Journal of Number Theory最新文献

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Elementary abelian Sylow subgroups of the multiplicative group 乘法群的初等阿贝尔西鲁子群

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.021

S. Morales , G. Polanco , P. Pollack

Erdős and Pomerance have shown that

φ (n)

typically has about

\frac{1}{2} {(\log \log n)}^{2}

distinct prime factors. More precisely,

ω (φ (n))

has normal order

\frac{1}{2} {(\log \log n)}^{2}

. Since

φ (n)

is the size of the multiplicative group

{(Z / n Z)}^{\times}

, this result also gives the normal number of Sylow subgroups of

{(Z / n Z)}^{\times}

. Recently, Pollack considered specifically noncyclic Sylow subgroups of

{(Z / n Z)}^{\times}

, showing that the count of those has normal order

\log \log n / \log \log \log n

. We prove that the count of noncyclic Sylow subgroups that are elementary abelian of a fixed rank

k \geq 2

has normal order

\frac{1}{k (k - 1)} \log \log n / \log \log \log n

. So for example, (typically) among the primes p for which the p-primary component of

{(Z / n Z)}^{\times}

is noncyclic, this component is

Z / p Z \oplus Z / p Z

about half the time. Additionally, we show that the count of p for which the p-Sylow subgroup of

{(Z / n Z)}^{\times}

is not elementary abelian has normal order

2 \sqrt{π} \sqrt{\log \log n} / \log \log \log n

Erdős和Pomerance已经证明φ(n)通常有大约12(log (log))2个不同的质因数。更准确地说，ω(φ(n))的正规阶是12(log log)2。由于φ(n)是乘法群（Z/nZ） x的大小，因此该结果也给出了（Z/nZ） x的Sylow子群的正常数目。最近，Pollack特别考虑了（Z/nZ） x的非循环Sylow子群，证明了这些子群的数量具有正阶log (n) /log (n) log (n)。证明了固定秩k≥2的初等阿贝尔的非循环Sylow子群的计数具有正态阶为1k(k−1)log (log)log (n) /log (log)log (log)log (n)。例如，（典型地）在（Z/nZ） x的p初级分量是非循环的素数p中，这个分量大约有一半的时间是Z/pZ⊕Z/pZ。此外，我们证明了（Z/nZ） x的p- sylow子群不是初等阿贝尔的p的计数具有正规阶2πlog (log) log (n) /log (n) log (n) log (n) log (n) log (n)。

引用次数: 0

Remarks on the Boston's unramified Fontaine-Mazur conjecture 论波士顿未被证实的方丹-马祖尔猜想

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.019

Yufan Luo

This paper studies Boston's generalization of the unramified Fontaine-Mazur conjecture for Galois representations. The first main result establishes that the conjecture can be verified by restricting to the cases of p-adic Galois representations and

F_{p} [[T]]

-adic representations. The second main result is a finiteness theorem for the associated unramified Galois deformation rings under certain conditions.

本文研究了伽罗瓦表示下未分枝的Fontaine-Mazur猜想的波士顿推广。第一个主要结果建立了该猜想可以通过限制p进伽罗瓦表示和Fp[[T]]进表示的情况来验证。第二个主要结果是在一定条件下相关的无分支伽罗瓦变形环的有限定理。

引用次数: 0

Compression and complexity for sumset sizes in additive number theory 可加数论中集合大小的压缩和复杂性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.025

Melvyn B. Nathanson

The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and

B_{h}

-sets). This paper considers the sets

R_{Z} (h, k)

and

R_{Z^{n}} (h, k)

of all sizes of h-fold sums of sets of k integers or of k lattice points, and the geometric and computational complexity of the sets

R_{Z} (h, k)

and

R_{Z^{n}} (h, k)

. For sumsets hA with large diameter, there is a compression algorithm to construct sets

A^{'}

with

| h A^{'} | = | h A |

and small diameter.

