \leq x$. Landau \cite{Hardy,Landau} proved that
\begin{align*}
\pi_k (x)\sim\tau_k (x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\quad(k\geq 2).
\end{align*}
For $k=1$, this result reduces to the prime number theorem, if, as usual, we take $0!=1$.

Conway and Guy \cite{Conway} introduced the conception of hyperharmonic numbers as
$$
h_n^{(r)}:=\sum_{j=1}^n h_{j}^{(r-1)} \quad (n,r \in \mathbb N:=\{1,2,3,\ldots\})\quad\hbox{with}\quad h_n^{(1)}=H_n:=\sum_{j=1}^{n} 1/j.
$$
Dil, Mez\H o, and Cenkci \cite{Dil1} introduced the notion of generalized hyperharmonic numbers as
$$
H_n^{(p,r)}:=\sum_{j=1}^n H_{j}^{(p,r-1)} \quad (n,p,r \in \mathbb N),
$$
and studied the Euler sums of hyperharmonic numbers. {\"O}m{\"u}r and
Koparal \cite{omur} introduced the generalized hyperharmonic numbers
$H_n^{(p,r)}$ independently and almost simultaneously from a combinatorial
point of view, and defined two $n\times n$ matrices $A_n$ and $B_n$ with
$a_{i,j}=H_i^{(j,r)}$ and $b_{i,j}=H_i^{(p,j)}$, respectively. \" Om\"
ur and Koparal also gave some interesting factorizations and determinant
properties of the matrices $A_n$ and $B_n$. The author \cite{Rusen}
proved that the generalized hyperharmonic numbers $H_n^{(p,r)}$ are
linear combinations of $n$'s power times generalized harmonic numbers.

The author \cite{LiRusen} introduced the conception of generalized alternating hyperharmonic numbers $H_n^{(p,r)}$. Define the notion of the generalized alternating hyperharmonic numbers of types {\rm I, II, and III}, respectively, as
\begin{align*}
&H_n^{(p,r,1)}:=\sum_{k=1}^n (-1)^{k-1} H_{k}^{(p,r-1,1)}\quad (H_n^{(p,1,1)}=H_n^{(p)}),\\
&H_n^{(p,r,2)}:=\sum_{k=1}^n H_{k}^{(p,r-1,2)} \quad   (H_n^{(p,1,2)}=\overline{H}_n^{(p)}:=\sum_{j=1}^n (-1)^{j-1}/j^p),\\
&H_n^{(p,r,3)}:=\sum_{k=1}^n (-1)^{k-1} H_{k}^{(p,r-1,3)} \quad (H_n^{(p,1,3)}=\overline{H}_n^{(p)}).
\end{align*}
Let $\mathbb N_{0}$ denote the set of nonnegative integers. If $p \in \mathbb N_{0}$, then $H_n^{(-p)}$ and $\overline{H}_n^{(-p)}$ are the sum $\sum_{j=1}^{n} j^p$ and $\sum_{j=1}^{n} (-1)^{j-1} j^p$, respectively. The author \cite{LiRusen} proved that Euler sums of the generalized alternating hyperharmonic numbers of types {\rm 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 $\sum_{p_{n}\leq x}p_{n}^{\alpha}f(n)^m$. The author \cite{Rusen-sumsoverprimes} gave explicit asymptotic formulas for sums over primes involving generalized hyperharmonic numbers of type $\sum_{p_{n}\leq x} p_{n}^{\alpha} (H_{n}^{(p,r)})^m$. The author \cite{Rusen-sumsoverprimes} also considered analogous results for numbers with $k$ prime factors.

The motivation of this paper arises from an exercise in Granville's book \cite{Granville} and the author's recent work \cite{LiRusen} on generalized alternating hyperharmonic numbers. This paper is a continuation of the previous paper of the author with the same title \cite{Rusen-sumsoverprimes}.
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.

\section{Some notation and lemmas}
We now recall some notation and lemmas.
\begin{lemma}[\cite{mezo}]\label{hyperharmonic}
For all $n \in \mathbb N$ and a fixed order $r\geq 1$, we have
\begin{align*}
h_n^{(r)}\sim\frac{1}{(r-1)!}n^{r-1}\log (n).
\end{align*}
\end{lemma}

\begin{lemma}[\cite{Rusen-sumsoverprimes}]\label{generalizedhyperharmonic}
For $r, n, p \in \mathbb N$ with $p\geq 2$, we have
\begin{align*}
H_n^{(p,r)}\sim\frac{1}{(r-1)!}n^{r-1}\zeta(p),
\end{align*}
where $\zeta(p):=\sum_{n=1}^\infty n^{-p}$ is the Riemann zeta function.
\end{lemma}


