Sums over Vanishing Determinants

Estimates for the growth of inverse determinant sums of quasi-orthogonal and number field lattices

Inverse determinant sums appear naturally as a tool for analyzing performance of space-time codes in Rayleigh fading channels. This work will analyze the growth of inverse determinant sums of a family of quasi-orthogonal codes and will show that the growths are in logarithmic class. This is considerably lower than that of comparable number field codes.

Kloosterman和的消失性质与Dyson猜想

Vanishing properties of Kloosterman sums and Dyson's conjectures

本文通过证明模5和模7情形下Kloosterman和的特殊消失性质，为Dyson秩猜想及Ramanujan同余式提供了新证明。

Small Solutions of Quadratic Congruences, and Character Sums with Binary Quadratic Forms

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.

Binary polynomial power sums vanishing at roots of unity

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$. We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.

由Frobenius行列式求三角数的和

Sums of triangular numbers from the Frobenius determinant

我们表明，分母公式的奇怪系列的仿射超代数，由卡茨和Wakimoto，证明了Zagier，从一个经典的行列式评价Frobenius。作为极限情况，我们得到了任意数表示为4m^2/d个三角形之和（当d整除2 m时）和4 m（m+1）/d个三角形之和（当d整除2 m或d整除2 m +2时）的精确公式。这扩展了Getz和Mahlburg，Milne和Zagier最近的结果。

关于有限域中$\,m\,$单位根的零和

On vanishing sums of $\,m\,$th roots of unity in finite fields

在早期的工作中，作者已经确定了所有可能的权重$n$，其中存在特征0中的$m$单位根$ζ_i$的消失和$ζ_1+\cdots +ζ_n=0$。本文在特征$p$的有限域上研究了同样的问题。对于给定的$m$和$p$，在整数$n_0$上获得结果，使得所有整数$n\geq n_0$都在"权重集“$W_p(m)$中。本文的主要结果$(1.3)$在适当的条件下保证了有限域上所有坐标不为零的$x_1^d+\cdots+x_n^d=0$解的存在性。

逆行列式和及衰落信道信息论与代数的联系

Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

本文主要研究了由错误概率的联合界所引起的逆行列式和，作为设计和分析代数空时分组码的一种工具。建立了研究这些和的一般框架，并研究了逆行列式和的渐近增长与分集复用增益权衡之间的联系。证明了除代数空时码逆行列式和的增长完全取决于单位群的增长。这减少了逆行列式和分析研究某些李群中的渐近积分。利用最近的遍历理论的方法，最著名的代数空时码的逆行列式和的一个完整的分类。该方法揭示了李群中分集复用增益权衡和点计数之间有趣而紧密的关系。

Thue-Morse序列的Hankel行列式不为零的组合证明

A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

1998年，Allouche，Peyrière，Wen和Wen建立了与$\{-1, 1\}$上的Thue-Morse序列相关的Hankel行列式总是非零的。他们的证明依赖于一组十六个递归关系。我们提出了一个替代的，纯粹的组合证明相同的结果。我们还重新证明了最近的结果Coons的非零的Hankel行列式与其他两个经典的整数序列。

Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.

根的单位根的最小消失和分类

Classifying minimal vanishing sums of roots of unity

本文通过手工分类重量不超过16的根的单位根的最小消失和，揭示了超出1998年分类之外的新现象，并开发算法探索更高重量的分类扩展。

Moments of Spinor L-Functions and symplectic Kloosterman sums

We compute the second moment of spinor $L$-functions at central points of Siegel modular forms on congruence subgroups of large prime level $N$ and give applications to non-vanishing.

$p$-adic理论中的部分托里克指数和

$p$-adic Theory for Partial Toric Exponential Sums

本文通过Dwork方法提出$p$-adic证明，展示部分$L$-函数可表示为交替乘积的扭曲Fredholm行列式，并构造$p$-adic上同调理论及公式，证明其具有唯一的$p$-adic单位根。

An analogue of Siegel's determinant

Siegel-Shidlovskii theory of $E$-functions involves a non-vanishing proof for the determinants attached to the linear forms $D^kR(t)$, derivatives of an auxiliary function $R(t)$. Let a non-zero function $F(t)$ satisfy $m$th order linear differential equation which we shall write using the differential operator $Δ=tD$ and let $L(t)$ be any non-zero linear form of the derivatives $Δ^i F(t)$ $(i=0,...,m-1; m\ge 2)$. The determinants $\det\mathcal A_k$ attached to the linear forms $Δ^kL(t)$ have certain simple properties that allow us to give a short proof for the non-vanishing of $\det\mathcal A…

Vanishing sums of roots of unity and the Favard length of self-similar product sets

We improve a special case of the Lam-Leung lower bound on the number of elements in a vanishing sum of $N$-th roots of unity. Using this result, we extend the Favard length estimates due to Bond, Łaba, and Volberg to a new class of rational product Cantor sets in $\mathbb{R}^2$.

