1 Introduction

Let n be a natural number, and let $\omega (n)$ count its number of distinct prime divisors. In [Reference Erdős and Kac5], Erdős and Kac used probabilistic means to establish that $\omega (n)$ obeys the Gaussian distribution with mean $\log \log n$ and variance $\log \log n$ over natural numbers. In particular, for any $\gamma \in \mathbb {R}$ , they proved

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{x} \left| \left\{ n \leq x \ : \ n \geq 3, \ \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \leq \gamma \right\} \right| = \Phi(\gamma),\end{align*} $$

where

(1.1) $$ \begin{align} \Phi(\gamma) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\gamma e^{-u^2/2} \ du, \end{align} $$

and where $|{\mathcal S}|$ denotes the cardinality of the set ${\mathcal S}$ . Following their work, various approaches to the Erdős–Kac theorem have been pursued. For example, in [Reference Murty, Murty and Pujahari13], Murty, Murty, and Pujahari proved an all-purpose Erdős–Kac theorem which applies to diverse settings. In [Reference Das, Kuo and Liu1], the authors provided a generalization of the Erdős–Kac theorem in number fields. Moreover, the third author, in [Reference Liu10], provided a generalization of the Erdős–Kac theorem over any countably generated abelian monoid. We extend this work to provide another generalization of the Erdős–Kac theorem over any subset of such abelian monoids satisfying some additional conditions.

Let ${\mathcal P}$ be a countable set of elements with a map

$$ \begin{align*}N: {\mathcal P} \rightarrow \mathbb{Z}_{>1}, \quad {\mathfrak p} \mapsto N({\mathfrak p}).\end{align*} $$

We call this map the Norm map. Let ${\mathcal M}$ be a free abelian monoid generated by elements of ${\mathcal P}$ . For each ${\mathfrak m} \in {\mathcal M}$ , we write

$$ \begin{align*}{\mathfrak m} = \sum_{{\mathfrak p} \in {\mathcal P}} n_{\mathfrak p}({\mathfrak m}) {\mathfrak p},\end{align*} $$

with $n_{\mathfrak p}({\mathfrak m}) \in \mathbb {Z}_{>0} \cup \{ 0 \}$ and $n_{\mathfrak p}({\mathfrak m}) =0$ for all but finitely many ${\mathfrak p}$ . We extend the map N to ${\mathcal M}$ as the following:

$$ \begin{align*} N : {\mathcal M} & \rightarrow \mathbb{Z}_{>0} \\ {\mathfrak m} = \sum_{{\mathfrak p} \in {\mathcal P}} n_{\mathfrak p}({\mathfrak m}) {\mathfrak p} & \longmapsto N({\mathfrak m}) := \prod_{{\mathfrak p} \in {\mathcal P}} N({\mathfrak p})^{n_{\mathfrak p}({\mathfrak m})}. \end{align*} $$

Thus, N can be extended to a monoid homomorphism from $({\mathcal M},+)$ to $(\mathbb {Z}_{>0},\cdot )$ . Let X be a countable subset of $\mathbb {Q}$ that contains the image $\text {Im}(N({\mathcal M}))$ with an extra condition: if $x_1,x_2 \in X$ , the fraction $x_1/x_2$ belongs to X, too. Without loss of generality, we assume $X = \mathbb {Q}$ or $X = \{ q^z : z \in \mathbb {Z} \}$ for some $q \in \mathbb {Z}_{>0}$ . For interested readers, the details behind X being limited to these two choices are presented in [Reference Liu10, Theorem 2].

Given ${\mathcal P}$ , ${\mathcal M}$ , X, and for sufficiently large $x \in X$ , we assume that the following condition holds:

(⋆) $$ \begin{align} {\mathcal M}(x) := \sum\limits_{\substack{{\mathfrak m} \in {\mathcal M} \\ N({\mathfrak m}) \leq x}} 1 = \kappa x + O(x^\theta), \quad \text{ for some } \kappa> 0 \text{ and } 0 \leq \theta < 1. \end{align} $$

Let $\mathcal {{\mathcal S}}$ be a subset of infinitely many elements in ${\mathcal M}$ . For $x \in X$ , $x>1$ , we define

$$ \begin{align*}{\mathcal S}(x) = \{ {\mathfrak m} \in {\mathcal S} \ : \ N({\mathfrak m}) \leq x \}.\end{align*} $$

We assume that ${\mathcal S}$ satisfies the following condition:

(1.2) $$ \begin{align} |{\mathcal S}(x^{1/2})| = o(|{\mathcal S}(x)|) \quad \text{as } x \rightarrow \infty. \end{align} $$

Let $\kappa '> 0$ , $0 < \alpha < 1$ , and $\gamma \in \mathbb {R}$ . Note that, $|S(x)| \ll x$ , (1.2) is satisfied when ${\mathcal S}(x) \sim \kappa ' x^\alpha (\log x)^\gamma $ , and (1.2) fails if ${\mathcal S}(x) \sim \kappa ' (\log x)^\gamma $ . Thus, ${\mathcal S}(x) \sim \kappa ' x^\alpha (\log x)^\gamma $ would be a good choice for the asymptotic size of ${\mathcal S}(x)$ so that (1.2) holds.

