Let $p>3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\det\left[(i^2+dj^2)^m\left(\frac{i^2+dj^2}{p}\right)\right]_{1\leqslant i,j \leqslant (p-1)/2},$$ and $$D_p^{(m)}= \det\left[(i^2-j^2)^m\left(\frac{i^2-j^2}{p}\right)\right]_{1\leq i,j\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun.
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\frac p4:\ (\frac kp)=-1\}|}&\text{if}\ p\equiv1\pmod4, \\(-1)^{\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where $$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$W_p=\det\left[\left(\frac{i^2-((p-1)/2)!j}p\right)\right]_{0\le i,j\le(p-1)/2}.$$ We …
I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.
For $0\leq α<1$ and prime number $p$ let $L(α,p)$ be the sum of the first $[αp]$ values of Legendre symbol modulo $p$. We study positivity of $L(α,p)$ and prove that for $|α-\frac13|<2\cdot 10^{-6}$ and for rational $α\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(α,p)\geq 0$ holds for majority of primes.
On a determinant involving linear combinations of Legendre symbols
School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西安利物浦大学数学与物理学院）；School of Mathematics, Nanjing University（南京大学数学系）；School of Physical and Mathematical Sciences, Nanjing Tech University（南京工业大学物理与数学科学学院）
Let $m$ be any positive integer and let $δ_1,δ_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=δ_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=δ_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any di…
In this paper, by the tools of circulant matrices and hyperelliptic curves over finite fields, we study some arithmetic properties of certain determinants involving the Legendre symbols and $k$-th residues.
Let $p>3$ be a prime. We investigate Legendre symbol of $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ , where $4c(a+b+c)k \equiv (4ac-b^2)\pmod{p}.$ Parallel results are given for $\displaystyle \prod_{i,j=1 \atop p\nmid f(i,j) }^{(p-1)/2} \bigg(\frac{ f(i,j) }{p}\bigg).$ Then we show that $ \displayst…
Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.
Large sieve inequality for sums of Legendre symbols over short intervals
Université Jean Monnet, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, ICJ UMR5208（让·莫内大学、里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、法国国家科学研究中心、ICJ UMR5208）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, University of Bristol（布里斯托大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$$ and $$[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which we…
In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(…
For each odd prime $p$, let $ζ_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix $\(\frac{1-ζ_p^{j^2k^2}}{1-ζ_p^{j^2}}\)_{1\le j,k\le (p-1)/2}$ can be written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with $a_p,b_p\in\frac12\Z$ and $$\begin{cases}ν_p(a_p)=ν_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8, \\ν_p(a_p)=ν_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv 7\pmod8,\end{cases}$$ where $ν_p(x)$ denotes the $p$-adic or…
In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.
