arXiv subjects

Wenzhong Lei

Publications and source records attributed to Wenzhong Lei.

3 recordsLinked to original sources

arXiv2026

The Normal Form of Smith's Matrices

For any integers $x$ and $y$, let $(x,y)$ and $[x,y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$, respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1,...,x_n\}$ be a set of $n$ distinct positive integers. Let $(f((x_i,x_j)))$ (abbreviated by $f(S)$) and $(f([x_i,x_j]))$ (abbreviated by $(f([S]))$) stand for the $n\times n$ matrices whose $(i,j)-$entry is $(f((x_i,x_j)))$ and $(f([x_i,x_j]))$ respectively. In 1989, Beslin and Ligh gave a description of the lower triangular decomposition of $((x_i,x_j))$. In 1992, Bourque and Ligh showed that if $S$ is factor closed (i.e., S contains all positive divisors of any element of S), then the GCD matrix $((x_i,x_j))$ divides the LCM matrix $([x_i,x_j])$ (written as $((x_i,x_j))|([x_i,x_j])$) in the ring $M_n(\mathbb{Z})$ of $n\times n$ matrices over the integers. In this paper, we will show the diagonalization of $((x_i,x_j))$ and its applications. Our main new contributions are Theorems 4.1 and 4.2, which extend previous results to gcd-closed sets satisfying condition $\mathcal{G}$.

math.NT↗

arXiv2026

Proof of the Gawron-Miska-Ulas conjecture concerning unboundedness of coefficients of power series expansion of $\prod_{n=0}^{\infty}(1-x^{2^{n}})^m$

It is well known that $F(x)=\prod_{n=0}^{\infty}(1-x^{2^n})$ is the generating function of the Prouhet-Thue-Morse sequence $\{(-1)^{σ_2(n)}\}_{n=0}^\infty$, where $σ_2(n)$ is the sum of (binary) digits of $n$. Let $m$ be an integer. In 2018, Gawron, Miska and Ulas initiated the study of arithmetic properties of power series expansion of the function $$F_m(x)=F(x)^m=\sum_{n=0}^{\infty}t_m(n) x^n,$$ and proposed a conjecture stating that for any given integer $m\ge 2$, the sequence $\{t_m(n)\}_{n=0}^{\infty}$ is unbounded. In this paper, we introduce a new method to investigate this conjecture. In fact, by making use of algebraic, $p$-adic and analytic methods, we show that the Gawron-Miska-Ulas conjecture is true.

math.NT↗

arXiv2026

Proofs of Lupu's conjectures for multiple zeta values and multiple $t$-values

Let $r\ge 1$ be an integer. For any multiple index $\mathbf{s}=(s_1,s_2,\cdots,s_r) \in\mathbb{Z}_{\geq 1}^r$ with $s_r>1$, the multiple zeta value (MZV for short) is defined by \begin{align*} ζ(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}} \end{align*} and the multiple $t$-value is defined by \begin{align*} t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}}, \end{align*} where if the index is empty, then we define the value $t(\emptyset):=1$. We denote by $\{a_1,\cdots,a_k\}^d$ the sequence formed by repeating the sequence $\{a_1,\cdots,a_k\}$ exactly $d$ times. Let $H(a,b)=ζ(\{2\}^a,3,\{2\}^b)$ and $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. In this paper, by using the Lai-Lupu-Orr integral expressions for $H(a,b)$ and $T(a,b)$ and the properties of Beta function and Gamma function, we show that for any nonnegative integers $a$ and $b$, we have \begin{align*} H(a,b):=\frac{-4π^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{ζ(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}} \end{align*} and \begin{align*} T(a,b)=\frac{-2}{(2a+1)!}\left(\fracπ{2}\right)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{ζ(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align*} This confirms two conjectures of Lupu proposed in [C. Lupu, Another look at Zagier's formula for multiple zeta values involving Hoffman elements, Math. Z. 301 (2022), 3127-3140].

math.NT↗