Journal of Number Theory最新文献

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A triple convolution sum of the divisor function 除数函数的三重卷积和

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.014

Bikram Misra , M. Ram Murty , Biswajyoti Saha

We study the triple convolution sum of the divisor function given by

\sum_{n \leq x} d (n) d (n - h) d (n + h)

for

h \neq 0

where

d (n)

denotes the number of positive divisors of n. Based on some algebraic and geometric considerations, Browning conjectured that the above sum is asymptotic to

c_{h} x {(\log x)}^{3}

, for a suitable constant

c_{h} \neq 0

, as

x \to \infty

. This conjecture is still unproved. Using sieve-theoretic results of Wolke and Nair (respectively), it is possible to derive the exact order of the sum. The lower bound of the correct order of magnitude can also be derived by very elementary arguments. In this article, using the Tauberian theory for multiple Dirichlet series, we prove an explicit lower bound and provide a new theoretical framework to predict Browning's conjectured constant

c_{h}

我们研究了当h≠0时∑n≤xd(n)d(n−h)d（n+h）给出的除数函数的三重卷积和，其中d(n)表示n的正除数的个数。基于一些代数和几何考虑，Browning推测上述和是渐近于chx(log x)3的，对于一个合适的常数ch≠0，当x→∞。这个猜想尚未得到证实。利用Wolke和Nair（分别）的筛理论结果，可以推导出和的精确阶数。正确数量级的下界也可以通过非常基本的论证推导出来。本文利用多重狄利克雷级数的陶伯利理论，证明了它的显下界，为预测勃朗宁猜想常数ch提供了一个新的理论框架。

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引用次数: 0

Transitive actions and purely periodic N-expansions 传递作用和纯周期n展开

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.013

Mohammad Javaheri

We study the dynamical properties of the semigroup

〈 f, g 〉

generated by the maps

f (x) = x + 1

and

g (x) = N / x

acting on the positive rational numbers

Q^{+}

, with a focus on their connection to N-expansions. For

N \in N ∖ {1, 2, 3, 4, 6}

, we establish the existence of an infinite subset

A \subseteq Q^{+}

on which

〈 f, g 〉

acts transitively: for any

x, y \in A

, there exists

σ \in 〈 f, g 〉

such that

y = σ (x)

. We use this result to show that, for the same values of N, there are infinitely many rational numbers with purely periodic N-expansions of any sufficiently large minimal period.

研究了正有理数Q+上的映射f(x)=x+1和g(x)=N/x所生成的半群< f,g >的动力学性质，重点讨论了它们与N展开的联系。对于N∈N∈{1,2,3,4,6}，建立了< f,g >可传递作用于一个无限子集A的存在性：对于任意x，y∈A，存在∑∈< f,g >使得y=σ(x)。我们用这个结果证明，对于相同的N值，存在无限多个具有任何足够大的最小周期的纯周期N展开的有理数。

引用次数: 0

Counting rational points on the sphere with bounded denominator 在有界分母的球面上计算有理点

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.015

Dubi Kelmer

We give an optimal bound for the remainder when counting the number of rational points on the n-dimensional sphere with bounded denominator for any

n \geq 2

对于任意n≥2的n维球面上有界分母有理点的个数，给出了余数的最优界。

引用次数: 0

Local monodromy of 1-dimensional p-divisible groups 一维p可分群的局部单性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.008

Tristan Phillips

Let G be a p-divisible group over a complete discrete valuation ring R of characteristic p. The generic fiber of G determines a Galois representation ρ. The image of ρ admits a ramification filtration and a Lie filtration. We relate these filtrations in the case G is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the étale part of G is irreducible, generalizing a result of Chai.