Minimizing GCD sums and applications to non-vanishing of theta functions and to Burgess' inequality

In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the sum $\sum_{m_1 , m_2 \leqslant N} w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the intertwined question of minimizing a weighted version of the usual multiplicative energy. We give three applications of our results. Firstly we obtain a logarithmic refinement of Burgess' bound on character sums $\displaystyle{\sum_{M<n\leqslant M+N}χ(…

On the Matrix-Element Expansion of a Circulant Determinant

The determinant of an $N \times N$ circulant matrix $M = {\rm CIRC}[x_0, x_1, ..., x_{N-1}$] can be expanded in the form det$ ~M= \sum C_{a_0 a_1 ...a_{N-1}} x_{a_0} x_{a_1}...x_{a_{N-1}}$. By using the generating function of a restricted, mod-$N$ partition function, we derive a formula for the coefficients in this expansion as finite sums over products of binomial coefficients with integer variables.

四个k次幂的和与差

Sums and differences of four k-th powers

我们证明了一个上界的数量表示的正整数$N$的总和为四个$k$次幂的整数大小最多$B$，使用一个新版本的行列式方法开发的希斯-布朗，随着最近的结果Salberger的密度上的整数点仿射曲面。更一般地，我们考虑任何积分对角形式的表示。上界的形式为$O_{N}(B^{c/\sqrt{k}})$，而在这种情况下，行列式方法的早期版本将产生$k^{-1/3}$阶的$B$指数。进一步证明了正整数$N$表示为非负整数的四个$k$次幂之和的个数对于$k \geq 3$至多为$O_ε(N^{1/k+2/k^{3/2}+ε})$，从而改进了Wisdom的界.

计数具有平方自由行列式的整数矩阵

Counting integer matrices with square-free determinants

研究整数矩阵集合中行列式为平方自由数的矩阵数量，并推导出相关渐进行为，同时改进方法计算矩阵行列式与欧拉函数的和。

A Spectral reciprocity formula and non-vanishing for L-functions on GL(4)xGL(2)

We develop a reciprocity formula for a spectral sum over central values of L-functions on GL(4)xGL(2). As an application we show that for any self-dual cusp form Pi for SL(4,Z), there exists a Maass form pi for SL(2,Z) such that L(1/2, Pi x pi) is nonvanishing. An important ingredient is a "balanced" Voronoi summation formula involving Kloosterman sums on both sides, which can also be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues.

通过狄利克雷 L 函数的特殊值研究三角行列式

Trigonometric Determinants via special values of Dirichlet $L$-Functions

本文通过狄利克雷 L 函数的特殊值研究三角行列式，并推导出正弦矩阵行列式的显式公式。

Sums of powers of integers via differentiation

For integer $k \geq 0$, let $S_k$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. For any given $k$, the power sum $S_k$ can in principle be determined by differentiating $k$ times (with respect to $x$) the associated exponential generating function $\sum_{k=0}^{\infty}S_k x^k/k!$, and then taking the limit of the resulting differentiated function as $x$ approaches $0$. In this paper, we exploit this method to establish a couple of seemingly novel recurrence relations, one of them involving the even-indexed power sums $S_2, S_4,\ldots, S_{2k}$,…

多重Ramanujan和的算术性质

Arithmetical properties of Multiple Ramanujan sums

本文引入了算术函数的多重Ramanujan和，它是D. R. Anderson和T. M.阿波斯托然后，我们发现的基本算术性质的多重Ramanujan和和研究几种类型的狄利克雷级数涉及的多重Ramanujan和。作为应用，我们评估高维矩阵的高维行列式，其中的条目是由多个Ramanujan和的值。

最优同调消失：特征和相消与 $\mathbb{F}_q[t]$ 上的 Patterson 猜想

Optimal homological vanishing: cancellation of character sums and Patterson's conjecture over $\mathbb{F}_q[t]$

本文证明辫子向量空间同调群的显式消失线，其斜率随 n 增大收敛至最优值，并应用于函数域上高阶高斯和偏差界（推广 Patterson 猜想）及 Galois 扩张中几乎所有特征和近平方根相消。

加权二进制数字和的汉克尔行列式

Hankel determinants of weighted binary sums of digits

研究加权二进制数字和的汉克尔行列式\(\mathcal{H}_\mathbf{w}(n)\)，推导递归式计算它，给出普通二进制数字和时的闭式解，还研究\(w_j = t^j\)时情况及相关一阶差分的行列式，推广了倍周期序列汉克尔行列式的结果。

The Legendre determinant form for Drinfeld modules in arbitrary rank

For each positive integer $r$, we construct a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\GL_r(\FF_q[θ])$, necessarily of least possible weight, via determinants using rigid analytic trivializations of the universal Drinfeld module of rank $r$ and deformations of vectorial Eisenstein series. Along the way, we deduce that the cycle class map from de Rham cohomology to Betti cohomology is an isomorphism for Drinfeld modules of all ranks over $\FF_q[θ]$.

