Journal of Number Theory最新文献

英文中文

Spherical varieties and non-ordinary families of cohomology classes 上同调类的球形变种和非普通族

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-24 DOI: 10.1016/j.jnt.2025.08.012

Rob Rockwood

We show that p-adic families of cohomology classes associated to symmetric spaces vary p-adically over small discs in weight space, without any ordinarity assumption. This generalises previous work of Loeffler, Zerbes and the author. Furthermore, we show that these families exhibit full variation in the cyclotomic direction, generalising previous constructions of Euler systems and p-adic L-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.

我们证明了与对称空间相关的上同调类的p进族在权空间中的小圆盘上以p进的方式变化，而不作任何序性假设。这概括了Loeffler， Zerbes和作者之前的工作。此外，我们证明了这些族在环切方向上表现出充分的变化，推广了以前的欧拉系统和p进l函数的结构。作为一个应用，我们证明了Loeffler-Skinner-Zerbes的lema - flach Euler系统在Coleman族内插。

引用次数: 0

Unconditional lower bounds for the sixth and eighth moments of the Riemann zeta function 黎曼函数的第六阶矩和第八阶矩的无条件下界

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.017

Timothy Page

Unconditional bounds on the sixth and eighth moments of the Riemann zeta function are improved by bounding twisted second and fourth moments that arise upon application of the Cauchy-Schwarz inequality and Hölder's inequality. An unconditional bound on the sixth moment of the derivative of the Riemann zeta function is also deduced.

利用Cauchy-Schwarz不等式和Hölder不等式，对Riemann zeta函数的第6和第8矩的无条件界进行了改进。推导出黎曼ζ函数导数的第六阶矩的无条件界。

引用次数: 0

Higher moments for non-normal fields with Galois group Ad and Sd 具有伽罗瓦群Ad和Sd的非正规场的高矩

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.018

Jiong Yang

Let K be a non-normal number field of degree d with Galois group

A_{d}

S_{d}

. Let

a_{K} (n)

be the number of integral ideals of norm n in K. We obtain an asymptotic formula for the summation

\sum_{n \leq x} a_{K}^{l} (n)

for any

l \geq 2

. As a consequence, we obtain such an asymptotic formula for any number field K of degree less or equal to 8 unconditionally.

设K为具有伽罗瓦群Ad或Sd的d次非正规数域。设k (n)为k中范数n的积分理想数，得到了任意l≥2时∑n≤xaKl(n)的求和的渐近公式。因此，对于任意小于或等于8次的数域K，我们无条件地得到了这样一个渐近公式。

引用次数: 0

Explicit formulas for Grassmannian polylogarithms in weights 4 and 5 权值为4和5的格拉斯曼多对数的显式公式

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.011

Steven Charlton , Herbert Gangl , Danylo Radchenko

We explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2 iterated integrals. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of

{GL}_{4} (C)

in terms of

{Li}_{4}

我们明确地将权重为4和权重为5的格拉斯曼多对数分别简化为深度为2的迭代积分。此外，利用这种权值4的减少，我们得到了一个显式的，尽管复杂的，所谓的4比形式，它给出了GL4(C)在Li4的连续上同调中的Borel类的表达式。

引用次数: 0

On the hook length biases of the 2- and 3-regular partitions 关于2规则分区和3规则分区的钩子长度偏差

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.016

Wenxia Qu , Wenston J.T. Zang

Let

b_{t, i} (n)

denote the total number of i hooks in the t-regular partitions of n. Singh and Barman (2024) [14] raised two conjectures on

b_{t, i} (n)

. The first conjecture is on the positivity of

b_{3, 2} (n) - b_{3, 1} (n)

for

n \geq 28

. The second conjecture states that when

k \geq 3

b_{2, k} (n) \geq b_{2, k + 1} (n)

for all n except for

n = k + 1

. In this paper, we confirm the first conjecture. Moreover, we show that for any odd

k \geq 3

, the second conjecture fails for infinitely many n. Furthermore, we verify that the second conjecture holds for

k = 4

and 6. We also propose a conjecture on the even case k, which is a modification of Singh and Barman's second conjecture.

设bt，i(n)表示n的t正则分区中i个钩子的总数。Singh和Barman(2024)[14]对bt，i(n)提出了两个猜想。第一个猜想是关于当n≥28时，b3,2(n)−b3,1(n)的正性。第二个猜想表明，当k≥3时，b2,k(n)≥b2时，除n=k+1外，所有n均为k+1(n)。在本文中，我们证实了第一个猜想。此外，我们证明了对于任意奇数k≥3，对于无穷多个n，第二个猜想不成立。进一步，我们证明了对于k=4和6，第二个猜想成立。我们还提出了一个关于偶数情况k的猜想，它是对Singh和Barman第二猜想的修正。

引用次数: 0

Resolution of Erdős' problems about unimodularity 解决Erdős的单模性问题

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.014

Stijn Cambie

Letting

δ_{1} (n, m)

be the density of the set of integers with exactly one divisor in

(n, m)

, Erdős wondered if

δ_{1} (n, m)

is unimodular for fixed n. We prove this is false in general, as the sequence

(δ_{1} (n, m))

has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs;

n = 1

. We also solve the question on unimodality of the density of integers whose

k^{t h}

prime is p.

