Journal of Number Theory最新文献

英文中文

Goldbach representations with several primes 几个素数的哥德巴赫表示

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-22 DOI: 10.1016/j.jnt.2025.07.008

Thi Thu Nguyen

We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of k primes. We extend the existing result for

k = 2

to a general k and obtain a better error term for all k larger than 3. Moreover, we prove an equivalence between the Riemann Hypothesis and a good average order in this case.

我们研究了整数作为k个素数和的哥德巴赫表示的平均阶数的渐近公式。我们将已有的k=2的结果推广到一般的k，并对所有大于3的k得到了更好的误差项。此外，在这种情况下，我们证明了黎曼假设与良好平均阶之间的等价性。

引用次数: 0

Numerical investigation of lower order biases in moment expansions of one parameter families of elliptic curves 椭圆曲线一参数族矩展开的低阶偏差数值研究

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-22 DOI: 10.1016/j.jnt.2025.07.003

Timothy Cheek , Pico Gilman , Kareem Jaber , Steven J. Miller , Vismay Sharan , Marie-Hélène Tomé

For a fixed elliptic curve E without complex multiplication,

a_{p} ≔ p + 1 - # E (F_{p})

O (\sqrt{p})

and

a_{p} / 2 \sqrt{p}

converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves

y^{2} = x^{3} + A (T) x + B (T)

with

A (T), B (T) \in Z [T]

and non-constant j-invariant, the second moment of

a_{p} (T)

p^{2} + O (p^{3 / 2})

. The size and sign of the lower order terms has applications to the distribution of zeros near the central point of Hasse-Weil L-functions and the Birch and Swinnerton-Dyer conjecture. S. J. Miller conjectured that the highest order term of the lower order terms of the second moment that does not average to zero is on average negative. Previous work on the conjecture has been restricted to a small set of highly nongeneric families. We create a database and a framework to quickly and systematically investigate biases in the second moment of any one-parameter family. When looking at families which have so far been beyond current theory, we find several potential violations of the conjecture for

p \leq 250, 000

and discuss new conjectures motivated by the data.

对于没有复数乘法的固定椭圆曲线E， ap +1−#E（Fp）是O(p)， ap/2p收敛于一个半圆分布。Michel证明了对于单参数椭圆曲线族y2=x3+ a (T)x+B(T)，其中a (T)，B(T)∈Z[T]，非常数j不变量，ap(T)的二阶矩为p2+O（p3/2）。低阶项的大小和符号可以应用于Hasse-Weil l -函数中心点附近的零分布以及Birch和Swinnerton-Dyer猜想。S. J. Miller推测，二阶矩的低阶项的最高阶项，如果平均值不为零，则平均为负。先前关于这一猜想的研究仅限于一小部分高度非属的家族。我们创建了一个数据库和一个框架，以快速系统地调查任何单参数族的第二时刻的偏差。当研究到目前为止已经超出当前理论的家庭时，我们发现了p≤250,000的猜想的几个潜在违反，并讨论了由数据激发的新猜想。

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引用次数: 0

New bounds in R.S. Lehman's estimates for the difference π(x)−li(x) R.S. Lehman对差值π(x)−li(x)估计的新界

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.002

Michael Revers

We denote by

π (x)

the usual prime counting function and let

l i (x)

the logarithmic integral of x. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for

π (x) - l i (x)

for some value x not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term

S_{3}

in his original paper. We are now able to completely eliminate the lower condition for the size-length η. For further numerical computations this enables us to establish sharper results on the positions for the sign changes. We illustrate with some numerical computations on the lowest known crossover regions near 10³¹⁶ and we discuss numerically on potential crossover regions below this value.