高阶双射与全局函数域上的加权莫比乌斯和

Higher Order Dualities over Global Function Fields and Weighted Möbius Sums over $\mathbb{F}_q{[T]}$

本文在全局函数域中建立了Alladi双射恒等式，扩展了Duan等人2021年的结果至更高阶，研究了受限于素因子子集的加权莫比乌斯和的渐进行为。

二次行列式方程的解计数

Counting solutions to the quadratic determinant equation

本文研究了二次行列式方程x₁x₂ - x₃x₄ = h在整数解中的渐进行为，通过组合与分析方法结合 Kloosterman 和界，证明了当h = N² + O(N)时的平方根消去误差项，验证了Dhanda-Haynes-Prasala的猜想。

二元加法方程的素数解与特征和混合矩

Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums

本文通过泊松求和与Kloosterman分数和均值估计，得到二元加法方程素数解的渐近公式，并应用于特征和混合四阶矩，获得非平凡界。

代数数对数矩阵的秩 II：矩阵系数猜想

Ranks of Matrices of Logarithms of Algebraic Numbers II: The Matrix Coefficient Conjecture

本文针对代数数对数方阵的行列式非零性这一数论问题，提出矩阵系数猜想，称行列式为零的该类矩阵经左右有理基变换后可得到含零系数的矩阵。

Small Gál sums and applications

In recent years, maximizing Gál sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the $L^1$-norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of non vanishing theta fun…

在临界值处消失的爱森斯坦级数的L-级数

L-series of Eisenstein series vanishing at critical values

本文利用有理函数、余切和与周期多项式的关系，刻画了L-级数在临界值处消失的爱森斯坦级数，并在权较大时确定了具有同时消失性质的子空间。

行列式曲面上的格点与自守拉普拉斯算子的谱

Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian

利用经典傅里叶分析和自守形式谱论，推导了扩张盒子[-X,X]^4中行列式方程ad-bc=r的整数解个数的渐近公式，并应用于加权和的研究。

中心点处Dirichlet $L$-函数的平均非零性

Average non-vanishing of Dirichlet $L$-functions at the central point

该研究突破广义黎曼假设限制，通过对模q本原特征及区间内q双重平均，无条件证明至少50.073%的Dirichlet L函数中心点值非零，采用三段软化子的软化法，依赖Kloosterman和估计，同时指出存在可修正的错误。

On the smallest number of terms of vanishing sums of units in number fields

Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$…

单位根的小和

Small sums of roots of unity

研究\(k\)项\(N\)次单位根非零和的最小值问题，给出\(N^{-\alpha_k}\)形式上界，\(\alpha_k\)随\(k\)增大，还探讨对部分\(N\)改进界的方法。

关于结构化余弦和及其应用

On structured cosine sums and applications

研究循环群\(\mathbb{Z}/N\mathbb{Z}\)上多重集\(S\)相关的余弦和，借助单位根消失和等理论得出余弦和消失准则及傅里叶刚性，应用于循环凯莱图得到零特征值等相关结果。

关于带有角度限制的L-函数中心值非消失性

Non-vanishing of central values of L-functions with angular restrictions

本文通过迹函数理论和 mollification 技术，研究了带有角度限制的L-函数中心值非消失性，利用Katz和Fouvry等人结果证明了正比例的非消失性。

Vanishing of Quartic and Sextic Twists of $L$-functions

Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,χ)$ as $χ$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on $E$. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Earlier work by David, Fearnley and Kisilevsky formulates analogous conjectures for characters of any odd prime order. In the composite order case, however, we need to justify …

Vanishing of Multiple Zeta Values over $\mathbb{F}_q[t]$ at Negative Integers

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper, we study the vanishing behavior of multizeta values over $\mathbb{F}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums $S_d(k)$ which are polynomials in $t$. By studying the $t$-valuation of $S_d(s)$ for $s < 0$, we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of "greedy element" by Carlitz.

有限域上高亏格曲线的多重 zeta 值的非零性

Non-vanishing of multiple zeta values for higher genus curves over finite fields

本文在间隙序列假设下证明有限域上高亏格曲线函数域的 ∞-进多重 zeta 值非零，并通过推广 Sheats–Thakur 方法计算相关幂和绝对值来完成证明。

On linear independence measures of the values of Mahler functions

In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Padé approximation polynomials. Based on the non-vanishing of these determinants, we apply the functional equations to get an infinite sequence of approximations which is used to produce the linear independence measures.