设δ1（n,m）为（n,m）中恰好有一个除数的整数集的密度，Erdős想知道δ1（n,m）对于固定n是否单模。我们一般证明这是错误的，因为序列（δ1(n,m)）具有超多项式的多个局部极值。然而，我们在它发生的单一情况下确认单峰性；n = 1。我们还解决了第k素数为p的整数密度的单模性问题。

引用次数: 0

Infinitude of the zeros of the Lerch zeta function on the half plane ℜ(s)>1 半平面上lach zeta函数零点的无穷大

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.019

Biswajyoti Saha , Dhananjaya Sahu

For

a \in (0, 1]

, the zeros of the Hurwitz zeta function

ζ (s, a) : = \sum_{n \geq 0} {(n + a)}^{- s}

have interesting features. There are no zeros in the half plane

ℜ (s) \geq 1 + a

, whereas there are infinitely many zeros in the strip

1 < ℜ (s) < 1 + a

, provided

a \neq 1 / 2, 1

. The existence of these infinitely many zeros was first proved by Davenport and Heilbronn for rational and transcendental values of a and then by Cassels for algebraic irrational values of a. In this article, we consider the analogous question for the zeros of the cognate Lerch zeta function

ζ_{z} (s, a) : = \sum_{n \geq 0} z^{n} {(n + a)}^{- s}

, where z is a complex number of unit modulus. When z is a root of unity, the question can be answered using a theorem of Zaghloul, which is an extension of a work of Chatterjee and Gun. In the general case, we need to further extend the method of Chatterjee and Gun.

对于a∈(0,1)，Hurwitz zeta函数ζ(s,a):=∑n≥0(n+a)−s的零点具有有趣的特征。在半平面（≥1+a）上不存在零，而在条形（1< 1< 1+a）上存在无穷多个零，只要a≠1/2,1。首先由Davenport和Heilbronn证明了a的有理性值和超越值的无穷多个零的存在性，然后由Cassels证明了a的代数无理性值的无穷多个零的存在性。在本文中，我们考虑了近亲Lerch zeta函数ζz（s,a）的零点的类似问题：=∑n≥0zn(n+a)−s，其中z是单位模的复数。当z是单位的根时，可以用Zaghloul的定理来回答这个问题，Zaghloul定理是Chatterjee和Gun的推广。在一般情况下，我们需要进一步扩展Chatterjee和Gun的方法。

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引用次数: 0

An extension of smooth numbers: Multiple dense divisibility 光滑数的推广：多重稠密可整除性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-23 DOI: 10.1016/j.jnt.2025.08.013

Garo Sarajian , Andreas Weingartner

The i-tuply y-densely divisible numbers were introduced by a Polymath project, as a weaker condition on the moduli than y-smoothness, in distribution estimates for primes in arithmetic progressions. We obtain the order of magnitude of the count of these integers up to x, uniformly in x and y, for every fixed natural number i.

一个Polymath项目引入了i-tuply y-密可整除数，作为等差数列中素数分布估计中模的一个弱条件。对于每一个固定的自然数i，我们得到这些整数的数量级，直到x，在x和y上一致。

引用次数: 0

Torsion of rational elliptic curves over the Zp-extensions of quadratic fields 二次域zp扩展上有理椭圆曲线的扭转

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-08 DOI: 10.1016/j.jnt.2025.08.009

Ömer Avcı

Let E be an elliptic curve defined over

Q

. For a quadratic number field K and an odd prime number p, let L be a

Z_{p}

-extension of K. We prove that

E {(L)}_{tors} = E {(K)}_{tors}

when

p > 5

. It enables us to classify the groups that can be realized as the torsion subgroup

E {(L)}_{tors}

, by using the classification of torsion subgroups over the quadratic fields.

设E是定义在q上的椭圆曲线，对于二次数域K和奇素数p，设L是K的zp扩展，证明当p>；5时，E(L)tors=E(K)tors。利用二次域上扭转子群的分类，可以对可实现为扭转子群E(L)子群的群进行分类。

引用次数: 0

Galois theory of quadratic rational functions with periodic critical points 具有周期临界点的二次有理函数的伽罗瓦理论

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-09-08 DOI: 10.1016/j.jnt.2025.08.010

Özlem Ejder

Given a number field k, and a quadratic rational function

f (x) \in k (x)

, the associated arboreal representation of the absolute Galois group of k is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for

f \in Z [x]

contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the nth level of the tree remains small as n goes to infinity.

In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of f, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs.

Furthermore, we study the iterated extension

k (f^{- \infty} (t))

generated by adjoining to

k (t)

all roots of

f^{n} (x) = t

for

n \geq 1

for a parameter t. We call the intersection of

k (f^{- \infty} (t))

with

\bar{k}

, the field of constants associated with f. When one of the two critical points of f is the image of the other, we show that the field of constants is contained in the cyclotomic extension of k generated by all 2-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function

\frac{1}{{(x - 1)}^{2}}

给定一个数字域k，一个二次有理函数f(x)∈k(x)， k的绝对伽罗瓦群的关联树表示是正则根二叉树的自同构群的子群。Boston和Jones推测，对于f∈Z[x]，这样一个表示的像包含一个密集的固定元素集合。当n趋于无穷时，如果自同构在树的第n层上的轨道数仍然很小，则该自同构就成立了。在这篇文章中，我们展示了许多二次有理函数，其相关的树伽罗瓦群不是密集的。这些例子来自于临界点位于单一周期轨道上的二次有理函数。为了证明我们的结果，我们对f的迭代单群（IMG）进行了详细的研究，这也使我们能够对Jones和Levy关于固定对的问题提供否定的答案。进一步，我们研究了由相邻于k(t)的fn(x)=t的所有根（n≥1）对参数t产生的迭代扩展k（f−∞(t)）。我们称k（f−∞(t)）与k¯的交集为与f相关的常数域。当f的两个临界点之一是另一个临界点的像时，我们证明了常数域包含在由所有单位的2次方根产生的k的环形扩展中。特别地，我们证明了Ejder、Kara和Ozman关于有理函数1(x−1)2的猜想。

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