International Journal of Number Theory最新文献

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Explicit evaluation of triple convolution sums of the divisor functions 除数函数三重卷积和的显式计算

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-27 DOI: 10.1142/s1793042124500544

B. Ramakrishnan, Brundaban Sahu, Anup Kumar Singh

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums $W_{d_{1}, d_{2}, d_{3}}^{r_{1}, r_{2}, r_{3}} (n) = \sum_{\binom{l_{1}, l_{2}, l_{3} \in ℕ}{d_{1} l_{1} + d_{2} l_{2} + d_{3} l_{3} = n}} σ_{r_{1}} (l_{1}) σ_{r_{2}} (l_{2}) σ_{r_{3}} (l_{3}),$ for odd integers $r_{1}, r_{2}, r_{3} \geq 1,$ and $d_{1}, d_{2}, d_{3}, n \in ℕ$ , where

本文利用模形式理论，给出了求卷积和 Wd1,d2,d3r1,r2,r3(n)=∑l1,l2 的一般方法、l3∈ℕd1l1+d2l2+d3l3=nσr1(l1)σr2(l2)σr3(l3)，对于奇整数 r1,r2,r3≥1，以及 d1,d2,d3,n∈ℕ，其中 σr(n) 是 n 的正除数的 r 次幂和。我们考虑了四种情况，即 (i) r1=r2=r3=1，(ii) r1=r2=1; r3≥3 (iii) r1=1; r2,r3≥3 和 (iv) r1,r2,r3≥3，并给出了各自卷积和的明确表达式。我们举例说明了每种情况下的卷积和，并进一步利用这些公式得到了某些正定二次型对正整数 n 的表示数的明确公式。现有公式为 W1,1,1(n) (见 [20]), W1,1,2(n), W1,2,2(n), W1,2,4(n) (见 [7]), W1,1,11,3,3(n), W1,1,31,3,3(n), W1,3,31,3,3(n), W3,1,11,3,3(n), W3,3,11,3,3(n) (见 [35])、Wd1,d2,d3(n),lcm(d1,d2,d3)≤6（在 [30] 中）和 lcm(d1,d2,d3)=7,8,9 （在 [31] 中），都是利用准模态理论得到的。

引用次数: 0

Near-miss identities and spinor genus classification of ternary quadratic forms with congruence conditions 具有全等条件的三元二次型的近似等式和旋量属分类

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-27 DOI: 10.1142/s1793042124500507

Kush Singhal

In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.

本文发现并证明了一些具有全等条件的积分三元二次型的表示数的近似等式。然后对相应晶格余弦的属和旋子属进行了分类。最后，给出了所有导体 2 的 2-adically unimodular 晶格余集的属和旋量属的完整分类。

引用次数: 0

Dense clusters of zeros near the zero-free region of ζ(s) ζ(s)无零区域附近密集的零群

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-25 DOI: 10.1142/s1793042124500520

William D. Banks

The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function

ζ (s)

of the form

σ > 1 - \frac{c}{{(log τ)}^{2 / 3} {(log log τ)}^{1 / 3}} (τ ≔ | t | + 100) .

For many decades, the general shape of the zero-free region has not changed (although explicit known values for

c

have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of

ζ (s)

lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for

L

-functions associated to nonquadratic Dirichlet characters

χ

modulo

q \geq 2

科罗博夫和维诺格拉多夫的方法为黎曼zeta函数ζ(s) 得出了一个无零区域，其形式为 σ>1-c(logτ)2/3(loglogτ)1/3(τ≔|t|+100)。几十年来，无零区域的一般形状一直未变（尽管已知的明确 c 值多年来有所改进）。在本文中，我们证明了如果无零区域不能被大幅拓宽，那么就存在无限多个靠近无零区域边缘的ζ(s)零点密集簇。我们的证明提供了关于这些簇的位置和其中包含的零点数量的具体信息。为了证明这一结果，我们引入并应用了 de la Vallée Poussin 最初方法的变体，并结合图兰的思想来控制幂和的实部。我们还证明了与非二次迪里夏特字符 χ modulo q≥2 相关的 L 函数的类似结果。

