Bounds on ternary cyclotomic coefficients
Bartlomiej Bzdega
给出了$A = \max_n |a_{pqr}(n)|$的一个新的上界，其中$a_{pqr}(n)$是三元分圆多项式的系数.我们还证明了这样一个多项式的两个连续系数相差最多一个。
We present a new bound on $A = \max_n |a_{pqr}(n)|$, where $a_{pqr}(n)$ are the coefficients of a ternary cyclotomic polynomial. We also prove that two consecutive coefficients of such a polynomial differ by at most one.
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Neighboring ternary cyclotomic coefficients differ by at most one
一个分圆多项式Phi_n（x）称为三元多项式，如果n=pqr，其中p，q和r是不同的奇素因子.三元分圆多项式是最简单的多项式，其系数的行为尚未完全理解。Eli Leher在2007年证明了相邻的三元分圆系数最多相差4。我们表明，事实上，他们最多相差一个。因此，出现在三元分圆多项式中的系数集合由连续整数组成。作为应用，我们以一种更简单的方式重新证明了Bachman在2004年关于具有最优大系数集的三元分圆多项式的一个结果.
Jumps of ternary cyclotomic coefficients
已知三元分圆多项式$Φ_{pqr}(x)=\sum_k a_{pqr}(k)x^k$的两个连续系数相差至多1。本文给出了$k$满足$|a_{pqr}(k)-a_{pqr}(k-1)|=1$的一个判据。利用这一点证明了$n$次三元分圆多项式的非零系数个数大于$n^{1/3}$ .
On a certain family of inverse ternary cyclotomic polynomials
研究了一类逆三元分圆多项式$Ψ_{pqr}$，其中$r\leφ(pq)$是$p$和$q$的正线性组合.我们导出了这类多项式的高度公式，并刻画了这类多项式族中的所有平坦多项式。
The family of ternary cyclotomic polynomials with one free prime
一个分圆多项式Φ n（x）称为三元多项式，如果n=pqr，p，q，r是不同的奇素数.三元分圆多项式是最简单的多项式，其系数的行为尚未完全理解。本文建立了多项式族Φ_{pqr}（x）中系数的一些结果，并给出了一些公式，其中p<q<r，p和q是固定的，r是自由素数。
Non-Beiter ternary cyclotomic polynomials with an optimally large set of coefficients
设l>=1为任意奇数，p，q，r为素数.本文证明了存在无穷多个三元分圆多项式Φ pqr}（x），使得它们的系数集由区间[-（p-l-2）/2，（p+l+2）/2]中的p个整数组成，其中l ^2 +3l+5<= p<q<r.已知不可能有更大的系数范围。Beiter猜想指出Φ_{pqr}的分圆系数a_{pqr}（k）满足|a_{pqr}（k）|<=（p+1）/2，因此上述族与Beiter猜想相矛盾。本文给出了已知的两类具有最优大系数集的三元分圆多项式（G. Bachman）满足Beiter猜想。
In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $Φ_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2} \equiv1 \mod p_{1}$ and $p_{3} \equiv1 \mod {p_{1}p_{2}}$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_{2}$, when $p_{1}$ is sufficiently large.
Ternary cyclotomic polynomials having a large coefficient
令$Φ_n(x)$表示$n$次分圆多项式。在1968年修女马里昂贝特证明$a_n(k)$，系数$x^k$在$Φ_n(x)$，满足$|a_n(k)|\le (p+1)/2$的情况下$n=pqr$与$p<q<r$素数（在这种情况下$Φ_n(x)$被称为是三元）。从那时起，一些结果对建立她的猜想已被证明（例如$|a_n(k)|\le 3p/4$）。然而，在这里我们证明，Beiter的猜想对每一个$p\ge 11$都是假的。我们还证明了，给定任何$ε>0$有无限多个三元组$(p_j,q_j,r_j)$与$p_1<p_2<... $连续素数，使$|a_{p_jq_jr_j}(n_j)|>(2/3-ε)p_j$为$j\ge 1$。
We study the number of non-zero terms in two specific families of ternary cyclotomic polynomial, we find formulas for the number of terms by writing the cyclotomic polynomial as a sum of smaller sub-polynomials and study the properties of these polynomial.
On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials
本文证明了对于任意奇素数p和整数h，存在足够大的素数q和r，使得对应的循环多项式Φ_{pqr}的高度为h，并揭示了其直径的可能取值。
An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.