Let f be a map from ${\mathcal S}$ to ${\mathcal M}$ . For each $\ell \in {\mathcal P}$ , we write

(1.3) $$ \begin{align} \frac{1}{|{\mathcal S}(x)|} \left| \{ {\mathfrak m} \in {\mathcal S}(x) \ : \ n_\ell(f({\mathfrak m})) \geq 1 \} \right| = \lambda_\ell + e_\ell(x), \end{align} $$

where $\lambda _\ell $ denotes the main term (and is independent of x) and $e_\ell = e_\ell (x)$ is the error term. For any sequence of distinct elements $\ell _1, \ell _2, \ldots , \ell _u \in {\mathcal P}$ , we write

(1.4) $$ \begin{align} \frac{1}{|{\mathcal S}(x)|} \left| \{ {\mathfrak m} \in {\mathcal S}(x) \ : \ n_{\ell_i}(f({\mathfrak m})) \geq 1 \text{ for all } i = 1, \ldots, u \} \right|= \lambda_{\ell_1} \ldots \lambda_{\ell_u} + e_{\ell_1 \ldots \ell_u}(x). \end{align} $$

In this article, we will use $e_{\ell _1 \ldots \ell _u}$ to abbreviate $e_{\ell _1 \ldots \ell _u}(x)$ .

Suppose there exists a $\beta $ with $0 < \beta \leq 1$ and $y = y(x) < x^\beta $ such that the following conditions are satisfied:

1. (a) $\left |\{ \ell \in {\mathcal P} \ : \ N(\ell )> x^\beta , n_\ell (f({\mathfrak m})) \geq 1 \} \right | = O_\beta (1)$ for each ${\mathfrak m} \in {\mathcal S}(x)$ . Here, $O_{Y}$ denotes that the big-O constant depends on the variable set Y.

2. (b) $\sum _{y < N(\ell ) \leq x^\beta } \lambda _\ell = o \left ( (\log \log x)^{1/2} \right )$ .

3. (c) $\sum _{y < N(\ell ) \leq x^\beta } |e_\ell | = o \left ( (\log \log x)^{1/2} \right )$ .

4. (d) $\sum _{N(\ell ) \leq y} \lambda _\ell = \log \log x + o \left ( (\log \log x)^{1/2} \right )$ .

5. (e) $\sum _{N(\ell ) \leq y} \lambda _\ell ^2 = o \left ( (\log \log x)^{1/2} \right )$ .

6. (f) For $r \in \mathbb {Z}_{>0}$ , let u be any integer picked from $\{ 1, 2, \ldots , r \}$ . We have

   $$ \begin{align*}\sum{\vphantom{\sum}}" |e_{\ell_1 \ldots \ell_u}| = o \left( (\log \log x)^{-r/2} \right),\end{align*} $$
   where $\sum {\vphantom {\sum }}"$ extends over all u-tuples $(\ell _1,\ell _2,\ldots ,\ell _u)$ with $N(\ell _i) \leq y$ for all $i \in \{ 1, 2, \ldots , u\}$ and all $\ell _i$ ’s are distinct.

For each ${\mathfrak m} \in {\mathcal M}$ , we define

$$ \begin{align*}\omega({\mathfrak m}) = \sum_{\substack{{\mathfrak p} \in {\mathcal P} \\ n_{\mathfrak p}({\mathfrak m}) \geq 1}} 1,\end{align*} $$

the number of elements of ${\mathcal P}$ that generates ${\mathfrak m}$ , counted without multiplicity. Using this definition and the above conditions, we obtain the following subset generalization of the Erdős–Kac theorem.

Theorem 1.1 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy (⋆). Let ${\mathcal S}$ be a subset of ${\mathcal M}$ . For any $x \in X$ , let ${\mathcal S}(x)$ be the set of elements in ${\mathcal S}$ with norm less than or equal to x. Assume that ${\mathcal S}$ satisfies condition (1.2). Let $f : {\mathcal S} \rightarrow {\mathcal M}$ . Suppose there exists a $\beta $ with $0 < \beta \leq 1$ and $y = y(x) < x^\beta $ such that the conditions (a)–(f) above hold. Then, for $\gamma \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|{\mathcal S}(x)|} \left| \left\{ {\mathfrak m} \in {\mathcal S}(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\omega(f({\mathfrak m})) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq \gamma\right\} \right| = \Phi(\gamma),\end{align*} $$

where $\Phi (\gamma )$ is defined in (1.1).

We list some well-studied applications of this general setting.

