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Li-Yuan Wang

Publications and source records attributed to Li-Yuan Wang.

At least 19 recordsLinked to original sources
arXiv2026

The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(\chi)=\left[G_N(\chi^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$ where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $\chi$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(\chi^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}\chi^{ki+kj}(x)e^{2\pi ix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.

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arXiv2025

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums

Inspired by Weil's classical result on the zeta function of projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix $$\det\left[J_p(\chi^{ki},\chi^{kj})\right]_{1\le i,j\le n-1},$$ where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $\chi$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(\chi^{ki},\chi^{kj})$ is the Jacobi sum. For example, let $\zeta_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer $$\theta_k=\sum_{x\in\mathbb{F}_p,x^k=1}\zeta_p^x$$ over $\mathbb{Q}$. Then we prove that $$\det \left[J_p(\chi^{ki},\chi^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$ where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$.

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arXiv2025

The Pell sequence and cyclotomic matrices involving squares over finite fields

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.

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arXiv2024

The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums

In this paper, we mainly consider arithmetic properties of the cyclotomic matrix $B_p(k)=\left[J_p(\chi^{ki},\chi^{kj})^{-1}\right]_{1\le i,j\le (p-1-k)/k}$, where $p$ is an odd prime, $1\le k<p-1$ is a divisor of $p-1$, $\chi$ is a generator of the group of all multiplicative characters of the finite field $\mathbb{F}_p$ and $J_p(\chi^{ki},\chi^{kj})$ is Jacobi sum over $\mathbb{F}_p$. By using the Gross-Koblitz formula and some $p$-adic tools, we first prove that $$p^{n-2}\det B_p(k)\equiv (-1)^{\frac{(n-1)(p+n-3)}{2}} \left(\frac{1}{k!}\right)^{n-2}\frac{1}{(2k)!}\pmod {p},$$ where $p-1=kn$. By establishing some theories on almost circulant matrices, we show that $$\det B_p(k)=(-1)^{\frac{(n-1)(p+n-1)}{2}}p^{-(n-1)}n^{n-2}a_p(k).$$ Here $a_p(k)$ is the coefficient of $t$ in the minimal polynomial of $\sum_{y\in U_k}(e^{2\pi{\bf i}y/p}-1)$, where $U_k$ is the set of all $k$-th roots of unity over $\mathbb{F}_p$. Also, for $k=1,2$ we obtain explicit expressions of $\det B_p(k)$.

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arXiv2024

On a determinant involving linear combinations of Legendre symbols

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x + \left(\frac{i-j}p\right) + \left(\frac ip\right)y + \left(\frac jp\right)z + \left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd prime $p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3\pmod4$.

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arXiv2024

Gaussian hypergeometric functions and cyclotomic matrices

Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $\chi$ be a generator of $\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be all nonzero squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix $$B_{q,2}(\chi^r):=\left[\chi^r(s_i+s_j)+\chi^r(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}.$$ We prove that if $q\equiv 3\pmod 4$, then $$\det (B_{q,2}(\chi^r))=\prod_{0\le k\le (q-3)/2}J_q(\chi^r,\chi^{2k})= \begin{cases} (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(\chi^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\ G_q(\chi^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2, \end{cases}$$ where $J_q(\chi^r,\chi^{2k})$ and $G_q(\chi^r)$ are the Jacobi sum and the Gauss sum over $\mathbb{F}_q$ respectively.

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arXiv2024

On a generalization of R. Chapman's "evil determinant"

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant".

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arXiv2024

On cyclotomic matrices involving Gauss sums over finite fields

Inspired by the works of L. Carlitz and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let $q=p^n$ be an odd prime power with $p$ prime and $n\in\mathbb{Z}^+$. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $\chi$ be a generator of the group of all mutiplicative characters of the finite field $\mathbb{F}_q$. For the Gauss sum $$G_q(\chi^{r})=\sum_{x\in\mathbb{F}_q}\chi^{r}(x)\zeta_p^{{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x)},$$ we prove that $$\det \left[G_q(\chi^{2i+2j})\right]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha_p}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}},$$ where $$\alpha_p= \begin{cases} 1 & \mbox{if}\ n\equiv 1\pmod 2, (p^2+7)/8 & \mbox{if}\ n\equiv 0\pmod 2. \end{cases}$$

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arXiv2023

On finite field analogues of determinants involving the Beta function

Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices concerning the Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let $q>1$ be a prime power and let $\chi$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. Then we prove that $$\det\left[J_q(\chi^i,\chi^j)\right]_{1\le i,j\le q-2}=(q-1)^{q-3},$$ where $J_q(\chi^i,\chi^j)$ is the Jacobi sum over $\mathbb{F}_q$. This is a finite analogue of $$\det [B(i,j)]_{1\le i,j\le n}=(-1)^{\frac{n(n-1)}{2}}\prod_{r=0}^{n-1}\frac{(r!)^3}{(n+r)!},$$ where $B$ is the Beta function. Also, if $q=p\ge5$ is an odd prime, then we show that $$\det \left[J_p(\chi^{2i},\chi^{2j})\right]_{1\le i,j\le (p-3)/2}=\frac{1+(-1)^{\frac{p+1}{2}}p}{4}\left(\frac{p-1}{2}\right)^{\frac{p-5}{2}}.$$

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arXiv2022

A note on additive complements of the squares

Let $\mathcal{S}=\{1^2,2^2,3^2,...\}$ be the set of squares and $\mathcal{W}=\{w_n\}_{n=1}^{\infty} \subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S} + \mathcal{W} \supset \{n \in \mathbb{N}: n \geq N_0\}$ for some $N_0$. Let $\mathcal{R}_{\mathcal{S},\mathcal{W}}(n) = \#\{(s,w):n=s+w, s\in \mathcal{S}, w\in \mathcal{W}\} $. In 2017, Chen-Fang \cite{C-F} studied the lower bound of $\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)$. In this note, we improve Cheng-Fang's result and get that $$\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/2}.$$ As an application, we make some progress on a problem of Ben Green problem by showing that $$\limsup_{n\rightarrow\infty}\frac{\frac{\pi^2}{16}n^2-w_n}{n}\ge \frac{\pi}{4}+\frac{0.193\pi^2}{8}.$$

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arXiv2021

$q$-Supercongruences from transformation formulas

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]\Phi_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar $q$-supercongruence conjectured by Guo and Schlosser.

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arXiv2021

Frobenius matrices and a variant of Zolotarev's theorem

In this paper, with the help of the theory of matrices and finite fields we generalize Zolotarev's theorem to an arbitrary finite dimensional vector space over $\mathbb{F}_q$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements.

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arXiv2020

Two $q$-supercongruences from Watson's transformation

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] introduced a new method called `creative microscoping', to prove many $q$-supercongruences in a unified way. In this paper, we apply this method and Watson's ${}_8\phi_7$ transformation formula to prove two $q$-supercongruences, which were recently conjectured by Guo and Schlosser.

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arXiv2020

Products of quadratic residues and related identities

In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.

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arXiv2020

Almost universal ternary sums of pentagonal numbers

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.

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