On Ternary Inclusion-Exclusion Polynomials
从组合的角度来看分圆多项式，会导致一个更大的多项式类，我们称之为包含-排除多项式。这为关于这些多项式的系数的某些类型的问题提供了更合适的设置。在建立了容斥多项式的一些基本性质之后，我们转而详细研究三元容斥多项式的结构。后一子类的例子是分圆多项式$Φ_{pqr}$，其中$p<q<r$是奇素数。我们的主要结果是，$Φ_{pqr}$的系数集仅仅是一串连续的整数，它只依赖于$r$模$pq$的剩余类.
Divisibility criteria and coefficient formulas for cyclotomic polynomials
本文给出多项式被分圆多项式整除的充要条件，推导拉马努金和的新公式，并揭示分圆多项式系数的新性质及指数相差素因子时的递推关系。
Lower bounds for coefficients of certain cyclotomic polynomials
本文证明了关于某些n的第n个分圆多项式系数最大绝对值下界的猜想，并指出该估计本质上是精确的。
In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.
Cyclotomic coefficients: gaps and jumps
我们改进了Hong，Lee，Lee and Park（2012）关于分圆多项式的间隙和Bzdogga（2014）关于分圆多项式系数间跳跃的几个最近结果。除了直接的改进，我们还介绍了一些从未在这方面使用过的新技术。
Coefficients of cyclotomic polynomials
令$a(n, k)$为$n$第分圆多项式的第$k$系数。最近，Ji，Li和Moree证明了对任意整数$m\ge1$，$\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$ .本文改进了这一结果，证明了对任意整数$s>t\ge0$，$$\{a(ns+t, k)| n, k\in\mathbb{N}\}=\mathbb{Z}.$$
Values of coefficients of cyclotomic polynomials II
设a（n，k）是第n个分圆多项式的第k个系数。前两个作者在第一部分中证明，如果m是一个素数幂，n和k的范围在非负整数，那么a（mn，k）假设每一个整数值。这里，这个结果被推广到m是任意的情况。证明使用了倒数分圆多项式的一些性质（参见arXiv：0709.1570）。
In this article, we provide a short and elementary proof of the following result: For $n \geq 3$ the middle coefficient of $Φ_n(x)$ is either zero (when $n$ is a power of $2$) or an odd integer.
Cyclotomic polynomials play an important role in several areas of mathematics and their study has a very long history, which goes back at least to Gauss (1801). In particular, the properties of their coefficients have been intensively studied by several authors, and in the last 10 years there has been a burst of activity in this field of research. This concise survey attempts to collect the main results regarding the coefficients of the cyclotomic polynomials and to provide all the relevant references to their proofs. Previous surveys on this topic were given by Lenstra (1979), Vaughan (1989),…
On Modular Inverses of Cyclotomic Polynomials and the Magnitude of their Coefficients
设p和r是两个素数，n，m是pr的两个不同因子，考虑n阶和m阶分圆多项式.在本文中，我们提出的下限和上界的系数的逆的其中一个模的其他。我们提到一个应用程序的环面为基础的密码。
This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with $t\geq 3$ odd, then the numbers $-(r-2),-(r-3),\ldots, r-2, r-1$ are all coefficients of the cyclotomic polynomial $Φ_{2n}$. Furthermore, if $1+p_r<p_1+p_2$ then $1-r$ is also a coefficient of $Φ_{2n}$.} In the experimental part, in two instances we present computational evidence for asymptotic symmetry between distribution of positive and negative coefficien…
The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $Φ_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m\ge 1$ we show that every integer occurs as a coefficient of $Φ^*_{mn}(x)$ for some $n\ge 1$. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/Φ_n^*(x),$ the inverse unitary cyclotomic polyno…
A unitary cyclotomic polynomial of order three is a polynomial of the form \[ Φ^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$, $q$ and $r$. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of $P=p^a$, $Q=q^b$ and $R=r^c$. We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of $\mathbb{Z}$.