Remark 1.4 We would like to highlight a result from the literature that bears close resemblance to Theorem 1.1. In [Reference Murty, Murty and Pujahari13, Theorems 3 and 4], Murty, Murty, and Pujahari establish a general Erdős–Kac theorem applicable to a broad range of settings. Their framework involves a sequence $a_m$ of nonzero integers with polynomial growth, defined over an infinite subset $\mathcal {S}$ of the natural numbers. To ensure the validity of their results, they impose three additional conditions: one coinciding with (1.3) and (1.4); a second implying that $\limsup _{\ell \to \infty } \lambda _\ell < 1$ (where $\ell $ ranges over rational primes); and a third stipulating that, for any $B>0$ , there exists $\eta $ such that for any rational number x,

$$ \begin{align*}\sum_{\substack{d \in \mathbb{N} \\ d \text{ square-free} \\ d \leq x^{\eta}}} |e_d| = O \left(\frac{1}{\mu_x^{B}} \right),\end{align*} $$

where , and

$$ \begin{align*}\mu_x = \sum_{\ell \leq x^\epsilon} \lambda_\ell \rightarrow \infty \quad \text{ as } x \rightarrow \infty,\end{align*} $$

with $0 < \epsilon < 1$ . Their Theorem 4 is formulated in terms of $\mu _x$ , and, invoking the distribution of primes, they derive Theorem 3.

In our setting, the sequence $a_m$ in [Reference Murty, Murty and Pujahari13] corresponds to $f(m)$ in our work, though we do not impose the explicit polynomial growth condition. Indeed, all of the conditions in [Reference Murty, Murty and Pujahari13] imply our conditions. For instance, polynomial growth for $a_m$ implies condition (a). Assuming their second condition and that $\lambda _\ell $ is asymptotic to $1/\ell $ and $\mu _x$ asymptotic to $\log \log x$ , a condition required to go from [Reference Murty, Murty and Pujahari13, Theorem 4] to [Reference Murty, Murty and Pujahari13, Theorem 3], we can deduce our conditions (b)–(e), where $y = x^\epsilon $ . Furthermore, their third condition implies our condition (f). Thus, Theorem 1.1 is a generalization of [Reference Murty, Murty and Pujahari13, Theorem 3]. In Remark 1.5, we present an application of our main theorem over natural numbers that cannot be obtained from the work in [Reference Murty, Murty and Pujahari13].

The article [Reference Murty, Murty and Pujahari13] offers a broad generalization of the Erdős–Kac theorem with far-reaching applicability, covering contexts, such as ideals in number fields, sums of modular forms, and the Ramanujan $\tau $ -function. Our results extend these ideas to a wider class of abelian monoids and their subsets. In Theorems 1.9 and 1.10, we apply Theorem 1.1 to new, proper subsets of abelian monoids. Moreover, in Theorems 1.11 and 1.12, we establish Erdős–Kac theorems for refined $\omega $ -functions, and in Theorems 1.13 and 1.14, we extend these results to generalized weighted $\omega $ -functions. These contributions, to the best of our knowledge, are new to the literature.

Remark 1.5 Let ${\mathcal S} = \mathbb {N}$ , ${\mathcal P}$ be the set of rational primes, $X = \mathbb {Q}$ , and $N(\ell )=\ell $ . For each $n \in \mathbb {N}$ , fix an increasing sequence of primes $q_n$ satisfying $q_n \ge e^{n^3}$ and define

$$\begin{align*}f(n) := n \cdot q_n. \end{align*}$$

As in the classical case, take $\beta =1$ and $y = x^{1/\log \log x} < x^\beta $ . For any $m \le x$ , all primes dividing $f(m)=m q_m$ are those dividing m and the single prime $q_m$ . Since no primes $> x$ divides m and $q_m$ adds at most one more prime, we have

$$\begin{align*}\big|\{ \ell \in {\mathcal P} : \ell> x^\beta,\, n_\ell(f(m)) \ge 1 \}\big| = O_\beta(1), \end{align*}$$

verifying (a).

Since each $q_m$ is distinct and sufficiently large, it does not affect the sums in conditions (b)–(f). In particular, for every prime $\ell \le x$ , divisibility of $f(m)$ by $\ell $ coincides with that of m, so that $\lambda _\ell = 1/\ell $ and $e_\ell (x) = O(1/x)$ , just as in the classical case. Consequently, summing over all $\ell \le y$ and applying standard estimates for $\sum _{\ell \le y} 1/\ell $ and $\sum _{\ell \le y} 1/\ell ^2$ ensures that conditions (b)–(f) are satisfied.

Thus, Theorem 1.1 holds for such an f discussed in this example.

Finally, we demonstrate that this example is not covered by [Reference Murty, Murty and Pujahari13, Theorem 3] by showing that $f(m)$ grows faster than any polynomial.