设G是特征为p的完全离散赋值环R上的一个p可除群。G的一般纤维决定了伽罗瓦表示ρ。ρ的象允许一个分支过滤和一个李过滤。在G是一维的情况下，我们将这些过滤联系起来，在这种情况下给出森定理的等特征版本。这个结果推广了Gross的结果。此外，我们证明了与G的可变部分相关的表示是不可约的，推广了Chai的结果。

引用次数: 0

Contour integrations and parity results of cyclotomic Euler sums and multiple polylogarithm function 分环欧拉和和多个多对数函数的轮廓积分和奇偶性结果

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.012

Hongyuan Rui , Ce Xu

In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We will prove parity results for cyclotomic Euler sums of arbitrary degree, explicit formulas for the parity of cyclotomic linear and quadratic Euler sums, as well as some formulas for the parity of cyclotomic cubic Euler sums and multiple polylogarithms. As a direct corollary, we derive known formulas concerning the parity of classical Euler sums and alternating Euler sums.

本文用级数的方法定义了扩展三角函数，并利用轮廓积分的方法研究了某些环切欧拉和与多个多对数函数的宇称性。我们将证明任意次环切欧拉和的宇称结果，环切线性和二次欧拉和宇称的显式公式，以及环切三次欧拉和和多个多对数宇称的一些公式。作为一个直接推论，我们导出了关于经典欧拉和和交替欧拉和的宇称性的已知公式。

引用次数: 0

Low lying zeros of Rankin-Selberg L-functions Rankin-Selberg l函数的低零

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-29 DOI: 10.1016/j.jnt.2025.11.009

Alexander Shashkov

<div><div>We study the low lying zeros of <math><mi>G</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>G</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> Rankin-Selberg L-functions. Assuming the Generalized Riemann Hypothesis, we compute the 1-level density of the low-lying zeroes of <math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>f</mi><mo>⊗</mo><mi>g</mi><mo>)</mo></math> averaged over families of Rankin-Selberg convolutions, where <math><mi>f</mi><mo>,</mo><mi>g</mi></math> are cuspidal newforms with even weights <math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and prime levels <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, respectively. The Katz-Sarnak density conjecture predicts that in the limit, the 1-level density of suitable families of L-functions is the same as the distribution of eigenvalues of corresponding families of random matrices. The 1-level density relies on a smooth test function ϕ whose Fourier transform <math><mover><mrow><mi>ϕ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math> has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions ϕ with <math><mi>supp</mi><mspace></mspace><mover><mrow><mi>ϕ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>⊂</mo><mo>(</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math>. When <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, we prove the density conjecture for <math><mi>supp</mi><mspace></mspace><mover><mrow><mi>ϕ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>⊂</mo><mo>(</mo><mo>−</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></math> when <math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, and <math><mi>supp</mi><mspace></mspace><mover><mrow><mi>ϕ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>⊂</mo><mo>(</mo><mo>−</mo><mfrac><mrow><mn>29</mn></mrow><mrow><mn>28</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>29</mn></mrow><mrow><mn>28</mn></mrow></mfrac><mo>)</mo></math> when <math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. A secondary term contributes to the 1-level density when the support of <math><mover><mrow><mi>ϕ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math> ex

我们研究了GL(2)×GL(2) Rankin-Selberg l -函数的低洼零点。假设广义黎曼假设，我们计算了L（s,f⊗g）在Rankin-Selberg卷积族上平均的低零的1能级密度，其中f，g分别是偶数权值k1，k2和素数阶N1，N2的反转新形式。根据Katz-Sarnak密度猜想，在极限情况下，l -函数的合适族的1级密度与随机矩阵的相应族的特征值分布相同。1级密度依赖于平滑测试函数φ，其傅里叶变换φ具有紧凑的支持。一般来说，我们用supppφ φ φ(−12,12)证明了测试函数φ的Katz-Sarnak密度猜想成立。当N1=N2时，我们证明了k1≠k2时，supϕ´´（−54,54）和k1=k2时，supϕ´´（−2928,2928）的密度猜想。当φ -的支持度超过（−1,1）时，二级项有助于1级密度，这使得这些结果特别有趣。使我们能够将φ -的支持扩展到（- 1,1）以外的主要思想是对由Petersson公式产生的Kloosterman和的乘积的分析。在k1=k2的情况下，我们也仔细地处理了极点的贡献。我们的工作提供了在中心点不消失的Rankin-Selberg l -函数比例的条件下界，以及Keating和Snaith关于中心l值的一个相关猜想。