Value distribution of cyclotomic polynomial coefficients
设a_n（k）是第n个分圆多项式Phi_n（x）的第k个系数。由于n的范围是整数，a_n（k）仅假定n个值。对于任意这样的值v，我们确定整数n的密度，使得a_n（k）= v，并且我们研究了a_n（k）的平均值。对于1/Phi_n（x）（取x=0）的第k个泰勒系数，第n个倒数分圆多项式的第k个系数，我们得到了类似的结果.我们提出了各种开放的问题。
On the coefficients of the cyclotomic polynomials of order three
我们说一个分圆多项式Phi_{n}（x）有三阶，如果n是三个不同素数的乘积，p<q<r。设A（n）是Phi_{n}（x）的系数的最大绝对值，M（p）是A（pqr）的最大值。1968年，马里恩·贝特修女证明了A（pqr）<=（p+1）/2。在2008年，Yves Gallot和Pieter Moree证明了这个猜想对所有p>=11都是假的，他们提出了修正的贝特猜想：M（p）<= 2 p/3。本文给出了修正的Beiter猜想成立的一个充分条件，并在p=7时予以证明.
Reciprocal cyclotomic polynomials
令$Ψ_n(x)$是首一多项式，其具有恰好所有的非本原单位根$n$作为其简单零点。一个是$Ψ_n(x)=(x^n-1)/Φ_n(x)$，其中$Φ_n(x)$是$n$的第分圆多项式。$Ψ_n(x)$的系数是整数，其与$Φ_n(x)$的系数一样，在绝对值上往往令人惊讶地小，例如，对于$n<561$，$Ψ_n(x)$的所有系数在绝对值上都是$\le 1$。我们建立了$Ψ_n(x)$的系数的各种性质。
Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses
对于固定素数$p$，存在分圆多项式$Φ_{pqr}(x)$的最大系数（绝对值）$M(p)$，其中$r$和$q$是满足$r>q>p$的自由素数。贝特修女在1968年发表了$M(p)\le(p+1)/2$。2009年，Gallot和Moree证明了$M(p)\ge 2p(1-ε)/3$对于每个$p$都足够大。本文应用Kloosterman和（“cloister man和”）等模逆分布的工具，对Beiter姐妹猜想的反例进行了量化，并锐化了$M(p)$的上述下界.
Contour Integration and Cyclotomic Apéry-Like Series Involving Generalized Binomial Coefficients
本文通过轮廓积分方法研究了一类涉及广义二项式系数的 cyclotomic 阿佩里型级数，推导出多重 polylogarithms 等表达式，并恢复了一些已知结果。
On a Class of Ternary Inclusion-Exclusion Polynomials
三元包含-排除多项式是\[ Q_{p,q,r}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \]形式的多项式，其中$p$、$q$和$r$是整数$\ge3$，并且成对互质。这类多项式包含，作为其主要子类，对应于限制$p$，$q$和$r$是不同的奇素数的三元分圆多项式。我们的目的是继续研究$Q_{p,q,r}$和$Q_{p,q,s}$的系数之间的关系，用$r\equiv s\pmod{pq}$。更具体地说，我们考虑$1\le s<\max(p,q)<r$的情况，并获得函数$A(p,q,r)$的递归估计-该函数给出$Q_{p,q,r}$的系数的最大绝对值。我们主要结果的一个简单推论是下面的绝对估计。如果$s\ge1$和$r\equiv\pm s\pmod{pq}$，则为$A(p,q,r)\le s$。
Borwein and Choi conjectured that a polynomial $P(x)$ with coefficients $\pm1$ of degree $N-1$ is cyclotomic iff $$P(x)=\pm Φ_{p_1}(\pm x)Φ_{p_2}(\pm x^{p_1})\cdots Φ_{p_r}(\pm x^{p_1p_2\cdots p_{r-1}})$$ where $N=p_1p_2\cdots p_{r}$ and the $p_i$ are primes, not necessarily distinct. Here $Φ_p(x):=(x^p-1)/(x-1)$ is the $p-$th cyclotomic polynomial. In \cite{1}, they also proved the conjecture for $N$ odd or a power of 2. In this paper we introduce a so-called $E-$transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to inve…
We present a method to deal with the values of polynomials of type $P(z)=\prod_{d\in D}(1-z^d)^{j_d}$ on the unit circle. We use it to improve the known bounds on various measures of coefficients of cyclotomic and similar polynomials.
Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients
第n阶分圆多项式的第k个系数的Ramanujan和c_n（k）与a_n（k）是关于第n阶本原单位根的完全对称表达式。对于固定的k，我们研究了c_n（k）的值分布（继A. Wintner）和a_n（k）（部分遵循H. Moller）。特别是我们反驳了1970年的猜想H。Moller关于a_n（k）的平均（在n上）。我们证明了Dence兄弟所考虑的原根中的某些对称函数与c_{p-1}（k）和a_{p-1}（k）在素数上的p范围的行为有关，并研究了它们的值分布。本文是第二作者的硕士论文项目的产物，在第一作者的监督下在Korteweg-de Vries研究所（阿姆斯特丹大学）进行，并以硕士论文风格撰写。我们感谢Yves Gallot提供的数字援助。
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach…
Binary Cyclotomic Polynomials: Representation via Words and Algorithms
本文研究二元分圆多项式，将其系数向量视为三元字母表上的字，设计了一种线性时间的最优算法以计算其紧凑表示，并推广到二维数值半群相关多项式。
On Gauss-Kraitchik formula for cyclotomic polynomials via symmetric functions
本文通过利用对称多项式与幂和多项式的关系，给出高斯-克雷奇克公式中 cyclotomic 多项式系数的显式上界。
Post-critically finite polynomials over cyclotomic closures of number fields
该研究针对数域K的分圆闭包，证明了次数不小于2的首一中心化后临界有限多项式的个数有限，属于数论领域的相关结论。
We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the minimal order of such a relation and the integrality of its coefficients.
On Nonzero Coefficients of Binary Cyclotomic Polynomials
本文研究两个不同素数乘积所对应分圆多项式的非零系数计数问题，将 Fouvry 的阈值由 12/25 改进至 9/20，并给出带显式常数的一致渐近公式。
Lucas-type congruences for cyclotomic $ψ$-coefficients
设p是任意素数，a是正整数。对于非负整数l、n和整数r，已知归一化分圆$ψ$系数$${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod p^a)}(-1)^k{n \choose k}{{(k-r)/p^a} \choose l}$$是整数.在本文中，我们表明，这个系数的行为类似于二项式系数，并满足一些卢卡斯型同余。这意味着Wan的一个同余往往是最优的，Sun和Davis的两个同余是真的。
A proof of the corrected Sister Beiter cyclotomic coefficient conjecture inspired by Zhao and Zhang
本文受赵和张工作的启发，通过补充大量细节完善了修正的Sister Beiter分圆多项式系数猜想$M(p)\le 2p/3$的证明，并利用该界改进了三元分圆多项式系数的已有估计，确定了三个新素数对应的精确最大值。
We determine necessary conditions for when powers corresponding to positive/negative coefficients of $Φ_{n}$ are in arithmetic progression. When $n = pq$ for any primes $q>p>2$, our conditions are also sufficient. Finally, we generalize the result when $n = pq$ to the so-called inclusion-exclusion polynomials first introduced by Bachman.
Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers
本文引入与正则除数系和任意正整数集相关的广义分圆多项式，证明其具有整数系数并可分解为经典分圆多项式的乘积，同时给出新的 Menon 型恒等式。
Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes $p$ and $p'$ with $p\ge 127$ we have $p'-p<\sqrt{p}+1.$ We also conjecture that every natural number occurs as maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture that always $\sqrt{p'}-\sqrt{p}<1$ holds. This is th…
We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.
Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then $f(a^n)=0$. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases $K=\mathbb{F}_p$ and $K=\mathbb{Q}$.