Since $q_n \ge e^{n^3}$ , we have

$$\begin{align*}\log f(n) = \log n + \log q_n \ge n^3, \end{align*}$$

so for any fixed $A>0$ and all sufficiently large n,

$$\begin{align*}\log f(n)> A \log n \quad \Longrightarrow \quad f(n) > n^A. \end{align*}$$

In the following part, we discuss several other instances where Theorem 1.1 can be applied. Note that, moving forward, we shall always assume that $\mathcal {P}, \ {\mathcal M}$ , and X satisfy the condition (⋆).

For a nonzero element $\mathfrak {m} \in \mathcal {M}$ , let the prime element factorization of $\mathfrak {m}$ be given as

$$ \begin{align*} \mathfrak{m} = s_1 {\mathfrak p}_1 + \cdots + s_r {\mathfrak p}_r, \end{align*} $$

where ${\mathfrak p}_i'$ s are its distinct prime elements and $s_i'$ s are their respective nonzero multiplicities. Here,

$$ \begin{align*}N({\mathfrak m}) = N({\mathfrak p}_1)^{s_1} \ldots N({\mathfrak p}_r)^{s_r}.\end{align*} $$

Let $h \geq 2$ be an integer. We say that $\mathfrak {m}$ is an h-free element if $s_i \leq h-1$ for all $i \in \{1, \ldots , r\}$ , and we say that $\mathfrak {m}$ is an h-full element if $s_i \geq h$ for all $i \in \{1, \ldots , r\}$ . Let $\mathcal {S}_h$ denote the set of h-free elements and $\mathcal {N}_h$ denote the set of h-full elements. The distributions of these elements are well-studied in the literature, and to demonstrate this, we introduce some terminologies.

Let the generalized $\zeta $ -function which is an analog of the classical Riemann $\zeta $ -function be given as

$$\begin{align*}\zeta_{\mathcal M}(s) := \sum_{\substack{\mathfrak{m}}} \frac1{(N(\mathfrak{m}) )^s} = \prod_{{\mathfrak p}} \Big( 1- N ({\mathfrak p})^{-s}\Big)^{-1} \ \text{for } \Re (s)>1, \end{align*}$$

where $\mathfrak {m}$ and ${\mathfrak p}$ , respectively, range through the nonzero elements in ${\mathcal M}$ and the prime elements in ${\mathcal P}$ . The absolute convergence of the above series is explained in [Reference Das, Kuo and Liu2].

Let $x \in X$ and let $\mathcal {S}_h(x)$ denote the set of h-free elements in ${\mathcal M}$ with norm less than or equal to x. Since condition (⋆) satisfies [Reference Knopfmacher7, Chapter 4, Axiom A], thus, by [Reference Knopfmacher7, Chapter 4, Proposition 5.5], we have

(1.5) $$ \begin{align} |\mathcal{S}_h(x)| = \frac{\kappa}{\zeta_{\mathcal M}(h)} x + O_h \big( R_{\mathcal{S}_h}(x) \big), \end{align} $$

where

(1.6) $$ \begin{align} R_{\mathcal{S}_h}(x) = \begin{cases} x^\theta & \text{ if } \frac{1}{h} < \theta, \\ x^\theta (\log x) & \text{ if } \frac{1}{h} = \theta, \\ x^{\frac{1}{h}} & \text{ if } \frac{1}{h}> \theta. \end{cases} \end{align} $$

Let $\gamma _h$ be a constant given by

(1.7)

Let $\mathcal {N}_h(x)$ denote the set of h-full elements in ${\mathcal M}$ with norm less than or equal to x. For the distribution of h-full elements, we have (see [Reference Das, Kuo and Liu2, Theorem 1.1])

(1.8)

where is the constant defined in (1.7), and where

(1.9) $$ \begin{align} R_{\mathcal{N}_h}(x) = \begin{cases} x^{\theta/h} & \text{ if } \frac{h}{h+1} < \theta, \\ x^{\frac{1}{h+1}} (\log x) & \text{ if } \frac{h}{h+i} = \theta \text{ for some } i \in \{ 1, \ldots, h-1 \}, \\ x^{\frac{1}{h+1}} & \text{ if } \frac{h}{h+1}> \theta \ \& \ \frac{h}{h+i} \neq \theta \text{ for any } i \in \{ 1, \ldots, h-1 \}. \end{cases} \end{align} $$

Let $x \in X$ . Let $\mathfrak {A}$ and $\mathfrak {B}$ be constants defined as

(1.10) $$ \begin{align} {\mathfrak A}:= \lim_{x \rightarrow \infty} \left( \sum_{\substack{{\mathfrak p} \in \mathcal{P} \\ N({\mathfrak p}) \leq x}} \frac{1}{N({\mathfrak p})} - \log \log x \right). \end{align} $$

The existence of the constant $\mathfrak {A}$ is explained in [Reference Liu11, Lemma 2]. We define the constants

