International Journal of Number Theory最新文献

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Riemann hypothesis for period polynomials for cusp forms on Γ0(N) Γ0(N)上尖顶形式周期多项式的黎曼假设

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-05-30 DOI: 10.1142/s1793042124500982

SoYoung Choi

We prove that for even integer $k$ , almost all of zeros of the period polynomial associated to a cusp form of weight $k$ on $Γ_{0} (N)$ are on the circle $| z | = 1 / \sqrt{N}$ under some conditions.

我们证明，对于偶数整数 k，在某些条件下，Γ0(N) 上与权重为 k 的尖顶形式相关的周期多项式的几乎所有零点都位于圆 |z|=1/N 上。

引用次数: 0

Lehmer-type bounds and counting rational points of bounded heights on Abelian varieties 阿贝尔变体上的雷默型边界和有界高的有理点计数

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-05-29 DOI: 10.1142/s1793042124501045

Narasimha Kumar, Satyabrat Sahoo

In this paper, we study Lehmer-type bounds for the Néron–Tate height of $\bar{K}$ -points on abelian varieties $A$ over number fields $K$ . Then, we estimate the number of $K$ -rational points on $A$ with Néron–Tate height $\leq log B$ for $B ≫ 0$ . This estimate involves a constant $C$ , which is not explicit. However, for elliptic curves and the product of elliptic curves over $K$ , we make the constant explicitly computable.

在本文中，我们研究了数域 K 上的无性变项 A 上 K̄ 点的奈伦-塔特高度的雷默型边界。然后，我们估计了 B≫0 时 A 上奈伦-塔特高度≤logB 的 K 有理点的数量。这个估计涉及一个常数 C，它并不明确。然而，对于椭圆曲线和 K 上的椭圆曲线乘积，我们可以明确地计算这个常数。

引用次数: 0

Arithmetic progressions in polynomial orbits 多项式轨道中的算术级数

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-05-29 DOI: 10.1142/s1793042124500970

Mohammad Sadek, Mohamed Wafik, Tuğba Yesin

Let $f$ be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit ${Orb}_{f} (t) = {t, f (t), f (f (t)), \dots}$ , where $t$ is an integer, using arithmetic progressions each of which contains $t$ . Fixing an integer $k \geq 2$ , we prove that it is impossible to cover ${Orb}_{f} (t)$ using $k$ such arithmetic progressions unless ${Orb}_{f} (t)$ is contained in one of these progressions. In fact, we show that the relative density of terms covered by $k$ such arithmetic progressions in ${Orb}_{f} (t)$ is uniformly bounded from above by a bound that depends solely on $k$ . In addition, the latter relative density can be made as close as desired to $1$ by an appropriate choice of $k$ arithmetic progressions containing $t$ if $k$ is allowed to be large enough.

我们考虑的问题是用算术级数覆盖轨道 Orbf(t)={t,f(t),f(f(t)),...}，其中 t 是整数，而每个算术级数都包含 t。固定整数 k≥2，我们证明除非 Orbf(t) 包含在其中一个算术级数中，否则不可能用 k 个这样的算术级数覆盖 Orbf(t)。事实上，我们证明了在 Orbf(t) 中，由 k 个这样的算术级数所覆盖的项的相对密度是由一个完全取决于 k 的约束从上均匀限定的。

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引用次数: 0

On mean values for the exponential sum of divisor functions 关于除数函数指数和的平均值

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-05-17 DOI: 10.1142/s1793042124500933

Wei Zhang

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris360 (2022) 419–424].

本文研究除数函数指数和的均值。我们改进了之前的结果 [M.Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad.Acad.Paris360 (2022) 419-424].

引用次数: 0

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 Q3 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 <disp-formula-group><math altimg="eq-00001.gif" display="block" overflow="scroll"><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mfrac linethickness="0"><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>∈</mo><mi>ℕ</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>n</mi></mrow></mfrac></mrow></munder><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math></disp-formula-group> for odd integers <math altimg="eq-00002.gif" display="inline" overflow="scroll"><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≥</mo><mn>1</mn><mo>,</mo><mspace width="0.25em"></mspace></math> and <math altimg="eq-00003.gif" display="inline" overflow="scroll"><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></math>, where <math altimg="eq-00004.gif" display="inline" overf

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

Uniformity of quadratic points 二次点的均匀性

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-04-27 DOI: 10.1142/s1793042124500532

Tangli Ge

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field.

在本文中，我们将 Dimitrov 等人[Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237-298] 的统一性结果扩展到维数二，并利用它得到了非全椭圆非双曲曲线所有二次点集合的心数的统一约束，该约束只取决于莫德尔-韦尔等级、曲线的属和数域的度。

引用次数: 0

The 𝔭-primary uniform boundedness conjecture for Drinfeld modules 德林菲尔德模块的𝔭主均匀有界猜想

IF 0.7 3区数学 Q3 MATHEMATICS

International Journal of Number Theory

Pub Date : 2024-04-27 DOI: 10.1142/s1793042124500611

Shun Ishii

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the $𝔭$ -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa’s result on the $p$ -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties.

在本文中，我们研究了关于无性变体扭转的统一有界性猜想的德林费尔德模块类比。结果，我们证明了任意秩的 Drinfeld 模块一维族的𝔭-主均匀有界猜想，这扩展了 Poonen 的一个结果。这一结果可以看作是加多雷-玉川关于无性变体一维族 p 主均匀有界猜想结果的德林费尔德模类似物。

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 Q3 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 Q3 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 Q3 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

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International Journal of Number Theory

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