我们用π(x)表示通常的素数计数函数，设li(x)为x的对数积分。1966年，R.S. Lehman提出了一种新的方法和一种有效的方法来寻找一个上界，在这个上界中，π(x) - li(x)对于某个值x不高于这个给定的上界，可以保证符号发生变化。在本文中，我们对误差项进行了进一步的改进，包括对Lehman原论文中著名的误差项S3的改进。我们现在能够完全消除尺寸长度η的较低条件。对于进一步的数值计算，这使我们能够在符号变化的位置上建立更清晰的结果。我们对10316附近已知的最低交叉区域进行了数值计算，并对低于该值的潜在交叉区域进行了数值讨论。

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引用次数: 0

Elasticity of orders with prime conductor 带主导体的阶的弹性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.013

Jared Kettinger , Grant Moles

Let R be an order in a number field whose conductor ideal

P : = (R : \overline{R})

is prime in the ring of integers

\overline{R}

. In this paper, we explore the factorization properties of such orders. Most notably, we give a complete characterization of the elasticity of R in terms of its class group. We conclude with an application to the computation of class groups of certain orders.

设R是数域中的一个阶，其导体理想P:=(R:R)在整数环R中是素数。在本文中，我们探讨了这类阶的分解性质。最值得注意的是，我们给出了R的弹性在其类群中的完整表征。最后给出了在一定阶类群计算中的一个应用。

引用次数: 0

Note on a theorem of Birch–Erdős and m-ary partitions 关于Birch-Erdős和m个分区定理的注意事项

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.009

Yuchen Ding , Honghu Liu , Zi Wang

Let

p, q > 1

be two relatively prime integers and

N

the set of nonnegative integers. Let

f_{p, q} (n)

be the number of different expressions of n written as a sum of distinct terms taken from

{p^{α} q^{β} : α, β \in N}

. Erdős conjectured and then Birch proved that

f_{p, q} (n) \geq 1

provided that n is sufficiently large. In this note, for all sufficiently large number n we prove

f_{p, q} (n) = 2^{\frac{{(\log n)}^{2}}{2 \log p \log q} (1 + O (\log \log n / \log n))} .

We also show that

\lim_{n \to \infty} f_{2, q} (n + 1) / f_{2, q} (n) = 1

. Additionally, we will point out the relations between

f_{2, q} (n)

and m-ary partitions.

设p，q>；1为两个相对素数，N为非负整数的集合。设fp，q(n)是n的不同表达式的个数，表示为{pαqβ:α，β∈n}中不同项的和。Erdős推测，然后Birch证明，只要n足够大，fp,q(n)≥1。在本文中，对于所有足够大的数n，我们证明了q(n)=2(log (n))22log (plg)) q(1+O(log （log)) /log (n)）。我们还证明了limn→∞(f2,q(n+1)/f2,q(n)=1。此外，我们将指出f2，q(n)和m-ary分区之间的关系。

引用次数: 0

Variants of Kohnen's conjecture for Hermitian modular forms 厄密模形式的Kohnen猜想的变体

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.001

Biplab Paul , Sujeet Kumar Singh

Let F be a Hermitian cusp form of weight k and of degree 2 over

Q (i)

with Fourier-Jacobi coefficients

ϕ_{m}

m \in N

. Motivated by a conjecture of W. Kohnen on the growth of Petersson norm of

ϕ_{m}

in the set-up of Siegel modular forms, we study analogous questions in the set-up of Hermitian modular forms. We first propose a conjecture in this set-up which is analogous to that of Kohnen. We then provide some evidence by proving the conjecture for cusp forms lying in the Hermitian-Maass subspace. We also study certain other related problems.

设F是权k和二阶Q(i)的厄米尖峰形式，其系数为Fourier-Jacobi， m∈N。基于W. Kohnen关于在Siegel模形式的建立中，ϕm的Petersson范数的增长的一个猜想，我们研究了在hermite模形式的建立中的类似问题。我们首先提出一个类似于柯南的猜想。然后我们通过证明位于hermitian - mass子空间中的尖形的猜想提供了一些证据。我们还研究了其他一些相关问题。

引用次数: 0

Exponential shrinking problem in multiplicative Diophantine approximation 乘式丢番图近似中的指数收缩问题

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.016

Qi Jia, Junjie Shi

Besides limsup set, the liminf set also appears widely in Diophantine approximation. It gives precise information about when a point can be well approximated compared with limsup set. Moreover, one usually uses liminf set to determine the dimension of limsup set from below. In this paper, we consider the liminf setting within the context of multiplicative Diophantine approximation. More precisely, let

{q_{n}}_{n \in N}

be a sequence of positive integers with exponential growth speed. For any

τ > 0

, define

K_{d} (τ) = {x \in {[0, 1]}^{d} : \prod_{i = 1}^{d} ‖ q_{n} x_{i} ‖ \leq q_{n}^{- τ} for all n ultimately} .