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引用次数: 0

A note on logarithmic equidistribution 关于对数等差数列的说明

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-25 DOI: 10.1142/s1793042124500647

Gerold Schefer

For every algebraic number $κ$ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of $f_{κ} (z) = log | z - κ |$ at the conjugates are essentially bounded from above by $- h (κ)$ . This completes a characterization on functions $f_{κ}$ initiated by Autissier and Baker–Masser, who cover the cases $κ = 2$ and $| κ | \neq 1$ , respectively. Using the same ideas we also prove analogues in the $p$ -adic setting.

对于单位圆上每一个不是统一根的代数数κ，我们证明存在一个高度趋于零的代数数严格序列，使得共轭处 fκ(z)=log|z-κ| 的求值平均值基本上从上而下受 -h(κ)约束。这就完成了 Autissier 和 Baker-Masser 对函数 fκ 的描述，他们分别涉及了 κ=2 和 |κ|≠1 的情况。利用同样的思想，我们还证明了 p-adic 设置中的类似情况。

引用次数: 0

Statistics for Iwasawa invariants of elliptic curves, II 椭圆曲线岩泽不变式的统计，II

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-18 DOI: 10.1142/s1793042124500556

Debanjana Kundu, Anwesh Ray

We study the average behavior of the Iwasawa invariants for Selmer groups of elliptic curves. These results lie at the intersection of arithmetic statistics and Iwasawa theory. We obtain lower bounds for the density of rational elliptic curves with prescribed Iwasawa invariants.

我们研究了椭圆曲线塞尔玛群岩泽不变式的平均行为。这些结果是算术统计和岩泽理论的交叉点。我们获得了具有规定岩泽不变式的有理椭圆曲线密度的下限。

引用次数: 0

Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set 稀疏集合中整数处 L 函数乘积的傅立叶系数振荡

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-10 DOI: 10.1142/s1793042124500854

Babita, Mohit Tripathi, Lalit Vaishya

Let $f$ be a normalized Hecke eigenform of weight $k$ for the full modular group $S L_{2} (ℤ)$ . In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg $L$ -functions $R (s, f \times f)$ , supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant $D < 0 .$ We improve previous results in the case when the reduced form is given by $𝒬 (x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2} .$

设 f 是全模态群 SL2(ℤ) 权重为 k 的归一化赫克特征形式。在本文中，我们得到了与 Rankin-Selberg L 函数 R(s,f×f) 的傅里叶系数相关的一般除数函数的高阶矩的渐近值，R(s,f×f) 在整数处由固定判别式 D<0 的原始积分正定二元二次方程形式（还原形式）表示。当还原形式为𝒬(x1,x2)=x12+x22 时，我们改进了以前的结果。

引用次数: 0

Double and triple character sums and gaps between the elements of subgroups of finite fields 有限域子群元素间的双重和三重特征和与间隙

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-10 DOI: 10.1142/s1793042124500842

Jiankang Wang, Zhefeng Xu

For an odd prime $p$ , let $𝔽_{p}$ be the finite field of $p$ elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any $a, b, c \in 𝔽_{p}^{*}$ , we also obtain new upper bounds of the following double character sum $T_{a, b, c} (χ, ℋ_{1}, ℋ_{2}) = \sum_{h_{1} \in ℋ_{1}} \sum_{h_{2} \in ℋ_{2}} χ (a + b h_{1} + c h_{2})$ and a triple character sum $S_{χ} (a, b, ℋ_{1}, ℋ_{2}, 𝒩) = \sum_{x \in 𝒩} \sum_{h_{1} \in ℋ_{1}} \sum_{h_{2} \in ℋ}$