\begin{lemma}[\cite{Rusen}]\label{lemrusen}
For $r, n, p \in \mathbb N$, we have
\begin{align*}
&H_n^{(p,r,2)}=\sum_{m=0}^{r-1} \sum_{j=0}^{r-1-m} a(r,m,j) n^{j} \overline{H}_n^{(p-m)}.
\end{align*}
The coefficients $a(r,m,j)$ satisfy the following recurrence formulas:
\begin{align*}
&a(r+1,r,0)=-\sum_{m=0}^{r-1} a(r,m,r-m-1)\frac{1}{r-m},\\
&a(r+1,m,\ell)=\sum_{j=\ell-1}^{r-1-m} \frac{a(r,m,j)}{j+1} \binom{j+1}{j-\ell+1}B_{j-\ell+1}\\
&\qquad\qquad \qquad(0\leq m \leq r-1, 1\leq \ell \leq r-m),
\end{align*}
\begin{align*}
&a(r+1,m,0)=-\sum_{y=0}^{m} \sum_{j=\max\{0, m-y-1\}}^{r-1-y}a(r,y,j)D(r,m,j,y)\quad (0\leq m \leq r-1),
\end{align*}
where
{\small
$$
D(r,m,j,y)=\sum_{\ell=\max\{0, m-y-1\}}^{j} \frac{1}{j+1} \binom{j+1}{j-\ell}B_{j-\ell}\binom{\ell+1}{m-y}(-1)^{1+\ell-m+y}.
$$
}
The Bernoulli numbers $B_n$ satisfy the following recurrence formula
$$
\sum_{j=0}^k\binom{k+1}{j}B_j=k+1\quad (k\ge 0).
$$
The initial value is $a(1,0,0)=1$.
\end{lemma}


\begin{definition}\label{def1}
For $m, j \in \mathbb N_{0}$, define the quantities $c(m,j)$, $d(m,j)$, $c_{1}(m,j)$, and $d_{1}(m,j)$ as
\begin{align*}
&c(m,j)=\frac{1}{m+1} \binom{m+1}{m+1-j}B_{m+1-j}, \\
&d(m,j)=\frac{1}{m+1} \sum_{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)} \sum_{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)= \sum_{k=j}^{m}\binom{k}{j}(-1)^{k-j}c_{1}(m,k).
\end{align*}
\end{definition}


\begin{definition}\label{def2}
Let $g(r):=(2r-(-1)^{r}-3)/{4}$. For $r \in \mathbb N$, define the boundary values of the quantities $b_{1}(r,m,j,k)$, $k=0, 1, 2, 3$ as
\begin{itemize}
    \item $b_{1}(1,0,0,2)=1, \quad b_{1}(1,0,0,3)=0$;
    \item $b_{1}(r,m,j,0)=b_{1}(r,m,j,1)=0 \quad (\hbox{r odd})$;
    \item $b_{1}(r,m,j,2)=b_{1}(r,m,j,3)=0 \quad (\hbox{r even})$;
    \item $b_{1}(r,m,j,3)=0 \quad (\hbox{r odd}, \quad m+j=g(r))$.
\end{itemize}
For $k=0,1,2,3$, the quantities $b_{1}(r,m,j,k)$ satisfy the following recurrence formulas:

When $r$ is odd,
\begin{itemize}
    \item $\displaystyle b_{1}(r+1,m,j,0)=\sum_{\ell=m}^{g(r)}b_{1}(r,\ell,j,2) c_{1}(\ell,m)\quad (1 \le m \le g(r), \quad 0 \le j \le g(r)-m)$;
    \item $\displaystyle b_{1}(r+1,0,j,0)=\sum_{\ell=0}^{g(r)-j}b_{1}(r,\ell,j,2) c_{1}(\ell,0)\quad (0 \le j \le g(r))$;
    \item $\displaystyle b_{1}(r+1,m,j,1)=\sum_{\ell=m-1}^{g(r)-1}b_{1}(r,\ell,j,3) c(\ell,m)\quad (1 \le m \le g(r), \quad 0 \le j \le g(r)-m)$;
    \item $\displaystyle b_{1}(r+1,0,j,1)=\sum_{m=0}^{g(r)} \sum_{\substack{j_{1}+\ell=j\\ 0 \le j_{1} \le g(r)-m\\ 1 \le \ell \le m }}b_{1}(r,m,j_{1},2) d_{1}(m,\ell)
        +\sum_{m=0}^{g(r)-j}b_{1}(r,m,j,2) d_{1}(m,0)+b_{1}(r,0,j,3)-\sum_{m=0}^{g(r)-1} \sum_{\substack{j_{1}+\ell=j\\ 0 \le j_{1} \le g(r)-m-1\\ 1 \le \ell \le m+1 }}b_{1}(r,m,j_{1},3) d(m,\ell)\quad (0 \le j \le g(r))$.
\end{itemize}