(1.11) $$ \begin{align} \mathfrak{C}_1 := {\mathfrak A} - \sum_{{\mathfrak p}} \frac{N({\mathfrak p})-1}{N({\mathfrak p})(N({\mathfrak p})^h -1)}. \end{align} $$

Let $\mathfrak {L}_h(r)$ be the convergent sum defined for $r> h$ as

(1.12) $$ \begin{align} \mathfrak{L}_h(r) := \sum_{\mathfrak p} \frac{1}{N({\mathfrak p})^{(r/h)-1} \left( N({\mathfrak p}) - N({\mathfrak p})^{1-1/h} + 1 \right)}, \end{align} $$

and let

(1.13) $$ \begin{align} \mathfrak{D}_1 := \mathfrak{A} - \log h + \mathfrak{L}_h(h+1) - \mathfrak{L}_h(2h). \end{align} $$

For the distribution of $\omega ({\mathfrak m})$ over h-free and h-full elements, in [Reference Das, Kuo and Liu2, Theorems 1.2 and 1.3], we proved

$$ \begin{align*}\sum_{\mathfrak{m} \in \mathcal{S}_h(x)} \omega({\mathfrak m}) = \frac{\kappa}{\zeta_{\mathcal M}(h)} x \log \log x + \frac{\kappa {\mathfrak C}_1}{\zeta_{\mathcal M}(h)} x + O_h \left( \frac{x}{\log x}\right)\end{align*} $$

and

$$ \begin{align*} \sum_{{\mathfrak m} \in \mathcal{N}_h(x)} \omega({\mathfrak m}) & = \kappa \gamma_{h} x^{1/h} \log \log x + \kappa \gamma_h \mathfrak{D}_1 x^{1/h} + O_h \left( \frac{x^{1/h}}{\log x} \right). \end{align*} $$

This proves that $\omega ({\mathfrak m})$ has average order $\log \log N({\mathfrak m})$ over h-free and over h-full elements. Using the study of moments, in [Reference Das, Kuo and Liu2], we showed that $\omega ({\mathfrak m})$ has normal order $\log \log ({\mathfrak m})$ over h-free and over h-full elements. In this article, we prove that $\omega ({\mathfrak m})$ obeys the Gaussian distribution over h-free and over h-full elements as well. We achieve this as applications of the following theorem which is a consequence of Theorem 1.1.

Theorem 1.8 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let ${\mathcal S}$ be a subset of ${\mathcal M}$ . For any $x \in X$ , let ${\mathcal S}(x)$ be the set of elements in ${\mathcal S}$ with norm less than or equal to x. Let $f : {\mathcal S} \rightarrow {\mathcal M}$ . Suppose ${\mathcal S}$ satisfy

$$ \begin{align*}|\mathcal{S}(x)| = C_\beta x^{\beta} + O_\beta \big( x^{\xi \beta} \big),\end{align*} $$

for some $0 < \beta \leq 1$ , for some $0 \leq \xi < 1$ , and for some fixed constant $C_\beta $ . Additionally, for a fixed prime ideal ${\mathfrak p}$ , we assume that the set

$$ \begin{align*}\mathcal{S}_{\mathfrak p}(x) := \left\{ \mathfrak{m} \in {\mathcal S}(x) \ : \ n_{\mathfrak p} (f(\mathfrak{m})) \geq 1 \right\}\end{align*} $$

satisfies

(1.14) $$ \begin{align} |\mathcal{S}_{\mathfrak p}(x)| & = \frac{C_{\beta} x^\beta}{N({\mathfrak p})} + \frac{C_{{\mathfrak p},\beta}' x^\beta}{N({\mathfrak p})^{1+\eta}} + O_\beta \left( \frac{x^{\xi \beta}}{N({\mathfrak p})^\xi}\right), \end{align} $$

for some $\eta> 0$ and where the constant $C_{{\mathfrak p},\beta }'$ is uniformly bounded in ${\mathfrak p}$ . Then, for $\gamma \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|{\mathcal S}(x)|} \left| \left\{ {\mathfrak m} \in {\mathcal S}(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\omega(f({\mathfrak m})) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq \gamma\right\} \right| = \Phi(\gamma),\end{align*} $$

where $\Phi (\gamma )$ is defined in (1.1).

As applications of the above theorem, we prove the Erdős–Kac theorem over h-free and over h-full elements as the following.

Theorem 1.9 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $h \geq 2$ be any integer. Let $\mathcal {{\mathcal S}}_h(x)$ denote the set of h-free elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{{\mathcal S}}_h(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{{\mathcal S}}_h(x) \ : N({\mathfrak m}) \geq 3, \ \ \frac{\omega({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

Theorem 1.10 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $h \geq 2$ be any integer. Let $\mathcal {N}_h(x)$ denote the set of h-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_h(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{N}_h(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\omega({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

Next, we show that Theorem 1.8 has more applications in the general context, by taking f to be a non-identity function.

