Ramanujan Journal最新文献

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New inequalities for p(n) and logp(n). 关于p（n）和logp（n）的新不等式。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2023-01-01 Epub Date: 2022-10-19 DOI: 10.1007/s11139-022-00653-6

Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, WenHuan Zeng

Let p(n) denote the number of partitions of n. A new infinite family of inequalities for p(n) is presented. This generalizes a result by William Chen et al. From this infinite family, another infinite family of inequalities for $log p (n)$ is derived. As an application of the latter family one, for instance obtains that for $n \geq 120$ , $\begin{matrix} p {(n)}^{2} > (1 + \frac{π}{\sqrt{24} n^{3 / 2}} - \frac{1}{n^{2}}) p (n - 1) p (n + 1) . \end{matrix}$

设p（n）表示n的分区数。给出了一个新的无限不等式族。这推广了William Chen等人的一个结果。从这个无穷大族中，导出了logp（n）的另一个无穷大不等式族。作为后一个族的应用，例如，得到了对于n≥120，p（n）2>（1+π24n3/2-1n2）p（n-1）p（n+1）。

引用次数: 3

A single-variable proof of the omega SPT congruence family over powers of 5. ωSPT同余族在5次幂上的单变量证明。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2023-01-01 Epub Date: 2023-06-28 DOI: 10.1007/s11139-023-00747-9

Nicolas Allen Smoot

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function $ω (q)$ . Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $Z [X]$ -module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $Z [X]$ . To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.

2018年，王柳泉和杨一凡证明了三阶模拟θ函数ω（q）对应的最小部分函数存在一个无穷同余族。他们的证明采用了需要20个初始关系的归纳的形式，并利用了同构于自由秩2Z[X]模的模函数空间。这种证明策略最初是由Paule和Radu开发的，用于研究与亏格1的模曲线相关的同余族。我们证明了与亏格0模曲线相关的王和杨同余族，可以通过同构于Z[X]的局部化的模函数环，使用单变量方法来证明。据我们所知，这是第一次将这样的代数结构应用于配分同余理论。我们的归纳更为复杂，并且依赖于表现出某种不规则的五元收敛的函数序列。然而，该证明最终仅基于对10个初始关系的直接验证，并且类似于Ramanujan和Watson的经典方法。

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引用次数: 5

On the infinite Borwein product raised to a positive real power. 关于无穷大的Borwein乘积被提升为一个正的实权。

IF 0.6 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2023-01-01 Epub Date: 2021-11-02 DOI: 10.1007/s11139-021-00519-3

Michael J Schlosser, Nian Hong Zhou

In this paper, we study properties of the coefficients appearing in the q-series expansion of $\prod_{n \geq 1} {[(1 - q^{n}) / (1 - q^{pn})]}^{δ}$ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $δ$ . We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients. For $p = 3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $δ$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p = 3$ case.

本文研究了任意素数p的无穷大Borwein乘积πn≥1[（1-qn）/（1-qpn）]δ的q级数展开中出现的系数的性质，该乘积被提升为任意正实幂δ。我们使用Hardy-Ramanujan-Rademacher圆方法给出了系数的渐近公式。对于p=3，我们给出了它们增长的估计，这使我们能够部分地证实第一作者关于当指数δ在正实数的指定范围内时观察到的系数的符号模式的早期猜想。我们进一步建立了无穷大Borwein乘积的立方体系数的一些消失性和可分性。最后，我们在附录中提出了几个关于无穷乘积精确符号模式的新猜想，这些猜想与我们在p=3情况下的猜想相似。

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引用次数: 0

Sequences in overpartitions. 多分区中的序列。

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2023-01-01 Epub Date: 2023-01-17 DOI: 10.1007/s11139-022-00685-y

George E Andrews, Ali K Uncu

This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.

本文致力于研究过分区中的序列及其与2-色分区的关系。为了达到这些目的，需要对一类一般的双级数进行广泛的研究。

引用次数: 3

On signatures of elliptic curves and modular forms 椭圆曲线的特征与模形式

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00678-x

A. Dąbrowski, J. Pomykala, Sudhir Pujahari

引用次数: 0

Diagonalizable Thue equations: revisited 可对角化的Thue方程：重新审视

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00682-1

N. Saradha, Divyum Sharma

引用次数: 0

Proof of a conjecture of Sun and its extension by Guo 郭对孙猜想的证明及其推广

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00668-z

Wei Xia

引用次数: 0

Sums of k-th powers and Fourier coefficients of cusp forms 顶点形式的k次幂和傅立叶系数

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00677-y

Zhining Wei

引用次数: 0

Expansions over Legendre polynomials and infinite double series identities Legendre多项式与无穷二重级数恒等式的展开

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00663-4

W. Chu, J. Campbell

引用次数: 1

An asymptotic expansion of the hyberbolic umbilic catastrophe integral 双曲脐带突变积分的渐近展开式

IF 0.7 3区数学 Q3 MATHEMATICS

Ramanujan Journal

Pub Date : 2022-12-29 DOI: 10.1007/s11139-022-00675-0

Chelo Ferreira, J. López, Ester Pérez Sinusía

引用次数: 0

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