Subset Sums of Quadratic Residues over Finite Fields

有限域上Dickson多项式的剩余类和

Residue sums of Dickson polynomials over finite fields

本文研究奇特征有限域上任意次数Dickson多项式的剩余类和问题，给出了像中不同剩余类之和的闭式表达式并刻画像集大小，首次实现次数无界多项式此类和的非平凡分类。

Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…

Graphical patterns in quadratic residues

Plots of quadratic residues display some features that are analyzed mathematically.

Quadratic residues and non-residues for infinitely many Piatetski-Shapiro primes

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by S. Wright in certain aspects.

On the distribution of sums of residues

We generalize and solve the $\roman{mod}\,q$ analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the $2^n$ sums of the form $\sum_{i=1}^n\varepsilon_ia_i$, where each $\varepsilon_i$ is $0$ or $1$. For all $q$, $n$, $k$ we determine the maximum, over all reduced residues $a_i$ and all sets $P$ consisting of $k$ arbitrary residues, of the number of these sums that belong to $P$.

Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$

Prescribing the binary digits of squarefree numbers and quadratic residues

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cu…

Sumsets in quadratic residues

We describe all sets $A \subseteq \F_p$ which represent the quadratic residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\hat{+} A$. Also, we consider the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$ and prove that $A$ has a structure in the situation.

Statistics of small prime quadratic non-residues

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

Quadratic residues that are not primitive roots

We prove that any prime $p$ satisfying $ϕ(p-1)\leq (p-1)/4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$.

Characterizations of quadratic, cubic, and quartic residue matrices

We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively.

On the sums of squares of exceptional units in residue class rings

Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ be…

The largest cycles consist by the quadratic residues and Fermat primes

This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.

Kloosterman sums with primes and solvability of one congruence with inverse residues. I

In the paper, we establish a new estimate for Kloosterman sum over primes with respect to an arbitrary modulus $q$. This estimate together with some recent results of the second author are applied to the problem of solvability of the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q} \] in prime variables $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-δ}$. Here $g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, where $a,b,m$ and $k\ge 3$ are arbitrary integers, $(ab,q)=1$. The main result of the paper gives an asymptotic formula for the number of solutions for the case when $q$ is coprime …

On the density of primes with a set of quadratic residues or non-residues in given arithmetic progression

Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$ is at least 2 and $U$ is the union formed by taking certain arithmetic progressions of length $s$ from each element of $\mathcal{A}$, we calculate the asymptotic density of the set of all prime numbers $p$ such that $U$ is a set of quadratic residues (respectively, quadratic non-residues) of $p$.

On consecutive quadratic non-residues: a conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than the square root of p. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds the square root of p.

差集与正指数和 II：循环群中的三次剩余

Difference sets and positive exponential sums II: cubic residues in cyclic groups

本文通过构造非负指数和，给出了循环群中差集避免三次剩余的集合基数的上界。

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.

Two_Generalizations_for_Quadratic_Residue_Codes_over_Finite_Fields

It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper generalizes the constructions for the quadratic residue code with length $p$ to be the length $n=p_{1}\cdots p_{g}$, and to be the case $m$-th residue codes with length $p$ over finite fields, where $m\geq 2$ is a positive integer. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, and then the corresponding …

整数余类的平方和

Sums of squares of integers from residue classes

本文研究了余类模m的整数余类的平方和性质，证明了当余数d≡±1 mod m时，该余类是s-平方universal，并确定了最小的s值。

Quadratic Residues and Non-residues in Arithmetic Progression

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this are also established, and some problems connected with these results that are worthy of further study are discussed.

Joint distribution in residue classes of the base-$q$ and Ostrowski digital sums

Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ α=[0;\overline{1,m}],\ m\geq 2, \] let $S_α(n)$ denote the sum of digits in the Ostrowski $α$-representation of $n$. Let $m_1,m_2\geq 2$ be integers with $$\gcd(q-1,m_1)=\gcd(m,m_2)=1.$$ We prove that there exists $δ>0$ such that for all integers $a_1,a_2$, \begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_α(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-δ}). \end{eqnarray*} The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equali…

Quadratic Non-residues in Short Intervals

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

Quartic residues and sums involving $\binom{4k}{2k}$

Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big…

Quadratic Residues and Non-Residues: Selected Topics

Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the Disquisitiones deals with quadratic residues and nonresidues. Beginning with these fundamental contributions of Gauss, the study of quadratic residues and nonresidues has subsequently led directly to many of the key ideas …

The distribution of spacings between quadratic residues, II

We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper extends the result to arbitrary q.

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.

Conditional bounds for the least quadratic non-residue and related problems

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for $L$-functions at $s=1$. In particular, we derive explicit upper and lower bounds for $L(1,χ)$ and $ζ(1+it)$, and deduce explicit bounds for the class number of imaginary quadratic fields. Finally, we improve the best known theoretical bo…

On sums of generalized Ramanujan sums

Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.

Quadratic residues and quartic residues modulo primes

In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{cases}-(A^2+4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=1, \\(-A^2-4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2+5ij+2j^2…

Evaluations of some quadratic Euler sums

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.