Let $k \geq 1$ be any integer. For each ${\mathfrak m} \in {\mathcal M}$ , we define the $\omega _k$ -function as

$$ \begin{align*}\omega_k({\mathfrak m}) = \sum_{\substack{{\mathfrak p} \in {\mathcal P} \\ n_{\mathfrak p}({\mathfrak m}) = k}} 1,\end{align*} $$

the number of elements of ${\mathcal P}$ that generates ${\mathfrak m}$ , with multiplicity k.

For an element ${\mathfrak m} \in {\mathcal M}$ , let ${\mathfrak m}_k$ be defined as

(1.15) $$ \begin{align} {\mathfrak m}_k = k \cdot \sum_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) = k}} {\mathfrak p}. \end{align} $$

Thus,

$$ \begin{align*}{\mathfrak m} = \sum_{k \geq 1} {\mathfrak m}_k.\end{align*} $$

We define the map $f_k : {\mathcal S} \rightarrow {\mathcal M}$ as

$$ \begin{align*}f_k({\mathfrak m}) = {\mathfrak m}_k.\end{align*} $$

Note that, with the above definition, we have

$$ \begin{align*}\omega(f_k({\mathfrak m})) = \omega({\mathfrak m}_k) = \sum_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}_k) \geq 1}} 1 = \sum_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) = k} } 1 = \omega_k({\mathfrak m}).\end{align*} $$

As more applications of Theorem 1.8, we deduce that $\omega _1({\mathfrak m})$ satisfies the Erdős–Kac theorems over the subset of h-free elements, where $h \geq 2$ . We prove the following theorem.

Theorem 1.11 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $h \geq 2$ be any integer. Let $\mathcal {{\mathcal S}}_h(x)$ denote the set of h-free elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{{\mathcal S}}_h(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{{\mathcal S}}_h(x) \ : N({\mathfrak m}) \geq 3, \ \ \frac{\omega_1({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

In the following, for convenience, we use $\mathcal {N}_1 = {\mathcal M}$ and $\mathcal {N}_1(x)$ to denote the set of elements in ${\mathcal M}$ with norm less than or equal to x. We call $\mathcal {N}_1$ the set of $1$ -full elements. To encompass the set of h-full elements defined above and ${\mathcal M}$ under one notation, we use $\mathcal {N}_k$ where $k \geq 1$ to denote the set of k-full elements. We show that $\omega _k({\mathfrak m})$ satisfies the Erdős–Kac theorem over the subset of k-full elements in the following.

Theorem 1.12 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $k \geq 1$ be any integer. Let $\mathcal {N}_k(x)$ denote the set of k-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_k(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{N}_k(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\omega_k({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

We extend our study to the generalized $\omega $ -function related to a sequence, inspired by the work of Elma and Martin [Reference Elma and Martin4] over natural numbers. Let ${\mathcal A} = (a_1,a_2,\ldots )$ be a sequence of complex numbers, and for some integer $k \geq 1$ , the following property is satisfied:

(1.16) $$ \begin{align} \sum_{\substack{{\mathfrak p} \in {\mathcal P} \\ N({\mathfrak p}) \leq x^{1/k}}} \sum_{i \geq k+1} \frac{|a_i|}{N({\mathfrak p})^{i/k}} = O(1). \end{align} $$

The above relation produces a growth condition on $a_i$ ’s depending on the minimum value of the norm map. In particular, if we assume that $N({\mathfrak p}) \geq b$ for all ${\mathfrak p} \in {\mathcal P}$ , then a sufficient condition for (1.16) to hold is

$$ \begin{align*}a_i \ll B^i \quad \text{ as } i \rightarrow \infty, \quad \text{ where } \quad 0 < B \leq b^{\frac{1}{k} - \alpha} \quad \text{ for some } \alpha> 0.\end{align*} $$

For this article, we shall assume $b = 2$ , that is, the minimum possible value of the norm of a prime element.

Let the generalized $\omega $ -function, $\omega _{\mathcal A}$ attached to the sequence ${\mathcal A}$ be defined as

$$ \begin{align*}\omega_{\mathcal A} ({\mathfrak m}) = \sum_{k \geq 1} a_k \ \omega(f_k({\mathfrak m})) = \sum_{k \geq 1} a_k \ \omega_k({\mathfrak m}),\end{align*} $$

where the sum is finite for each ${\mathfrak m}$ .

With the above definitions and restrictions, for an integer $k \geq 1$ , we prove the following Erdős–Kac theorem for $\omega _{\mathcal A}({\mathfrak m})$ over k-full elements in ${\mathcal M}$ .

