Searcharxiv⌕ Search

arXiv subjects

Wentang Kuo

Publications and source records attributed to Wentang Kuo.

10 recordsLinked to original sources
arXiv2025

On the distribution of the total number of generators of $h$-free and $h$-full elements in an abelian monoid

Let $\mathfrak{m}$ be an element of an abelian monoid, with $\Omega(\mathfrak{m})$ denoting the total number of prime elements generating $\mathfrak{m}$. We study the moments of $\Omega(\mathfrak{m})$ over subsets of $h$-free and $h$-full elements, establishing the normal order of $\Omega(\mathfrak{m})$ within these subsets. This work continues the study on the distribution of generalized arithmetic functions over $h$-free and $h$-full elements in abelian monoids as introduced in the authors' previous work.

math.NT↗
arXiv2025

Generalizations of Erd\H{o}s-Kac theorem with applications

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erd\H{o}s and Kac established that $\omega(n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this paper, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of $h$-free and $h$-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.

math.NT↗
arXiv2025

A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erd\H{o}s and Kac established that $\omega(n)$ obeys the Gaussian distribution over natural numbers. In 2004, the third author generalized their theorem to all abelian monoids. In this work, we extend the work of the third author to any subset of the set of ideals of a number field satisfying some additional conditions. Finally, we apply this theorem to prove the Erd\H{o}s-Kac theorem over $h$-free and over $h$-full ideals of the number field.

math.NT↗
arXiv2025

On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid

This work introduces the first in-depth study of h-free and h-full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let m be an element of an abelian monoid, with {\omega}(m) denoting the number of distinct prime elements generating m. We study the moments of {\omega}(m) over subsets of h-free and h-full elements, establishing the normal order of {\omega}(m) within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.

math.NT↗
arXiv2024

On the number of prime factors with a given multiplicity over h-free and h-full numbers

Let $k$ and $n$ be natural numbers. Let $ω_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $ω_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $ω_1(n)$ has normal order $\log \log n$ over $h$-free numbers, $ω_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions $ω_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $ω_k(n)$ with $k > h$ do not have normal order over $h$-full numbers.

math.NT↗
arXiv2024

Distribution of $ω(n)$ over $h$-free and $h$-full numbers

Let $ω(n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $ω(n)$ has normal order $\log \log n$ over naturals. In this work, we establish the first and the second moments of $ω(n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $ω(n)$ has normal order $\log \log n$ over these subsets.

math.NT↗
arXiv2024

On the number of irreducible factors with a given multiplicity in function fields

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $ω_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and the second moments of $ω_k(f)$ with $k \geq 1$. Moreover, we prove that the function $ω_1(f)$ has normal order $\log (\text{deg}(f))$ and also satisfies the Erdős-Kac Theorem. Finally, we prove that the functions $ω_k(f)$ with $k \geq 2$ do not have normal order.

math.NT↗
arXiv2020

The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8 \times C_p]$

Let $p$ be a prime such that the multiplicative order $m$ of $2$ modulo $p$ is even. We prove that the integral group ring $\mathbb{Z}[Q_8 \times C_p]$ has the multiplicative Jordan decomposition property when $m$ is congruent to $2$ modulo $4$. There are infinitely many such primes and these primes include the case $p \equiv 3 \pmod{4}$. We also prove that $\mathbb{Z}[Q_8 \times C_5]$ has the multiplicative Jordan decomposition property in a new way.

math.RA↗
arXiv2015

On a problem of Sidon for polynomials over finite fields

Let $ω$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(ω) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in ω, a+b = n, 0 0$ for all $n$ sufficiently large and, for all $ε> 0$, $$ \lim_{n \rightarrow \infty} \frac{r_n(ω)}{n^ε} = 0. $$ P. Erdős proved this conjecture by showing the existence of a sequence $ω$ of positive integers such that $$ \log n \ll r_n(ω) \ll \log n. $$ In this paper, we prove an analogue of this conjecture in $\mathbb{F}_q[T]$, where $\mathbb{F}_q$ is a finite field of $q$ elements. More precisely, let $ω$ be a sequence in $\mathbb{F}_q[T]$. Given a polynomial $h\in\mathbb{F}_q[T]$, we define $$ r_h(ω) = |\{(f,g) \in \mathbb{F}_q[T]\times \mathbb{F}_q[T] : f,g\in ω, f+g =h, \text{deg } f, \text{deg } g \leq \text{deg } h, f\ne g\}|. $$ We show that there exists a sequence $ω$ of polynomials in $\mathbb{F}_q [T]$ such that $$ \text{deg } h \ll r_h(ω) \ll \text{deg } h $$ for $\text{deg } h$ sufficiently large.

math.NT↗
arXiv2015

Sidon basis in polynomial rings over finite fields

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is not $2$ or $3$. In this paper, we prove an $\mathbb{F}_q[t]$-analogue of results related to the conjecture of Erdős on the existence of infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We prove that there exists a $B_2[2]$ sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is an asymptotic basis of order $3$. We also prove that for any $\varepsilon> 0$, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is a Sidon basis of order $3 + \varepsilon$. In other words, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$ such that any $n \in \mathbb{F}_q[t]$ of sufficiently large degree can be expressed as a sum of four elements of the sequence, where one of them has a degree less than or equal to $\varepsilon \text{deg } n.$

math.NT↗