关于整数作为三角形数和广义五边形数的加权和表示的周期性符号变化

On periodic sign changes for weighted representations of integers as colored sums of triangular and generalized pentagonal numbers

研究整数表示为三角形数与三色广义五边形数之和的表示数的符号变化和消失，通过分析Dedekind-eta函数相关q级数展开的系数，并利用Hardy-Ramanujan-Rademacher圆法给出渐近公式。

整数平方和表示研究的新进展

Recent progress in the study of representations of integers as sums of squares

在这篇文章中，我们收集了最近的结果表示整数作为一个偶数平方和的启发下，Kac和Wakimoto的图解。我们开始与一个草图米尔恩的证明这两个apturtures。我们还展示了另一种途径来推导这两个矩阵从米尔恩的行列式公式的总和$4s^2$，分别$4s(s+1)$，三角形数。这种方法的灵感来自Zagier的证明Kac-Wakimoto公式通过模形式。我们以第一作者和蔡美儿最近的照片结束了调查。

谱型二重序列的Zeta行列式

Zeta determinant for double sequences of spectral type

我们研究谱函数，特别是zeta函数，与一类复数序列，称为谱型。我们调查的zeta函数的可分解性与一个双序列相对于一些简单的序列，我们提供了一种技术，用于获得首项在罗朗展开在零的zeta函数与一个双序列。我们将这种技术具体化到谱型序列的求和的情形，并给出两个应用：第一个应用于数论中出现的某些特殊函数，第二个应用于积空间上拉普拉斯算子的函数行列式。

$\mathbb{F}_p$ 上的椭圆曲线与 Legendre 矩阵的行列式

Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices

本文研究以 Legendre 符号为元素的行列式与有限域椭圆曲线的关系，证实了 Sun 提出的若干猜想，并证明了存在无穷多个奇素数使得特定矩阵的行列式为零。

通过在SL_+(2,Z)上 telescoping 评估格点求和：一个证明∑1/(|x|²|y|²|x+y|²)=π/4和Zagier恒等式的简短证明

Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity

本文通过在SL_+(2,Z)上引入telescoping方法，证明了格点求和∑1/(|x|²|y|²|x+y|²)=π/4，并给出了Zagier恒等式的简短证明。

Surprising symmetries in distribution of prime polynomials

The primes or prime polynomials (over finite fields) are supposed to be distributed `irregularly' , despite nice asymptotic or average behavior. We provide some conjectures/guesses/hypotheses with `evidence' of surprising symmetries in prime distribution. At least when the characteristic is $2$, we provide conjectural rationality and characterization of vanishing for families of interesting infinite sums over irreducible polynomials over finite fields. The cancellations responsible do not happen degree by degree or even for degree bounds for primes or prime powers, so rather than finite fields…

精确平方和公式、雅可比椭圆函数、连分数和Schur函数的无限族

Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions

在本文中，我们得到了许多无限族的显式精确公式涉及平方或三角数，其中两个推广雅可比的4和8平方的身份，分别为4n^2或4 n（n+1）平方，而不使用尖点形式。事实上，我们同样推广到无限家庭的所有雅可比的明确规定的程度2，4，6，8兰伯特级数展开的经典θ函数。此外，我们还将Jacobi关于2个正方形、2个三角形、6个正方形、6个三角形的特殊分析分别推广到12个正方形、12个三角形、20个正方形、20个三角形。这些结果，取决于新的扩展的权力的各种产品的经典θ函数，出现在设置的雅可比椭圆函数，相关的连分数，经常C-分数，汉克尔或Turanian决定，傅里叶级数，兰伯特级数，包含/排除，拉普拉斯展开式公式的决定，和舒尔功能。这些无穷族恒等式的舒尔函数形式类似于麦克唐纳的η-函数恒等式。此外，麦克唐纳工作中出现的幂4 n（n+1）、2n^2+n、2n^2-n也出现在我们分析的适当位置。我们的一般方法的一个特殊情况产生的两个Kac-Wakimoto证明的身份涉及表示一个正整数由4n^2或4 n（n+1）三角数的和，分别。

Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz number…

Fox-Neuwirth 胞腔、量子洗牌代数与结式的特征和

Fox-Neuwirth cells, quantum shuffle algebras, and character sums of the resultant

本文利用拓扑方法计算多重穿孔平面辫群同调并证明混合辫群同调的消没范围，从而给出给定次数首一无平方因子多项式对上结式特征和的上界，回答了 Ellenberg 和 Shusterman 在二次情形下的问题。