When $r$ is even,
\begin{itemize}
    \item $\displaystyle b_{1}(r+1,m,j,2)=\sum_{\ell=m-1}^{g(r)}b_{1}(r,\ell,j,0) c(\ell,m)\quad (1 \le m \le g(r)+1, \quad 0 \le j \le g(r)+1-m)$;
    \item $\displaystyle b_{1}(r+1,0,j,2)=-\sum_{m=0}^{g(r)} \sum_{\substack{j_{1}+\ell=j\\ 0 \le j_{1} \le g(r)-m\\ 1 \le \ell \le m+1 }}b_{1}(r,m,j_{1},0) d(m,\ell)
+\sum_{m=0}^{g(r)-j}b_{1}(r,m,j,1) d_{1}(m,0)+b_{1}(r,0,j,0)+\sum_{m=0}^{g(r)} \sum_{\substack{j_{1}+\ell=j\\ 0 \le j_{1} \le g(r)-m\\ 1 \le \ell \le m }}b_{1}(r,m,j_{1},1) d_{1}(m,\ell)\quad (0 \le j \le g(r)+1)$;
    \item $\displaystyle b_{1}(r+1,m,j,3)=\sum_{\ell=m}^{g(r)}b_{1}(r,\ell,j,1) c_{1}(\ell,m)\quad (1 \le m \le g(r), \quad 0 \le j \le g(r)-m)$;
    \item $\displaystyle b_{1}(r+1,0,j,3)=\sum_{\ell=0}^{g(r)-j}b_{1}(r,\ell,j,1) c_{1}(\ell,0)\quad (0 \le j \le g(r))$.
\end{itemize}
\end{definition}

\begin{lemma}[\cite{LiRusen}]\label{lemhnpr1}
For $r, n, p \in \mathbb N$, we have
\begin{align*}
H_n^{(p,r,1)}
&=\sum_{m=0}^{\frac{2r-(-1)^{r}-3}{4}}
\sum_{j=0}^{\frac{2r-(-1)^{r}-3}{4}-m}\biggl(b_{1}(r,j,m,0)(-1)^{n-1}H_n^{(p-m)}
+b_{1}(r,j,m,1)\overline{H}_n^{(p-m)}\\
&\quad+b_{1}(r,j,m,2)H_n^{(p-m)}
+b_{1}(r,j,m,3)(-1)^{n-1}\overline{H}_n^{(p-m)}\biggr)n^{j},
\end{align*}
\begin{align*}
H_n^{(p,r,3)}
&=\sum_{m=0}^{\frac{2r-(-1)^{r}-3}{4}}
\sum_{j=0}^{\frac{2r-(-1)^{r}-3}{4}-m}\biggl(b_{1}(r,j,m,0)(-1)^{n-1}\overline{H}_n^{(p-m)}
+b_{1}(r,j,m,1)H_n^{(p-m)}\\
&\quad+b_{1}(r,j,m,2)\overline{H}_n^{(p-m)}
+b_{1}(r,j,m,3)(-1)^{n-1}H_n^{(p-m)}\biggr)n^{j}.
\end{align*}
\end{lemma}

\begin{lemma}[\cite{Jakimczuk2005,Jakimczuk2010}]\label{lemJakimczuk2005}
Let $\sum_{i=1}^{\infty} a_{i}$ and $\sum_{i=1}^{\infty} b_{i}$ be two series of positive terms such that $a_{i}\sim b_{i}$. Then if $\sum_{i=1}^{\infty} b_{i}$ is divergent, the following result holds:
\begin{align*}
\sum_{i=1}^{n} a_{i}\sim\sum_{i=1}^{n} b_{i}.
\end{align*}
\end{lemma}

\begin{lemma}[\cite{Hardy,Rusen-sumsoverprimes}]\label{lemHardy}
Let $p_{n,k}$ denote the $n$th squarefree number with just $k$ prime factors and $q_{n,k}$ denote the $n$th number with just $k$ prime factors. Then the following asymptotic formulas hold:
\begin{align*}
&p_{n,k}\sim q_{n,k}\sim(k-1)!\frac{n\log(n)}{(\log\log(n))^{k-1}},\\
&p_{n,k}(\log\log(p_{n,k}))^{k-1}\sim q_{n,k}(\log\log(q_{n,k}))^{k-1}\sim(k-1)!n\log(n).
\end{align*}
For $k=1$, we have $p_{n}\sim n\log(n)$.
\end{lemma}