Theorem 1.13 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ . Let ${\mathcal A} = (a_1,a_2,\ldots )$ be a sequence of complex numbers. Let $k \in \mathbb {Z}_{>0}$ be such that $a_k \neq 0$ and the following property holds:

$$ \begin{align*}a_i \ll B^i \quad \text{ as } i \rightarrow \infty, \quad \text{ where } \quad 0 < B \leq 2^{\frac{1}{k} - \alpha} \quad \text{ for some } \alpha> 0.\end{align*} $$

Let $\mathcal {N}_k(x)$ denote the set of k-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_k(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{N}_k(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\frac{1}{a_k} \omega_{\mathcal A}({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

In the next result, we state the Erdős–Kac theorem for $\frac {1}{a_1} \omega _{\mathcal A}({\mathfrak m})$ over h-free elements. The set of h-free elements in ${\mathcal M}$ has a positive density and thus the proof of the theorem follows similarly to the case of $1$ -full elements in Theorem 1.13. To avoid repetition, we do not provide the proof in this article.

Theorem 1.14 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ . Let $h \geq 2$ be any integer and let $\mathcal {S}_h(x)$ denote the set of h-free elements in ${\mathcal M}$ with norm less than or equal to x. Let ${\mathcal A} = (a_1,a_2,\ldots )$ be a sequence of complex numbers, satisfying $a_1 \neq 0$ and

$$ \begin{align*}a_i \ll B^i \quad \text{ as } i \rightarrow \infty, \quad \text{ where } \quad 0 < B \leq 2^{1 - \alpha} \quad \text{ for some } \alpha> 0.\end{align*} $$

Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{S}_h(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{S}_h(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\frac{1}{a_1} \omega_{\mathcal A}({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

We provide applications of Theorem 1.13 to other well-known functions, in particular, the prime counting $\Omega $ -function and the divisor counting function. Note that our method provides a novel approach to study the prime divisor counting functions, that is, the $\omega $ -function and the $\Omega $ -function, and the divisor counting function in a single framework. As these functions are of prime importance to the various mathematicians, our work contributes significantly to the literature.

For each ${\mathfrak m} \in {\mathcal M}$ , we define

$$ \begin{align*}\Omega({\mathfrak m}) = \sum_{\substack{{\mathfrak p} \in {\mathcal P} \\ n_{\mathfrak p}({\mathfrak m}) \geq 1}} n_{\mathfrak p}({\mathfrak m}),\end{align*} $$

the number of elements of ${\mathcal P}$ that generates ${\mathfrak m}$ , counted with multiplicity. We say that ${\mathfrak d} \in {\mathcal M}$ is a divisor of ${\mathfrak m}$ and denote it as ${\mathfrak d} | {\mathfrak m}$ if $n_{\mathfrak p}({\mathfrak d}) \leq n_{\mathfrak p}({\mathfrak m})$ for all ${\mathfrak p} \in {\mathcal P}$ . We define

$$ \begin{align*}d({\mathfrak m}) = \sum_{\substack{{\mathfrak d} \in {\mathfrak m} \\ {\mathfrak d} | {\mathfrak m}}} 1,\end{align*} $$

the number of divisors of ${\mathfrak m}$ . When ${\mathcal M} = \mathbb {Z}_{>0}$ , $\Omega (n)$ counts the total number of primes diving the natural number n, and $d(n)$ counts the total number of divisors of n.

Notice that

$$ \begin{align*}\Omega({\mathfrak m}) = \sum_{k \geq 1} k \left(\sum_{\substack{{\mathfrak p} \in {\mathcal P} \\ n_{\mathfrak p}({\mathfrak m}) = k}} 1 \right) = \sum_{k \geq 1} k \cdot \omega_k({\mathfrak m}).\end{align*} $$

For ${\mathcal A} = (1,2,3,\ldots ,n,\ldots )$ and for any integer $k \geq 1$ , we have $a_i = i \ll 2^{i/(2k)}$ as $i \rightarrow \infty $ , satisfying the hypothesis of Theorem 1.13. Thus, we obtain the following corollary, called the Erdős–Kac theorem for $\Omega ({\mathfrak m})$ over k-full elements.

Corollary 1.15 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $k \geq 1$ be any integer. Let $\mathcal {N}_k(x)$ denote the set of k-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_k(x)|} \bigg| \left\{ {\mathfrak m} \in \mathcal{N}_k(x) \ : \ N({\mathfrak m}) \geq 3, \ \frac{\frac{1}{k} \Omega({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \right\} \bigg| = \Phi(a).\end{align*} $$

Note that, using the study of moments, in [Reference Das, Kuo and Liu3], we showed that $\Omega ({\mathfrak m})$ has normal order $\log \log ({\mathfrak m})$ over h-free and normal order $h \log \log ({\mathfrak m})$ over h-full elements.