引用次数: 0

Corrigendum to “Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields” [J. Number Theory 277 (2025) 105–123] “有限域上乘阶模一元不可约多项式的可整除性”的修正[J]。数论277 (2025)105-123]

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-12 DOI: 10.1016/j.jnt.2025.11.002

Joaquim Cera Da Conceição

引用次数: 0

On irrationals with Lagrange value exactly 3 在拉格朗日值为3的无理数上

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-09 DOI: 10.1016/j.jnt.2025.11.006

Zhe Cao , Harold Erazo , Carlos Gustavo Moreira

Text

For

c > 0

, let

X_{c}

denote the set of

x \in R ﹨ Q

such that

| x - \frac{p}{q} | < \frac{1}{c q^{2}}

has only finitely many rational solutions

\frac{p}{q}

. It is a classical fact, known since the 1950s, that

X_{c}

is uncountable for

c > 3

and countable for

c < 3

. However, the cardinality of

X_{3}

does not appear to be present in the literature. We prove that

X_{3}

is uncountable.

More generally, we show that for any

n \in N \cup {\infty}

, the set of

x \in R ﹨ Q

with Lagrange value exactly 3 and such that

| x - \frac{p}{q} | < \frac{1}{3 q^{2}}

has exactly n rational solutions

\frac{p}{q}

is also uncountable.

Video

For a video summary of this paper, please visit https://youtu.be/VyKB99-kVeY.

对于c>；0，设Xc表示x∈RQ的集合，使得|x−pq|<；1cq2只有有限多个有理解pq。这是一个经典的事实，从20世纪50年代就知道，Xc对于c>；3是不可数的，对于c<；3是可数的。然而，X3的基数似乎没有出现在文献中。我们证明X3是不可数的。更一般地，我们证明了对于任意n∈n∪{∞}，x∈RQ的拉格朗日值恰好为3且使得|x−pq|<；13q2恰好有n个有理解的集合pq也是不可数的。观看本文的视频摘要，请访问https://youtu.be/VyKB99-kVeY。

引用次数: 0

On ideal class groups of totally degenerate number rings 关于完全退化数环的理想类群

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-05 DOI: 10.1016/j.jnt.2025.11.001

Ruben Hambardzumyan , Mihran Papikian

Let

χ (x) \in Z [x]

be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring

Z [x] / (χ (x))

. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of

χ (x)

tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of

χ (x)

is 2 or 3.

设χ(x)∈Z[x]是一个一元多项式，其根是不同整数。研究了环Z[x]/(χ(x))的理想类单群和理想类群。我们得到了这些对象的阶的公式，并类比于Brauer-Siegel定理，研究了当χ(x)的判别式趋于无穷时它们的渐近行为。最后，我们描述了当χ(x)的度为2或3时理想类群的结构。

引用次数: 0

On the non-vanishing of Hilbert Poincaré series 关于希尔伯特·庞卡罗级数的不灭性

IF 0.7 3区数学 Q3 MATHEMATICS

Journal of Number Theory

Pub Date : 2025-12-03 DOI: 10.1016/j.jnt.2025.11.003

Mingkuan Zhang , Yichao Zhang

We prove that if ν has small norm with respect to the level and the weight, the ν-th Hilbert Poincaré series does not vanish identically. We also prove Selberg's identity on Kloosterman sums in the case of number fields, which implies certain vanishing and non-vanishing relations of Hilbert Poincaré series when the narrow class number is 1. Finally, we pass to the adelic setting and interpret the problem via Hecke operators.

我们证明了如果ν对水平和权值有小范数，则ν-波昂卡罗级数不完全消失。我们还证明了在数域的Kloosterman和上的Selberg恒等式，这意味着当窄类数为1时，Hilbert poincar级数的某些消失和不消失关系。最后，我们传递到阿德利克设置，并通过赫克算子解释问题。

引用次数: 0

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