\begin{lemma}[\cite{Rusen-sumsoverprimes}]\label{lemRusen-sumsoverprimes}
For $m, n, k, x \in \mathbb N$, we have
\begin{align*}
&\sum_{\ell=1}^{x}\ell^{m}(\log(\ell))^{n}\sim\frac{x^{m+1}(\log(x))^{n}}{m+1},\\
&\sum_{\ell=1}^{x}\frac{\ell^{m}(\log(\ell))^{n}}{(\log\log(\ell))^{k}}
\sim\frac{x^{m+1}(\log(x))^{n}}{(m+1)(\log\log(x))^{k}}.
\end{align*}
\end{lemma}


\section{Sums over primes involving generalized alternating hyperharmonic numbers of type \texorpdfstring{$H_n^{(p,r,1)}$}{Hn{(p,r,1)}}}
Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type $H_n^{(p,r,1)}$.
\begin{lemma}\label{generalizedalternatinghyperharmonicpr1}
Let $y, p \in \mathbb N$ with $p\geq 2$, the following asymptotic formulas hold:
\begin{align*}
&H_n^{(1,2y+1,1)}\sim \frac{1}{2^y \cdot y!} n^y \log(n),\quad
H_n^{(p,2y+1,1)}\sim \frac{1}{2^y \cdot y!} n^y \zeta(p),\\
&H_n^{(1,2y,1)}\sim \frac{1}{2^y \cdot (y-1)!} n^{y-1} (-1)^{n-1} \log(n),\\
&H_{2n}^{(p,2y,1)}\sim -\frac{1}{2 \cdot (y-1)!} n^{y-1} (\zeta(p)-\overline{\zeta}(p)),\\
&H_{2n-1}^{(p,2y,1)}\sim \frac{1}{2 \cdot (y-1)!} n^{y-1} (\zeta(p)+\overline{\zeta}(p)),
\end{align*}
where $\overline{\zeta}(s)$ is the well-known alternating zeta function
$$
\overline{\zeta}(s):=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^{s}}=(1-2^{1-s})\zeta(s) \quad \hbox{with} \quad
\overline{\zeta}(1)=\log 2.
$$
\end{lemma}
\begin{proof}
By applying Definition \ref{def2} and Lemma \ref{lemhnpr1}, we have the following identities:\\
when $r$ is odd,
\begin{align*}
&H_n^{(p,r,1)}
=\sum_{m=0}^{g(r)}
\sum_{j=0}^{g(r)-m}\biggl(b_{1}(r,j,m,2)H_n^{(p-m)}
+b_{1}(r,j,m,3)(-1)^{n-1}\overline{H}_n^{(p-m)}\biggr)n^{j};
\end{align*}
when $r$ is even,
\begin{align*}
&H_n^{(p,r,1)}
=\sum_{m=0}^{g(r)}
\sum_{j=0}^{g(r)-m}\biggl(b_{1}(r,j,m,0)(-1)^{n-1}H_n^{(p-m)}
+b_{1}(r,j,m,1)\overline{H}_n^{(p-m)}\biggr)n^{j}.
\end{align*}

When $r$ is odd, by $b_{1}(r,m,j,3)=0 \quad (m+j=g(r))$, 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 $p\geq 2$, we know that the main term of $H_n^{(p,r,1)}$ is $$\biggl(b_{1}(r,g(r),0,0)(-1)^{n-1}H_n^{(p-m)}
+b_{1}(r,g(r),0,1)\overline{H}_n^{(p)}\biggr) n^{g(r)}.$$

By applying Definition \ref{def2}, we can obtain the following recursive formulas:

When $r$ is odd with $r\geq 3$,
\begin{align*}
&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)}.
\end{align*}

When $r$ is even,
\begin{align*}
&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}.
\end{align*}

Let $y\in \mathbb N$. 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:
\begin{align*}
&b_{1}(2y+1,y,0,2)=\frac{1}{2^y \cdot y!},\\
&b_{1}(2y+1,y-1,0,3)=\frac{1}{2^{y+1} \cdot (y-1)!},\\
&b_{1}(2y,y-1,0,0)=b_{1}(2y,y-1,0,1)=\frac{1}{2^{y} \cdot (y-1)!}.
\end{align*}
Thus we get the desired results.
\end{proof}

