A note on odd partition numbers

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik Pub Date : 2024-05-24 DOI:10.1007/s00013-024-01999-7

Michael Griffin, Ken Ono

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引用次数: 0

Abstract

Ramanujan’s partition congruences modulo $\ell \in \{5, 7, 11\}$ assert that

$$\begin{aligned} p(\ell n+\delta _{\ell })\equiv 0\pmod {\ell }, \end{aligned}$$

where $0<\delta _{\ell }<\ell $ satisfies $24\delta _{\ell }\equiv 1\pmod {\ell }.$ By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \ge 5,$ we give a new proof of the conclusion that there are infinitely many m for which $p(\ell m+\delta _{\ell })$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies $m<(\ell ^2-1)/24,$ representing a significant improvement to the previous bound.

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关于奇数分区号的说明

Ramanujan's partition congruences modulo$\ell \in \{5, 7, 11\}$ assert that $$begin{aligned} p(\ell n+\delta _{\ell })equiv 0\pmod {\ell }, \end{aligned}$$where$0<；\滿足(24/delta _{\ell }\equiv 1\pmod {\ell }.\通过证明苏巴老的猜想，拉杜证明了在奇偶性方面不存在这样的全等。在每个算术级数中都有无穷多个奇数（或偶数）分割数。对于素数 \(\ell \ge 5,$，我们给出了一个新的证明，即有无穷多个 m 的 $p(\ell m+\delta _\{ell })$是奇数。这个证明使用了第二作者和拉姆齐对马祖尔在他关于爱森斯坦理想的经典论文中的一个结果的概括。我们还完善了斯图姆（Sturm）关于模形式全等的经典判据，从而证明了最小的这样的 m 满足 $m<(\ell ^2-1)/24,$ ，这是对之前界限的显著改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Archiv der Mathematik 数学-数学

CiteScore

1.10

自引率

0.00%

发文量

117

审稿时长

4-8 weeks

期刊介绍： Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.