有限整数集和的研究主要集中在具有小集合的集合（Freiman定理和相关工作）和具有大集合的集合（Sidon集合和bh集合）。本文考虑了k个整数集或k个格点集的h倍和的各种大小的集合RZ（h,k）和RZn(h,k)，以及集合RZ（h,k）和RZn（h,k）的几何复杂度和计算复杂度。对于直径较大的sumsets hA，有一种压缩算法来构造直径较小的|hA ‘ |=|hA|的集合a ’。

引用次数: 0

Some new results on the higher energies 关于高能量的一些新结果

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.018

I.D. Shkredov

We obtain a generalization of the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms.

我们得到了最近关于避免长度为3的等差数列的Kelley-Meka结果的推广。在我们的证明中，我们发展了高能量理论。此外，我们还讨论了长等差数列的情况，以及一类一般的范数，其中包括高能量范数和高尔斯范数。

引用次数: 0

Analytic rank growth of elliptic curves over cyclic extensions 椭圆曲线在循环扩展上的解析秩增长

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.026

Gyeongwon Oh , Peter J. Cho

Let E be an elliptic curve defined over

Q

. For an odd prime l, we consider the family of degree l cyclic extensions K over

Q

. When we view the elliptic curve E as a curve over K, the analytic rank of the L-function

L_{K} (s, E)

of E over K may increase compared to that of the L-function

L_{Q} (s, E)

of E over

Q

. Under the generalized Riemann hypothesis, we demonstrate the rarity of significant increases in analytic ranks.

设E是定义在q上的椭圆曲线，对于奇素数l，我们考虑l次循环扩展K / q族。当我们把椭圆曲线E看作是K上的曲线时，E / K上的l函数LK（s，E）的解析秩可能比E / q上的l函数LQ（s，E）的解析秩增加。在广义黎曼假设下，我们证明了解析秩显著增加的稀有性。

引用次数: 0

Artin's conjecture for Abelian varieties with Frobenius condition 具有Frobenius条件的阿贝尔变的Artin猜想

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-21 DOI: 10.1016/j.jnt.2025.09.017

Florian Hess , Leonard Tomczak

Let A be an abelian variety over a number field K of dimension r,

a_{1}, \dots, a_{g} \in A (K)

and

F / K

a finite Galois extension. We consider the density of primes

p

of K such that the quotient

\bar{A} (k (p)) / 〈 {\bar{a}}_{1}, \dots, {\bar{a}}_{g} 〉

has at most

2 r - 1

cyclic components and

p

satisfies a Frobenius condition with respect to

F / K

, where

\bar{A}

is the reduction of A modulo

p

k (p)

is the residue field of

p

and

〈 {\bar{a}}_{1}, \dots, {\bar{a}}_{g} 〉

is the subgroup generated by the reductions

{\bar{a}}_{1}, \dots, {\bar{a}}_{g}

. We develop a general framework to prove the existence of this density under the Generalized Riemann Hypothesis.

设A是维度为r， a1，…，ag∈A(K)的数域K上的阿贝尔变分，F/K是有限伽罗瓦扩展。我们考虑K的素数p的密度，使得商A¯(K (p))/ < A¯1，…，A¯g >最多有2r−1个循环分量，并且p满足关于F/K的Frobenius条件，其中A¯是A模p的约简，K (p)是p和< A¯1，…的残域，A¯g >是由约简A¯1，…，A¯g产生的子群。在广义黎曼假设下，我们建立了一个证明该密度存在的一般框架。

引用次数: 0

Irreducibility of the characteristic polynomials of random tridiagonal matrices 随机三对角矩阵特征多项式的不可约性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-03 DOI: 10.1016/j.jnt.2025.09.008

Lior Bary-Soroker , Daniele Garzoni , Sasha Sodin

Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard–Varjú (for polynomials with independent coefficients), and with those of Eberhard and Ferber–Jain–Sah–Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.