Now we state our main theorems of this section.
\begin{theorem}\label{maintheoremhnpr11}
For $\alpha, m, q, y \in \mathbb N$ with $q\geq 2$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}\sim\frac{x^{\alpha+my+1}(\log(x))^{\alpha+m}}{(2^y \cdot y!)^{m}(\alpha+my+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}\sim\frac{x^{\alpha+my+1}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(q,2y+1,1)})^{m}\sim\frac{\zeta(q)^{m}x^{\alpha+my+1}(\log(x))^{\alpha}}{(2^y \cdot y!)^{m}(\alpha+my+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_\ell^{(q,2y+1,1)})^{m}\sim\frac{\zeta(q)^{m} x^{\alpha+my+1}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log(x))^{my+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} ((-1)^{\ell-1}H_\ell^{(1,2y,1)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} ((-1)^{\ell-1}H_\ell^{(1,2y,1)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(1,2y,1)})^{m}\sim \frac{x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (-H_{2\ell}^{(1,2y,1)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(1,2y,1)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (-H_{2\ell}^{(1,2y,1)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(q,2y,1)})^{m}\sim\frac{(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(q,2y,1)})^{m}\sim\frac{(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (-H_{2\ell}^{(q,2y,1)})^{m}\sim\frac{(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (-H_{2\ell}^{(q,2y,1)})^{m}\sim\frac{(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonicpr1}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}\sim
\sum_{\ell\leq x}\frac{\ell^{\alpha+my}(\log(\ell))^{\alpha+m}}
{(2^y \cdot y!)^{m}}
\sim\frac{x^{\alpha+my+1}(\log(x))^{\alpha+m}}
{(2^y \cdot y!)^{m}(\alpha+my+1)}.
\end{align*}
We can prove thirteen additional asymptotic formulas in a similar manner.
\end{proof}

\begin{theorem}\label{maintheoremhnpr12}
For $\alpha, m, k, q, y \in \mathbb N$ with $q\geq2$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}\sim\frac{((k-1)!)^{\alpha} x^{\alpha+my+1}(\log(x))^{\alpha+m}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}\sim\frac{x^{\alpha+my+1}(\log\log(x))^{(my+1)(k-1)}}{(2^y \cdot y!)^{m}((k-1)!)^{my+1}(\alpha+my+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{q,2y+1,1)})^{m}\sim\frac{((k-1)!)^{\alpha} \zeta(q)^{m} x^{\alpha+my+1}(\log(x))^{\alpha}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q,2y+1,1)})^{m}\sim\frac{\zeta(q)^{m} x^{\alpha+my+1}(\log\log(x))^{(my+1)(k-1)}}{(2^y \cdot y!)^{m}((k-1)!)^{my+1}(\alpha+my+1)(\log(x))^{my+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} ((-1)^{\ell-1}H_\ell^{1,2y,1)})^{m}\sim\frac{((k-1)!)^{\alpha} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} ((-1)^{\ell-1} H_\ell^{(1,2y,1)})^{m}\sim\frac{ x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2^y \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{1,2y,1)})^{m}\sim \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (-H_{2\ell}^{(1,2y,1)})^{m}\\ \sim\frac{((k-1)!)^{\alpha} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{(1,2y,1)})^{m}\sim \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (-H_{2\ell}^{(1,2y,1)})^{m}\\ \sim\frac{ x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2 \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{(q,2y,1)})^{m}\sim\frac{((k-1)!)^{\alpha}(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{(q,2y,1)})^{m}\sim\frac{ (\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2 \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (-H_{2\ell}^{(q,2y,1)})^{m}\sim \frac{((k-1)!)^{\alpha}(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (-H_{2\ell}^{(q,2y,1)})^{m}\sim\frac{ (\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2 \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonicpr1}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(1,2y+1,1)})^{m}
\sim
\sum_{\ell\leq x}\frac{((k-1)!)^{\alpha} \ell^{\alpha+my}(\log(\ell))^{\alpha+m}}
{(2^y \cdot y!)^{m}(\log\log(\ell))^{\alpha(k-1)}}\\
&\sim\frac{((k-1)!)^{\alpha} x^{\alpha+my+1}(\log(x))^{\alpha+m}}
{(2^y \cdot y!)^{m}(\alpha+my+1)(\log\log(x))^{\alpha(k-1)}}.
\end{align*}
We can prove eleven additional asymptotic formulas in a similar manner.
\end{proof}