Hausdorff dimension of

K_{d} (τ)

is presented in this note.

除限集外，限集在丢芬图近似中也广泛出现。与limsup集相比，它给出了一个点何时可以很好地逼近的精确信息。此外，人们通常使用liminf set从下面确定limsup集合的维数。在本文中，我们考虑了乘式丢番图近似下的阈值设置。更精确地说，设{qn}n∈n是一个正整数序列，其增长速度为指数级。对于任何τ祝辞0,defineKd(τ)= {x∈[0,1]d:∏i = 1 d为qnxi为≤qn−τ为allnultimately}。本文给出了Kd（τ）的豪斯多夫维数。

引用次数: 0

Proof of the complete presence of a modulo 4 bias for the semiprimes 半素数的模4偏置完全存在的证明

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.004

Miroslav Marinov , Nikola Gyulev

In 2016, Dummit, Granville, and Kisilevsky showed that the proportion of semiprimes (products of two primes) not exceeding a given x, whose factors are congruent to 3 modulo 4, is more than a quarter when x is sufficiently large. They have also conjectured that this holds from the very beginning, that is, for all

x \geq 9

. Here we give a proof of this conjecture. For

x \geq 1.1 \cdot 10^{13}

we take an explicit approach based on their work. We rely on classical estimates for prime counting functions, as well as on very recent explicit improvements by Bennett, Martin, O'Bryant, and Rechnitzer, which have wide applications in essentially any setting involving estimations of sums over primes in arithmetic progressions. All

x < 1.1 \cdot 10^{13}

are covered by a computed assisted verification.

2016年，Dummit， Granville和Kisilevsky证明，当x足够大时，不超过给定x的因子等于3模4的半素数（两个素数的乘积）的比例大于四分之一。他们还从一开始就推测这是成立的，即对所有x≥9。这里我们给出这个猜想的一个证明。对于x≥1.1⋅1013，我们根据他们的工作采取明确的方法。我们依靠素数计数函数的经典估计，以及Bennett， Martin， O'Bryant和Rechnitzer最近的显式改进，这些改进基本上在任何涉及算术数列中素数和估计的设置中都有广泛的应用。所有x<；1.1·1013都由计算机辅助验证覆盖。

