T. Y. Lam, K. H. Leung
Consider the $m$-th roots of unity in {\bf C}, where $m>0$ is an integer. We address the following question: For what values of $n$ can one find $n$ such $m$-th roots of unity (with repetitions allowed) adding up to zero? We prove that the answer is exactly the set of linear combinations with non-negative integer coefficients of the prime factors of $m$.
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For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$.
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 $f_i$.
Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely often. The corresponding configurations were suggested by examining exact minimum values computed for n <= 221000. These minima can be explained at least in part by selection of the best example from multiple families of competing configurations related to close rational approximations.
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。
In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $ζ_1+\cdots +ζ_n=0$ of $m$th roots of unity $ζ_i$ in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic $p$. For given $m$ and $p$, results are obtained on integers $n_0$ such that all integers $n\geq n_0$ are in the ``weight set'' $W_p(m)$. The main result $(1.3)$ in this paper guarantees, under suitable conditions, the existence of solutions of $x_1^d+\cdots+x_n^d=0$ with all coordinates not equal to zero over a finite field.
Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets
本文研究了在 $2-\cdots-2,1-\cdots-1$ 指数下 $q$-多重调和和的显式表达式，探讨了指数组比值变化时的更一般结果。
We study sums of arithmetic functions, defined on Gaussian integers and taken over those pairs of integers whose coordinates give rise to a singular system.
We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
We consider the problem of whether or not certain mock theta functions vanish at the roots of unity with an odd order. We prove for any such function $f(q)$ that there exists a constant $C>0$ such that for any odd integer $n>C$ the function $f(q)$ does vanish at the primitive $n$-th roots of unity. This leads us to conjecture that $f(q)$ does not vanish at the primitive $n$-th roots of unity for any odd positive integer $n$.
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。
We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the $p$-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the…
We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the $p$-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.
We prove that the clique number of the Paley graph is at most $\sqrt{p/2} + 1$, and that any supposed additive decompositions of the set of quadratic residues can only come from co-Sidon sets.
We derive eight identities of symmetry in three variables related to generalized twisted Euler polynomials and alternating generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the $p$-adic integral expression of the generating function for the generalized twisted Euler polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the alternating generalized twiste…
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 $\mathbb{R}^2$.
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。
The Minimal Absolute Value of Sums of Fifth Roots of Unity
本文确定了可重复选取的n个五次单位根的非零和的最小绝对值，并刻画了达到最小值的和的结构。
The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.
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(…
We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by H. Bachmann, Y. Takeyama, and K. Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.
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.
We study the extension of a result of Loxton (1972) on representation of algebraic integers as sums of roots of unity to Kummer extensions.
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。
Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Minac, and Tan, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.
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。
We study the sum of the finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices at roots of unity with $r=1,2,3$. And we give the equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic $q$-series on $1\text{-}2\text{-}3$ indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.
For an abelian number field K containing a primitive p-th root of unity (p an odd prime) and satisfying certain technical conditions, we parametrize the Z_p[G(K/Q)]-annihilators of the "minus" part A_K^- of the p-class group by means of modules of Jacobi sums. Using a reflection theorem and Bloch-Kato's reciprocity law, we then determine the Fitting ideal of the "plus" part A_K^+ in terms of "twisted" Gauss sums.
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 $k\geq 3$…
Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least three terms such that $f(\ze_n)$ is a minimal vanishing sum of the $n$-th roots of unity.
On Matrices Arising in Finite Field Hypergeometric Functions
针对有限域超几何函数变换公式中出现、元素为单位根或零的一类矩阵，本文确定其特征多项式、特征值、特征向量与k次幂，并指出特征值由Jacobi和的乘积构成。
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。
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.
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。
The Ramanujan sum c_n(k) and a_n(k), the kth coefficient of the nth cyclotomic polynomial, are completely symmetric expressions in terms of primitive nth roots of unity. For fixed k we study the value distribution of c_n(k) (following A. Wintner) and a_n(k) (partly following H. Moller). In particular we disprove a 1970 conjecture of H. Moller on the average (over n) of a_n(k). We show that certain symmetric functions in primitive roots considered by the Dence brothers are related to the behaviour of c_{p-1}(k) and a_{p-1}(k) as p ranges over the primes and study their value distribution as wel…
On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums
本文研究了p次环形矩阵的算术性质，证明了其行列式与特定多项式系数的关系。
Let $N$ be a positive integer. In this paper we shall study the special values of multiple polylogarithms at $N$th roots of unity, called multiple polylogarithm values (MPVs) of level $N$. These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations f…
Periodic vanishings of the Legendre-17 signed partition numbers
本文研究Legendre-17符号分区数的周期性消失现象，推导了相关级数公式并证明其在特定算术进展上恒为零。
We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of $G$. To do so, we introduce an algebro-combinatorial model for these modules and develop for them a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the $p$-adic theory or to the natural grading parameter in the representa…
It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as an infinite product $\prod_p δ_p$ of local factors $δ_p$ reflecting the degree of the splitting field of $X^p-r$ at the primes $p$, multiplied by a somewhat complicated factor that corrects for the `entanglement' of these splitting fields. We show how the correction factors arising in Artin's original primitive root problem and some of its generalizations ca…
Ramanujan sums and rectangular power sums
该研究探讨了由拉马努金和与幂和对称函数构造的对称函数，证明了u=0和u=1时的Schur正性并给出对称群表示的分解，还猜想u=2时同样具有Schur正性。
We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, to higher dimensional abelian groups and integer polytopes. We consider the finite Weyl group $\mathcal{W}$, generated by the reflections with respect to the coordinate hyperplanes, as well as all permutations of the coordinates; further, we let $\mathcal G$ be the group generated by $\mathcal{W}$ as well as all integer translations in $\mathbb Z^d$. …
We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic.
The additive monoid $R_+(x)$ is defined as the set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(α)$ for complex numbers $α.$ Particularly interesting is the case of algebraic $α$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbb C\backslash \mathbb Z_+$. We prove that this object is a ring if and only if $α$ is an alg…
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