\section{Sums over primes involving generalized alternating hyperharmonic numbers of type \texorpdfstring{$H_n^{(p,r,2)}$}{Hn{(p,r,2)}}}
Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type $H_n^{(p,r,2)}$.
\begin{lemma}\label{generalizedalternatinghyperharmonic}
For $r, n, p \in \mathbb N$, we have
\begin{align*}
H_n^{(p,r,2)}\sim\frac{1}{(r-1)!}n^{r-1}\overline{\zeta}(p).
\end{align*}
\end{lemma}
\begin{proof}
By using Lemma \ref{lemrusen}, 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 \cite{Rusen-sumsoverprimes} proves that  $a(r,0,r-1)=\frac{1}{(r-1)!}$ and $\overline{H}_n^{(p)}\sim\overline{\zeta}(p)$. Thus we get the desired result.
\end{proof}


Now we show our main theorems of this section.
\begin{theorem}\label{maintheoremhnr21}
For $\alpha, m, q, k, r \in \mathbb N$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(q,r,2)})^{m}\sim\frac{\overline{\zeta}(q)^{m}x^{\alpha+m(r-1)+1}(\log(x))^{\alpha}}{((r-1)!)^{m}(\alpha+m(r-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_\ell^{(q,r,2)})^{m}\sim\frac{\overline{\zeta}(q)^{m}x^{\alpha+m(r-1)+1}}{((r-1)!)^{m}(\alpha+m(r-1)+1)(\log(x))^{m(r-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q,r,2)})^{m}\sim\frac{((k-1)!)^{\alpha}\overline{\zeta}(q)^{m} x^{\alpha+m(r-1)+1}(\log(x))^{\alpha}}{((r-1)!)^{m}(\alpha+m(r-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q,r,2)})^{m}\sim\frac{\overline{\zeta}(q)^{m} x^{\alpha+m(r-1)+1}(\log\log(x))^{(m(r-1)+1)(k-1)}}{((k-1)!)^{m(r-1)+1}((r-1)!)^{m}}\\
        \times\frac{1}{(\alpha+m(r-1)+1)(\log(x))^{m(r-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonic}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q,r,2)})^{m}\sim
\sum_{\ell\leq x}\frac{((k-1)!)^{\alpha}\overline{\zeta}(q)^{m}\ell^{\alpha+m(r-1)}(\log(\ell))^{\alpha}}
{((r-1)!)^{m}(\log\log(\ell))^{\alpha(k-1)}}\\
&\sim\frac{((k-1)!)^{\alpha}\overline{\zeta}(q)^{m} x^{\alpha+m(r-1)+1}(\log(x))^{\alpha}}
{((r-1)!)^{m}(\alpha+m(r-1)+1)(\log\log(x))^{\alpha(k-1)}}.
\end{align*}
We can prove three other asymptotic formulas in a similar manner.
\end{proof}

\begin{theorem}\label{maintheoremhnr22}
For $q_{1}, q_{2}, \alpha, \beta, m, k, s, n, r_{1}, r_{2} \in \mathbb N$ with $q_{1}\geq2$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q_{1},r_{1})})^{m} (h_\ell^{(s)})^{n} (H_\ell^{(q_{2},r_{2},2)})^{\beta} \sim\frac{((k-1)!)^{\alpha} \zeta(q_{1})^{m}\overline{\zeta}(q_{2})^{\beta} (\log(x))^{\alpha+n}}{((r_{1}-1)!)^{m}((s-1)!)^{n}((r_{2}-1)!)^{\beta}}\\ \times\frac{x^{\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1} }{(\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q_{1},r_{1})})^{m} (h_\ell^{(s)})^{n} (H_\ell^{(q_{2},r_{2},2)})^{\beta} \sim\frac{\zeta(q_{1})^{m}\overline{\zeta}(q_{2})^{\beta} }{(\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1)}\\ \times\frac{x^{\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1} (\log\log(x))^{(m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1)(k-1)} }{((r_{1}-1)!)^{m}((s-1)!)^{n}((r_{2}-1)!)^{\beta}((k-1)!)^{m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1}}\\ \times\frac{1}{(\log(x))^{m(r_{1}-1)+n(s-2)+\beta(r_{2}-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{hyperharmonic}, \ref{generalizedhyperharmonic}, \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonic}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(q_{1},r_{1})})^{m} (h_\ell^{(s)})^{n} (H_\ell^{(q_{2},r_{2},2)})^{\beta}\\
&\sim
\sum_{\ell\leq x}\frac{((k-1)!)^{\alpha} \zeta(q_{1})^{m}\overline{\zeta}(q_{2})^{\beta}\ell^{\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)}(\log(\ell))^{\alpha+n}}
{((r_{1}-1)!)^{m}((s-1)!)^{n}((r_{2}-1)!)^{\beta}(\log\log(\ell))^{\alpha(k-1)}}\\
&\sim
\frac{((k-1)!)^{\alpha} \zeta(q_{1})^{m}\overline{\zeta}(q_{2})^{\beta} (\log(x))^{\alpha+n}}
{((r_{1}-1)!)^{m}((s-1)!)^{n}((r_{2}-1)!)^{\beta}(\log\log(x))^{\alpha(k-1)}}\\
&\times\frac{x^{\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1} }{(\alpha+m(r_{1}-1)+n(s-1)+\beta(r_{2}-1)+1)}.
\end{align*}
We can prove the other asymptotic formula in a similar manner.
\end{proof}