在一定Dedekind zeta函数的Riemann假设条件下，证明了一类大维随机三对角矩阵的特征多项式是不可约的，其概率指数接近于1；有理数上的伽罗瓦群是对称群或交替群。这是Breuillard-Varjú（对于具有独立系数的多项式）以及Eberhard和Ferber-Jain-Sah-Sawhney（对于完全随机矩阵）的结果的对应。我们还分析了一类相关的随机三对角矩阵，其中伽罗瓦群要小得多。

引用次数: 0

M-functions and screw functions: Applications to Goldbach's problem and zeros of the Riemann zeta-function m函数和螺旋函数：在哥德巴赫问题和黎曼ζ函数零点中的应用

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.013

Kohji Matsumoto , Masatoshi Suzuki

We study the M-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the M-functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such M-functions and screw functions in generalized settings by axiomatizing them.

研究了描述哥德巴赫计数函数求和函数渐近公式中次主项值分布的极限定理的m函数。其中一个新的方面是由m函数的一些公式提供的黎曼假设的充分条件，这是以前工作中的必要条件。另一个新的方面是次要主项与螺旋函数之间的关系，这为黎曼假设提供了另一个充分必要条件。我们通过公理化的方法研究了广义环境下的m函数和螺旋函数。

引用次数: 0

Hilbert modular Eisenstein congruences of local origin 局部原点的希尔伯特模爱森斯坦同余

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.009

Dan Fretwell , Jenny Roberts

Let F be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight

k \geq 3

Hilbert eigenforms of level

mp

and Hilbert Eisenstein series of level

m

, for arbitrary ideal

m

and prime ideal

p ∤ m

O_{F}

. Such congruences have their moduli coming from special values of Hecke L-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving general results about this.

设F是一个任意的全实场。在标准条件下，我们证明了对于of的任意理想m和素数理想p∤m， mp层的平行权k≥3个希尔伯特特征形式与m层的希尔伯特爱森斯坦级数之间存在一定的爱森斯坦同余。这种同余的模来自于Hecke l函数的特殊值及其欧拉因子，并且我们的结果允许特征形式具有非平凡的Hecke特征。在此之后，我们考虑了新形式何时可以满足这种同余的问题，并证明了关于它的一般结果。

引用次数: 0

Asymptotics and limiting distributions of several overpartition statistics 几个过划分统计量的渐近性和极限分布

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-10-02 DOI: 10.1016/j.jnt.2025.09.010

Helen W.J. Zhang, Ying Zhong

This paper primarily is dedicated to studying the asymptotics and limiting distributions of several statistics in overpartitions. As a preliminary result, we use asymptotic methods to prove that the number of distinct parts and distinct integers in overpartitions is asymptotically normal, extending the results of Corteel and Hitczenko. Furthermore, we investigate the asymptotic and distributional properties of two types of crank statistics for overpartitions, originally introduced by Bringmann and Lovejoy. Utilizing the Hardy-Ramanujan circle method, we derive asymptotic formulas for the moments of these two cranks, as well as for the symmetrized moments proposed by Jennings-Shaffer. Building on these, we employ the probabilistic method of moments to prove that both two cranks asymptotically follow a logistic distribution when appropriately normalized. Consequently, our results recover the asymptotic formulas for the positive moments first obtained by Zapata Rolon using Wright's circle method.

本文主要研究过分区中几种统计量的渐近性和极限分布。作为初步结果，我们利用渐近方法证明了过分割中不同部分和不同整数的数目是渐近正态的，推广了Corteel和Hitczenko的结果。此外，我们研究了两类由Bringmann和Lovejoy最初引入的过分区曲柄统计量的渐近性和分布性。利用Hardy-Ramanujan圆方法，我们导出了这两个曲柄的矩的渐近公式，以及Jennings-Shaffer提出的对称矩的渐近公式。在此基础上，我们采用矩的概率方法来证明两个曲柄在适当归一化时渐近地遵循逻辑分布。因此，我们的结果恢复了Zapata Rolon首先用Wright圆法得到的正矩的渐近公式。

引用次数: 0

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