引用次数: 0

Ramification filtration via deformations, II 通过变形的分枝过滤，2

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.005

Victor Abrashkin

<div><div>Let <math><mi>K</mi></math> be a field of formal Laurent series with coefficients in a finite field of characteristic p. For <math><mi>M</mi><mo>∈</mo><mi>N</mi></math>, let <math><msub><mrow><mi>G</mi></mrow><mrow><mo><</mo><mi>p</mi><mo>,</mo><mi>M</mi></mrow></msub></math> be the maximal quotient of the Galois group of <math><mi>K</mi></math> of period <math><msup><mrow><mi>p</mi></mrow><mrow><mi>M</mi></mrow></msup></math> and nilpotent class <p and let <math><msub><mrow><mo>{</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mo><</mo><mi>p</mi><mo>,</mo><mi>M</mi></mrow><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msubsup><mo>}</mo></mrow><mrow><mi>v</mi><mo>⩾</mo><mn>0</mn></mrow></msub></math> be the filtration by ramification subgroups in upper numbering. We use the identification <math><msub><mrow><mi>G</mi></mrow><mrow><mo><</mo><mi>p</mi><mo>,</mo><mi>M</mi></mrow></msub><mo>=</mo><mi>G</mi><mo>(</mo><mi>L</mi><mo>)</mo></math> of nilpotent Artin-Schreier theory: here <math><mi>G</mi><mo>(</mo><mi>L</mi><mo>)</mo></math> is the group obtained from a suitable profinite Lie <math><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>M</mi></mrow></msup></math>-algebra <math><mi>L</mi></math> via the Campbell-Hausdorff composition law. We develop new techniques to obtain a “geometrical” construction of the ideals <math><msup><mrow><mi>L</mi></mrow><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msup></math> such that <math><mi>G</mi><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msup><mo>)</mo><mo>=</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mo><</mo><mi>p</mi><mo>,</mo><mi>M</mi></mrow><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msubsup></math>. Given <math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⩾</mo><mn>1</mn></math>, we construct a decreasing central filtration <math><mi>L</mi><mo>(</mo><mi>w</mi><mo>)</mo></math>, <math><mn>1</mn><mo>⩽</mo><mi>w</mi><mo>⩽</mo><mi>p</mi></math>, on <math><mi>L</mi></math>, an epimorphism of Lie <math><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>M</mi></mrow></msup></math>-algebras <math><mover><mrow><mi>V</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>:</mo><msup><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>†</mo></mrow></msup><mo>⟶</mo><mover><mrow><mi>L</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>:</mo><mo>=</mo><mi>L</mi><mo>/</mo><mi>L</mi><mo>(</mo><mi>p</mi><mo>)</mo></math>, and a unipotent action Ω of <math><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>M</mi></mrow></msup></math> on <math><msup><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>†</m

设K是具有特征p的有限域中的系数的形式劳伦级数的域。对于M∈N，设G<；p，M是周期为pM的K的伽罗瓦群和幂零类<；p的最大商，并设{G<；p，M(v)}v小于0是由分枝子群在上编号中的过滤。我们使用幂零Artin-Schreier理论的辨识G<；p，M=G(L)：这里G(L)是由合适的无限Lie Z/ pm代数L根据Campbell-Hausdorff复合定律得到的群。我们开发了新的技术来获得理想L(v)的“几何”构造，使得G(L(v))=G<p，M(v)。给定v0小于1，我们在L上构建一个递减的中心过滤L(w), 1≤w≤p， Lie Z/pM-代数V¯：L¯†L¯：=L/L(p)的外射，以及Z/pM在L¯†上的单能作用Ω，它诱导在L¯上的恒等作用。设dΩ=B†，其中B†∈DiffL¯†，L¯†[v0]是由B†（L¯†）的元素生成的L¯†的理想。我们的主要结果表明，分支理想L（v0）是由V¯B†（L¯†[v0]）生成的L¯理想的原像。在最后一节中，我们将此应用于L¯（v0）生成器的显式构造。本文证明了ΓK的分支子群的一个几何起源，可用于进一步发展非阿贝尔局部类场论。

引用次数: 0

Bushnell-Reiner zeta functions over two-dimensional semilocal rings 二维半局部环上的Bushnell-Reiner ζ函数

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-08-21 DOI: 10.1016/j.jnt.2025.07.010

Sean B. Lynch

Lustig gave an infinite product formula for the zeta function of a commutative two-dimensional regular local ring with finite residue field. We extend this to the noncommutative setting with a method based on filtration by an invertible ideal. One application gives an abstract two-dimensional analogue of Hey's formula. Another application provides effective formulae for zeta functions over Rump's two-dimensional regular semiperfect rings. In the appendices, we supplement these two-dimensional applications with requisite one-dimensional calculations.

Lustig给出了具有有限剩余域的二维可交换正则局部环的zeta函数的无穷积公式。我们用一种基于可逆理想过滤的方法将其推广到非交换情况。一个应用给出了Hey公式的抽象二维模拟。另一个应用为Rump的二维正则半完美环上的zeta函数提供了有效的公式。在附录中，我们用必要的一维计算来补充这些二维应用。

引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Journal of Number Theory

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