\section{Sums over primes involving generalized alternating hyperharmonic numbers of type \texorpdfstring{$H_n^{(p,r,3)}$}{Hn{(p,r,3)}}}
Now we provide the asymptotic formulas for the generalized alternating hyperharmonic numbers of type $H_n^{(p,r,3)}$.
\begin{lemma}\label{generalizedalternatinghyperharmonicpr3}
Let $y, p \in \mathbb N$, the following asymptotic formulas hold:
\begin{itemize}
    \item $\displaystyle H_n^{(p,2y+1,3)}\sim \frac{1}{2^y \cdot y!} n^y \overline{\zeta}(p)$;
    \item $\displaystyle H_n^{(1,2y,3)}\sim \frac{1}{2^y \cdot (y-1)!} n^{y-1} \log(n)$;
    \item $\displaystyle H_{2n}^{(p,2y,3)}\sim \frac{1}{2 \cdot (y-1)!} n^{y-1} (\zeta(p)-\overline{\zeta}(p))\quad (p\geq 2)$;
    \item $\displaystyle H_{2n-1}^{(p,2y,3)}\sim \frac{1}{2 \cdot (y-1)!} n^{y-1} (\zeta(p)+\overline{\zeta}(p))\quad (p\geq 2)$.
\end{itemize}
\end{lemma}
\begin{proof}
By applying Definition \ref{def2} and Lemma \ref{lemhnpr1}, we have the following identities:\\
when $r$ is odd,
\begin{align*}
&H_n^{(p,r,3)}
=\sum_{m=0}^{g(r)}
\sum_{j=0}^{g(r)-m}\biggl(b_{1}(r,j,m,2)\overline{H}_n^{(p-m)}
+b_{1}(r,j,m,3)(-1)^{n-1}H_n^{(p-m)}\biggr)n^{j};
\end{align*}
when $r$ is even,
\begin{align*}
&H_n^{(p,r,1)}
=\sum_{m=0}^{g(r)}
\sum_{j=0}^{g(r)-m}\biggl(b_{1}(r,j,m,0)(-1)^{n-1}\overline{H}_n^{(p-m)}
+b_{1}(r,j,m,1)H_n^{(p-m)}\biggr)n^{j}.
\end{align*}


When $r$ is odd, by $b_{1}(r,m,j,3)=0 \quad (m+j=g(r))$, 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 $p\geq 2$, we know that the main term of $H_n^{(p,r,3)}$ is $$\biggl(b_{1}(r,g(r),0,0)(-1)^{n-1}\overline{H}_n^{(p-m)}
+b_{1}(r,g(r),0,1)H_n^{(p)}\biggr) n^{g(r)}.$$

Let $y\in \mathbb N$. By applying Lemma \ref{generalizedalternatinghyperharmonicpr1}, we have the following explicit formulas:
\begin{align*}
&b_{1}(2y+1,y,0,2)=\frac{1}{2^y \cdot y!},\\
&b_{1}(2y+1,y-1,0,3)=\frac{1}{2^{y+1} \cdot (y-1)!},\\
&b_{1}(2y,y-1,0,0)=b_{1}(2y,y-1,0,1)=\frac{1}{2^{y} \cdot (y-1)!}.
\end{align*}
Thus we get the desired results.
\end{proof}

Now we show our main theorems of this section.
\begin{theorem}\label{maintheoremhnpr31}
For $\alpha, m, p, q, y \in \mathbb N$ with $q\geq 2$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(p,2y+1,3)})^{m}\sim\frac{\overline{\zeta}(p)^{m} x^{\alpha+my+1}(\log(x))^{\alpha}}{(2^y \cdot y!)^{m}(\alpha+my+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_\ell^{(p,2y+1,3)})^{m}\sim\frac{\overline{\zeta}(p)^{m} x^{\alpha+my+1}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log(x))^{my+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(1,2y,3)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_\ell^{(1,2y,3)})^{m}\sim\frac{x^{\alpha+m(y-1)+1}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(q,2y,3)})^{m}\sim\frac{(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_{2\ell-1}^{(q,2y,3)})^{m}\sim\frac{(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell}^{\alpha} (H_{2\ell}^{(q,2y,3)})^{m}\sim\frac{(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)}$;
    \item $\displaystyle \sum_{p_{\ell}\leq x} p_{\ell}^{\alpha} (H_{2\ell}^{(q,2y,3)})^{m}\sim\frac{(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonicpr3}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell}^{\alpha} (H_\ell^{(p,2y+1,3)})^{m}\sim
\sum_{\ell\leq x}\frac{\overline{\zeta}(p)^{m}\ell^{\alpha+my}(\log(\ell))^{\alpha}}
{(2^y \cdot y!)^{m}}
\sim\frac{\overline{\zeta}(p)^{m} x^{\alpha+my+1}(\log(x))^{\alpha}}
{(2^y \cdot y!)^{m}(\alpha+my+1)}.
\end{align*}
We can prove seven other asymptotic formulas in a similar manner.
\end{proof}