Flat Cyclotomic Polynomials: A New Approach
我们建立了一个新的理论，分析任何分圆多项式的系数考虑它作为一个gcd的简单多项式。利用这一理论，我们推广了一个称为周期性的结果，并给出了两个新的平坦分圆多项式族。一个是3阶的，由布罗德赫斯特证明：$Φ_{pqr}(x)$是平坦的，其中$p<q<r$是素数，并且存在正整数$w$使得$r\equiv\pm w\pmod{pq}$，$p\equiv1\pmod w$和$q\equiv1\pmod{wp}$。另一个是第一个4阶的一般族：$Φ_{pqrs}(x)$对于素数$p,q,r,s$是平坦的，其中$q\equiv-1\pmod p$，$r\equiv\pm1\pmod{pq}$和$s\equiv\pm1\pmod{pqr}$。最后，我们证明了这第二个家庭的自然延伸到5阶是从来没有平坦的，这表明没有平坦的分圆多项式的5阶。
Coefficients of a relative of cyclotomic polynomials
设$N=p_1p_2... p_n$是$n$不同素数的乘积。将$P_N(x)$定义为多项式$(1-x^N)\prod_{1\leq i<j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i})$。(When $n=2$，$P_{pq}(x)$是$pq$次分圆多项式，并且当$n=3$，$P_{pqr}(x)$是$(1-x)$乘以$pqr$次分圆多项式时.）设多项式的高度是其系数之一的最大绝对值。众所周知，$Φ_{pq}(x)$的高度是1，并且Gallot和Moree表明，当$n=3$时，对于$P_{pqr}(x)$也是如此。我们发现，$P_N(x)$的系数主要取决于$p_j^{-1} \pmod {p_i}$形式的剩余和的相对顺序。这使我们能够明确地描述当$n=3$时$P_N(x)$的系数，并表明当$n=4$时$P_N(x)$的高度至多为2。我们还表明，对于任何$n$存在$P_N(x)$的高度为1，但在一般情况下，$P_N(x)$的最大高度是一个函数，只依赖于$n$的增长率$2^{n^2/2+O(n\log n)}$。
We present an algorithm that computes the product of two n-bit integers in O(n log n (4\sqrt 2)^{log^* n}) bit operations. Previously, the best known bound was O(n log n 6^{log^* n}). We also prove that for a fixed prime p, polynomials in F_p[X] of degree n may be multiplied in O(n log n 4^{log^* n}) bit operations; the previous best bound was O(n log n 8^{log^* n}).
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。
On computing factors of cyclotomic polynomials
当n> 1时，n阶分圆多项式满足高斯恒等式。奥里菲耶、拉瑟尔和卢卡斯也有类似的身份。这些恒等式都涉及某些整系数整系数多项式。我们展示了如何通过简单的算法来计算这些系数，这些算法需要O（n^2）的算术运算并在整数上工作。给出了多项式的显式公式和生成函数，并通过数值例子说明了它在整数分解中的应用。
Gaps in binary cyclotomic polynomials
本文研究二元环形多项式Φ_{pq}的间隙，证明第二间隙为r-1和p-r-1的最大值，并确定特定条件下间隙数量。
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $Φ_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $Φ_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $Φ_n(e^{2πi/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $Φ_n(x)$. We also obtain a formula for $Φ_n'(e^{2πi/m}) / Φ_n(e^{2πi/m})$ with $n \ne m$, which is …
The scaled inverse of a nonzero element $a(x)\in \mathbb{Z}[x]/f(x)$, where $f(x)$ is an irreducible polynomial over $\mathbb{Z}$, is the element $b(x)\in \mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c \pmod{f(x)}$ for the smallest possible positive integer scale $c$. In this paper, we investigate the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $Φ_{p^s}(x)$ or $Φ_{p^s q^t}(x)$, where $p, q$ are primes with $p<q$ and $s, t$ are positive integers. Our main results are that the coefficient size of the scaled inverse of $(x^i-x^j)$ is bounded by $p-1$ with the scale $p$ modul…
It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $Φ_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show t…
Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas
本文结合伽罗瓦轨道与有限频率展开，研究分圆余弦求值，揭示值层面伽罗瓦不变性缺乏稳定范围刚性，并证明特定模式族在足够大水平下坍缩为仿射函数，最终用拉马努金和给出精确轨道公式。
Sharp Diameter Bounds for Nonnegative Cyclotomic Multiples
本文证明非负循环分圆倍式的支撑直径下界为(p-1)N/p，并刻画等式情形，证实Steinberger猜想，进而建立Coven-Meyerowitz直径界及其改进。
Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficien…
For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements a…
The size of coefficients of certain polynomials related to the Goldbach conjecture
最近的工作Borwein，崔，和第二作者研究了一个集合的多项式密切相关的哥德巴赫猜想：多项式$F_N$是整除的$N$次分圆多项式当且仅当没有表示$N$作为两个奇素数的总和。这些多项式的系数稳定，作为$N$增长，以一个固定的序列$a(m)$ ;他们推导出上限和下界为$a(m)$，和一个渐近公式的总和功能$A(M)$的序列，无论是假设一个著名的猜想哈代和利特尔伍德。本文改进了这些结果：在同样的假设下得到了$a(m)$的一个渐近公式，并无条件地建立了$A(M)$的渐近公式.
Bilinear sums with $GL(2)$ coefficients and the exponent of distribution of $d_3$
本文利用 delta 符号方法估计带 $GL(2)$ 系数的双线性和，将三元除数函数 $d_3$ 的分布指数提升至 $1/2+1/30$，改进了多项先前结果。
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