对于奇素数 p，让 𝔽p 成为 p 元素的有限域。本文的主要目的是建立关于有限域乘法子群元素间差距的新结果。对于任意 a,b,c∈𝔽p∗，我们还得到了以下双特征和 Ta,b,c(χ,ℋ1、ℋ2)=∑h1∈ℋ1∑h2∈ℋ2χ(a+bh1+ch2)和三重特征和 Sχ(a,b,ℋ1,ℋ2,𝒩)=∑x∈𝒩∑h1∈ℋ1∑h2∈ℋ2χ(x+ah1+bh2)，其中𝒩={1,...,N}，且乘法子群ℋ1,ℋ2⊆𝔽p∗分别为阶 H1 和 H2。

引用次数: 0

The transcendence of growth constants associated with polynomial recursions 与多项式递推相关的增长常数的超越性

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-06 DOI: 10.1142/s1793042124500672

Veekesh Kumar

Let

P (x) : = a_{d} x^{d} + \dots + a_{0} \in ℚ [x]

a_{d} > 0

, be a polynomial of degree

d \geq 2

. Let

(x_{n})

be a sequence of integers satisfying

x_{n + 1} = P (x_{n}) for all n = 0, 1, 2, \dots and x_{n} \to \infty as n \to \infty .

Set $α : = {lim}_{n \to \infty} x_{n}^{d^{- n}}$ . Then, under the assumption $a_{d}^{1 / (d - 1)} \in ℚ$ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J.57 (2022) 569–581], either $α$ is transcendental or

设 P(x):=adxd+⋯+a0∈ℚ[x]，ad>0，为阶数 d≥2 的多项式。设 (xn) 为一整数序列，满足 xn+1=P(xn)for all n=0,1,2,...，且 xn→∞as n→∞.设 α:=limn→∞xnd-n。那么，在 ad1/(d-1)∈ℚ的假设下，在最近的一个结果 [A. Dubickas, Transcendency of n.Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J.57 (2022) 569-581] 的最新结果中，要么 α 是超越的，要么 α 可以是一个整数或二次皮索单元，α-1 是它在ℚ 上的共轭。在本文中，我们在不假设 ad1/(d-1) 在 ℚ 中的情况下研究了这种 α 的性质，并证明了要么 α 是超越数，要么 αh 是 Pisot 数，而 h 是数域 ℚα,ad-1d-1 的伽罗瓦闭的扭转子群的阶。本文提出的其他结果研究了在 (n,q1,...,qk)∈ℕ×(K×)k 中不等式 ||q1α1n+⋯+qkαkn+β||<𝜃n 的解，考虑了 β 是有理数还是无理数。这里，K 代表一个数域，𝜃∈(0,1)。符号 ||x|| 表示 x 与其在 ℤ 中最接近的整数之间的距离。

引用次数: 0

A note on Schmidt’s subspace type theorems for hypersurfaces in subgeneral position 关于次一般位置超曲面的施密特子空间类型定理的说明

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-06 DOI: 10.1142/s1793042124500490

Lei Shi, Qiming Yan

In this paper, motivated by Nochka weights and the replacing hypersurfaces technique, we give an improvement of Schmidt’s subspace type theorem for hypersurfaces which are located in subgeneral position.

在本文中，受诺奇卡权重和置换超曲面技术的启发，我们给出了施密特子空间类型定理对位于亚一般位置的超曲面的改进。

引用次数: 0

A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture 关于厄尔多斯-图兰猜想的鲁兹萨数的新上限

IF 0.7 3区数学 Q2 Mathematics

International Journal of Number Theory

Pub Date : 2024-04-06 DOI: 10.1142/s179304212450074x

Yuchen Ding, Lilu Zhao

In this paper, we show that the Ruzsa number

R_{m}

is bounded by

192

for any positive integer

m

, which improves the prior bound

R_{m} \leq 288

given by Chen in 2008.

在本文中，我们证明了对于任意正整数 m，鲁兹萨数 Rm 的边界为 192，这改进了陈在 2008 年给出的先前边界 Rm≤288。

引用次数: 0

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