arXiv 1206.43552012-06-14math.NT

Variations on a question concerning the degrees of divisors of x^n-1

Lola Thompson

备注：: Formerly part of arXiv:1111.5403

英文摘要：

In this paper, we examine a natural question concerning the divisors of the polynomial x^n-1: "How often does x^n-1 have a divisor of every degree between 1 and n?" In a previous paper, we considered the situation when x^n-1 is factored in Z[x]. In this paper, we replace Z[x] with F_p[x], where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On Karatsuba's Problem Concerning the Divisor Function $τ(n)$

We study an asymptotic behavior of the sum $\sum\limits_{n\le x}\frac{\D τ(n)}{\D τ(n+a)}$. Here $τ(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.

2010-11-05math.NT

Alternating sums concerning multiplicative arithmetic functions

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordellès and Cloitre. We formulate certain open problems.

2016-12-29math.NT

An improvement of an inequality of Ochem and Rao concerning odd perfect numbers

Let $Ω(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $ω(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $ω(N)$ and $Ω(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $Ω\geq \frac{8}{3}ω(N)-\frac{7}{3}$ and if $3 |N$ then $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$

2017-10-29math.NT

关于Dirichlet除数问题中误差项的积分

On the integral of the error term in the Dirichlet divisor problem

得到了几个结果有关的功能$Δ_k(x)$，它代表的误差项在一般狄利克雷除数问题。这些包括该函数的积分估计，以及相应的均方积分。详细研究了$Δ_2(x)$的均方积分。

2005-10-06math.NT

关于含整数小素因子的广义除数函数的奇偶性结果

Parity results concerning the generalized divisor function involving small prime factors of integers

针对含整数小素因子的广义除数函数相关和式$S_{-k}(x,y)$，结合多种方法推导其渐近行为，揭示$k+1$素性对结果的关键影响，完善相关理论研究。

2024-12-04math.NT

L函数的狄利克雷级数的条件收敛性

The conditional convergence of the Dirichlet series of an L-function

以Dirichlet除数问题为模型，给出了关于L函数的狄利克雷级数条件收敛的一个猜想.

2009-03-05math.NT

更多相关论文收起更多论文

The reciprocal sum of divisors of Mersenne numbers

We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ fo…

2020-08-13math.NT

与素数II相关的Divisor和的高相关性：素数定理中误差项的变化

Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem

我们计算截断除数和$λ_{R}(n)$的三重相关。$λ_{R}(n)$在某些平均值上的行为就像素数计数冯曼戈尔特函数$Λ(n)$所做的或被要求做的那样。我们还计算了混合（因子为$Λ(n)$）相关性。结果的时刻到第三次，因此，影响的素数分布在短时间内，是相同的，我们得到的（在第一篇论文与此标题）通过使用更简单的近似$Λ_{R}(n)$。然而，当使用$λ_{R}(n)$时，奇异级数近似中的误差通常比$Λ_{R}(n)$允许的误差小得多。在Dirichlet $L$ -函数的广义Riemann假设下，得到了素数定理中误差项变分的$Ω_{\pm}$ -结果.以前，我们对GRH的了解仅限于$Ω$ -该变化绝对值的结果。在这项工作的最后一部分的一个重要组成部分是最近的结果由于蒙哥马利和Soundararajan这使得我们有可能免除一个大的误差项在评估某一奇异系列平均。我们相信我们关于$λ_{R}(n)$和$Λ_{R}(n)$的结果可以应用于有关素数的各种问题。

2004-12-18math.NT

新曼多项式及其因数的算法方面

Algorithmic aspects of Newman polynomials and their divisors

Université de Lorraine（洛林大学）

研究确定哪些整数多项式能整除新曼多项式，发现一个度数为10的多项式不整除任何新曼多项式，改进了关于所有整数多项式均能整除新曼多项式的上界。

2026-07-17math.NT

Variations on a theorem of Davenport concerning abundant numbers

Let σ(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/σ(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/σ(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including φ(n), τ(n), and μ(n). They also apply to the representation function for sums of two s…

2013-06-03math.NT

Shifts of the prime divisor function of Alladi and Erdős

We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prove that no unbounded sequences occur, analogous to the Catalan-Dickson conjecture, and give evidence towards the analogue of the Erdős-Granville-Pomerance-Spiro conjecture on the pre-image of $s(n)$.

2017-10-30math.COmath.NT

Expander graphs, gonality and variation of Galois representations

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

2011-09-28math.NT

A variation on the theme of Nicomachus

We prove some conjectures of K. Stolarsky concerning the first and third moment of the Beatty sequences with the golden section and its square.

2017-07-29math.CAmath.NT

关于可加函数的和函数与正则变分

On summatory functions of additive functions and regular variation

综述了关于一类可加函数的和函数的渐近性的结果和问题。这类函数的问题涉及卡拉马塔的定期变化。给出了$F(n) = \sum_{p^α||n}αh(p)$型可加函数的Abel和Tauberian的一些新结果.

2003-11-20math.CAmath.NT

Variation of the canonical height for a family of polynomials

A theorem of Tate asserts that, for an elliptic surface E/X defined over a number field k, and a section P of E, there exists a divisor D on X such that the canonical height of the specialization of P to the fibre above t differs from the height of t relative to D by at most a bounded amount. We prove the analogous statement for a one-parameter family of polynomial dynamical systems. Moreover, we compare, at each place of k, the local canonical height with the local contribution to the height relative to D, and show that the difference is analytic near the support of D, a result which is analo…

2011-05-05math.DSmath.NT