Energy bounds, bilinear sums and their applications in function fields

We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear sums and to some questions regarding smooth and square-free polynomials in residue classes.

Applications of Lerch's theorem and permutations concerning quadratic residues

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary $k$-th power residue modulo $p$ and primitive roots of a power of $p$. Finally, we discuss some permutation problems con…

Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $ ε\in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-ε} \mbox{ and } \frac{ϕ(p-1)}{p-1} \leq \frac{1}{2} - ε, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.

Quadratic residues and difference sets

It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, …

Sumsets of reciprocals in prime fields and multilinear Kloosterman sums

We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.

二次 Weyl 群多重 Dirichlet 级数的留数

Residues of quadratic Weyl group multiple Dirichlet series

本文研究有限不可约根系的 Chinta-Gunnells 平均在单短根极除子处的留数，给出由正交根子系平均表示的显式公式，并推广到有理函数域与 Gauss 域上的二次 Weyl 群多重 Dirichlet 级数。

Characterizations of the $d$th-power residue matrices over finite fields

In a recent paper of the author with D. Dummit and H. Kisilevsky, we constructed a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and proved a simple criterion characterizing such matrices. We then analyzed the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively. In this paper, the goal is to construct and study the fi…

关于四次多项式的二次特征和

On quadratic character sums over quartics

本文研究了多项式变元的二次特征和问题，推导出了四次和三次多项式变元的二次特征和的变换公式。

最小二次剩余与二次型表示的整数

The least quadratic residue and integers represented by quadratic forms

该研究针对模n的最小非平凡既约二次剩余ℓ(n)，从n的大小与素因子数维度建立近乎最优界，构造出ℓ(n)异常大的模，并将结果应用于推导有界判别式二元二次型表示不超过N的正整数的速率界。

关于大二次特征和的注记

Note on large quadratic character sums

本文在广义黎曼假设下，证明了二次狄利克雷特征和存在大值的Omega结果，深化了对该特征和极值行为的理解。

大二次特征和的结构

Structure of large quadratic character sums

本文基于Lamzouri的工作，研究大二次特征和的分布，并获得了具有大特征和的二次特征的结构结果。

Fekete polynomials, quadratic residues, and arithmetic

Fekete polynomials associate with each prime number $p$ a polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo $p$. These polynomials were first considered in the 19th century in relation to the studies of Dirichlet $L$-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomial…

$(1-2u^3)$-constacyclic codes and quadratic residue codes over $\mathbb{F}_{p}[u]/\langle u^4-u\rangle$

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^4$ is def…

Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues

In 2000, A. Tripathi used generating functions to obtain a formula for the number of non-negative solutions (x,y) of the equation ax + by = n where a, b and n are given positive integers. We generalize this procedure for the number of solutions of the equation ax + by + cz = n. The formula leads us to a surprising connection between the number of solutions of the equation ax + by + cz = n and quadratic residues. As a consequence of our work, we are able to prove the equivalence between two fundamental results by Gauss and Sylvester in the nineteenth century which are generally viewed to be ind…

Sums of fractions modulo $p$

Let $\F_p$ be the field of residue classes modulo a large prime $p$. The present paper is devoted to the problem of representability of elements of $\F_p$ as sums of fractions of the form $x/y$ with $x,y$ from short intervals of $\F_p$.

实二次域的算术 Dijkgraaf-Witten 不变量、二次剩余图与密度公式

Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

本文针对实二次域整数环，利用素数间的二次剩余图简化了 Hirano 的模 2 算术 Dijkgraaf-Witten 不变量公式，回答了 Ken Ono 的问题，并进一步给出了相应不变量的密度公式。

两平方和强烈偏向于二次剩余

Sums of two squares are strongly biased towards quadratic residues

本文研究了算术级数中两平方和的偏向问题，发现其偏向比素数更强，在自然密度意义上对几乎所有整数成立，且仅需 GRH 和 Chowla 型猜想即可证明。

Quadratic residues and a new infinity of orders for which a conjecture of Ryser about Circulant Hadamard matrices holds

For every positive integer $k$ such that $k>1,$ there are an infinity of odd integers $h$ with $ω(h) =k$ distinct prime divisors such that there do not exist a Circulant Hadamard matrix $H$ of order $n=4h^2.$ Moreover, our main result implies that for all of the odd numbers $h$, with $1< h < 10^{13}$ there is no Circulant Hadamard matrix of order $n=4h^2.$

大二次特征和

Large quadratic character sums

本文在广义黎曼假设下证明二次狄利克雷特征和的Omega结果，其精确度与之前大素数模所有特征的结果相当。

幂剩余、数字展开与相对类数

Power residues, digit expansions and relative class numbers

本文综述了模素数幂剩余分布与相对类数的联系，重点在二次和六次幂剩余，狄利克雷类数公式及几何在推广中起关键作用。

Explicit evaluation of quadratic Euler sums

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

Some Evaluation of Quadratic Euler Sums

In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta values, polylogarithm functions and linear sums. Furthermore, some relationships between Euler sums and integrals of polylogarithm functions are established.

Double Sums of Kloosterman Sums in Finite Fields

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.