\begin{theorem}\label{maintheoremhnpr32}
For $\alpha, m, k, p, q, y \in \mathbb N$ with $q\geq2$, we have
\begin{itemize}
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{p,2y+1,1)})^{m}\sim\frac{((k-1)!)^{\alpha} \overline{\zeta}(p)^{m} x^{\alpha+my+1}(\log(x))^{\alpha}}{(2^y \cdot y!)^{m}(\alpha+my+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(p,2y+1,1)})^{m}\sim\frac{\overline{\zeta}(p)^{m} x^{\alpha+my+1}(\log\log(x))^{(my+1)(k-1)}}{(2^y \cdot y!)^{m}((k-1)!)^{my+1}(\alpha+my+1)(\log(x))^{my+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{1,2y,3)})^{m}\sim\frac{((k-1)!)^{\alpha} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha+m}}{(2^y \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} ( H_\ell^{(1,2y,3)})^{m}\sim\frac{ x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2^y \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-2)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{(q,2y,3)})^{m}\sim\frac{((k-1)!)^{\alpha}(\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_{2\ell-1}^{(q,2y,3)})^{m}\sim\frac{ (\zeta(q)+\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2 \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$;
    \item $\displaystyle \sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_{2\ell}^{(q,2y,3)})^{m}\sim\frac{((k-1)!)^{\alpha}(\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log(x))^{\alpha}}{(2 \cdot (y-1)!)^{m}(\alpha+m(y-1)+1)(\log\log(x))^{\alpha(k-1)}}$;
    \item $\displaystyle \sum_{p_{\ell,k}\leq x} p_{\ell,k}^{\alpha} (H_{2\ell}^{(q,2y,3)})^{m}\sim\frac{ (\zeta(q)-\overline{\zeta}(q))^{m} x^{\alpha+m(y-1)+1}(\log\log(x))^{(m(y-1)+1)(k-1)}}{(2 \cdot (y-1)!)^{m}((k-1)!)^{m(y-1)+1}}\\ \times\frac{1}{(\alpha+m(y-1)+1)(\log(x))^{m(y-1)+1}}$.
\end{itemize}
\end{theorem}
\begin{proof}
By using Lemmas \ref{lemJakimczuk2005}, \ref{lemHardy}, \ref{lemRusen-sumsoverprimes}, and \ref{generalizedalternatinghyperharmonicpr3}, we have
\begin{align*}
&\sum_{\ell\leq x} p_{\ell,k}^{\alpha} (H_\ell^{(p,2y+1,3)})^{m}
\sim
\sum_{\ell\leq x}\frac{((k-1)!)^{\alpha} \overline{\zeta}(p)^{m}\ell^{\alpha+my}(\log(\ell))^{\alpha}}
{(2^y \cdot y!)^{m}(\log\log(\ell))^{\alpha(k-1)}}\\
&\sim\frac{((k-1)!)^{\alpha} \overline{\zeta}(p)^{m} x^{\alpha+my+1}(\log(x))^{\alpha}}
{(2^y \cdot y!)^{m}(\alpha+my+1)(\log\log(x))^{\alpha(k-1)}}.
\end{align*}
We can prove seven other asymptotic formulas in a similar manner.
\end{proof}

\section{Acknowledgment}
The author would like to thank the anonymous referee for many helpful comments.

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\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 11L20, 11N25, 11N37.

\noindent \emph{Keywords:} sum over primes,
generalized alternating hyperharmonic number,
asymptotic formula, number with $k$ prime factors.

\bigskip
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\noindent (Concerned with sequence
\seqnum{A000040}.)

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\vspace*{+.1in}
\noindent
Received January 7 2024;
revised versions received March 11 2024; March 12 2024.
Published in {\it Journal of Integer Sequences}, March 17 2024.

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