For the $d({\mathfrak m})$ function, one can easily deduce that

$$ \begin{align*}d({\mathfrak m}) = \sum_{\substack{{\mathfrak d} \in {\mathfrak m} \\ {\mathfrak d} | {\mathfrak m}}} 1 = \prod_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) \geq 1}} (n_{\mathfrak p}({\mathfrak m}) + 1).\end{align*} $$

Thus,

$$ \begin{align*}\log d({\mathfrak m}) &= \sum\limits_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) \geq 1}} \log (n_{\mathfrak p}({\mathfrak m}) + 1) =\sum_{k \geq 1} \log(k+1) \left(\sum\limits_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) = k}} 1 \right)\\ & = \sum_{k \geq 1} \log(k+1) \cdot \omega_k({\mathfrak m}).\end{align*} $$

Taking ${\mathcal A} = (\log 2, \log 3, \log 4, \ldots , \log (n+1), \ldots )$ and for any integer $k \geq 1$ , we have $a_i = \log (i+1) \ll 2^{i/(2k)}$ as $i \rightarrow \infty $ , satisfying the hypothesis of Theorem 1.13. Thus, we obtain the following corollary, called the Erdős–Kac theorem for $\log d({\mathfrak m})$ over k-full elements.

Corollary 1.16 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $k \geq 1$ be any integer. Let $\mathcal {N}_k(x)$ denote the set of k-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}&\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_k(x)|} \bigg| \bigg\{ {\mathfrak m} \in \mathcal{N}_k(x) : N({\mathfrak m}) \geq 3, \ \frac{\frac{1}{\log (k+1)} \log d({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \bigg\} \bigg|\\&\quad = \Phi(a).\end{align*} $$

Next, we show that Theorem 1.13 can also be applied to some new functions. Let $\omega _T({\mathfrak m})$ denote the difference in the number of prime elements in the factorization of ${\mathfrak m}$ with odd multiplicity and the number of prime elements in the factorization of ${\mathfrak m}$ with even multiplicity, that is,

(1.17) $$ \begin{align} \omega_T({\mathfrak m}) = \sum_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) \ \text{odd}}} 1 - \sum_{\substack{{\mathfrak p} \\ n_{\mathfrak p}({\mathfrak m}) \ \text{even}}} 1 = \sum_{{\mathfrak p}} (-1)^{n_{\mathfrak p}({\mathfrak m}) - 1} = \sum_{k \geq 1} (-1)^{k-1} \omega_k({\mathfrak m}). \end{align} $$

Taking ${\mathcal A} = (1, -1, 1, -1, \ldots , (-1)^{n-1}, \ldots )$ and for any integer $k \geq 1$ , we have $(-1)^{i-1} \ll 2^{i/(2k)}$ as $i \rightarrow \infty $ . Thus, as another application of Theorem 1.13, we obtain the following corollary.

Corollary 1.17 Let ${\mathcal P}$ , ${\mathcal M}$ , and X satisfy condition (⋆). Let $x \in X$ and $k \geq 1$ be any integer. Let $\mathcal {N}_k(x)$ denote the set of k-full elements in ${\mathcal M}$ with norm less than or equal to x. Then, for $a \in \mathbb {R}$ , we have

$$ \begin{align*}&\lim_{x \rightarrow \infty} \frac{1}{|\mathcal{N}_k(x)|} \bigg| \bigg\{ {\mathfrak m} \in \mathcal{N}_k(x) : N({\mathfrak m}) \geq 3, \ \frac{(-1)^{k-1} \omega_T({\mathfrak m}) - \log \log N({\mathfrak m})}{\sqrt{\log \log N({\mathfrak m})}} \leq a \bigg\} \bigg|\\&\quad = \Phi(a).\end{align*} $$

Finally, in Section 7, we provide various applications of our general setting. Let ${\mathcal A} = (a_1, a_2, \ldots )$ be a sequence of complex numbers satisfying any of the following types:

1. (1) if $a_i = 1$ for all $i \in \mathbb {Z}_{>0}$ , that is, $\omega _{\mathcal A}({\mathfrak m}) = \omega ({\mathfrak m})$ ;

2. (2) if $a_i = i$ for all $i \in \mathbb {Z}_{>0}$ , that is, $\omega _{\mathcal A}({\mathfrak m}) = \Omega ({\mathfrak m})$ ;

3. (3) if $a_i = \log (i+1)$ for all $i \in \mathbb {Z}_{>0}$ , that is, $\omega _{\mathcal A}({\mathfrak m}) = \log d({\mathfrak m})$ ;

4. (4) if $a_i = 1$ for all odd i and $a_i = - 1$ for all even i, that is, $\omega _{\mathcal A}({\mathfrak m}) = \omega _T({\mathfrak m})$ ;

5. (5) if $a_i = 0$ for all $i \neq k$ and $a_k = 1$ , that is, $\omega _{\mathcal A}({\mathfrak m}) = \omega _k({\mathfrak m})$ .

For such ${\mathcal A}'$ s, we apply Theorems 1.13 and 1.14 to prove the Erdős–Kac theorems for the $\omega _{\mathcal A}$ -function in number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad applicability of our approach.