{"title":"几个过划分统计量的渐近性和极限分布","authors":"Helen W.J. Zhang, Ying Zhong","doi":"10.1016/j.jnt.2025.09.010","DOIUrl":null,"url":null,"abstract":"<div><div>This paper primarily is dedicated to studying the asymptotics and limiting distributions of several statistics in overpartitions. As a preliminary result, we use asymptotic methods to prove that the number of distinct parts and distinct integers in overpartitions is asymptotically normal, extending the results of Corteel and Hitczenko. Furthermore, we investigate the asymptotic and distributional properties of two types of crank statistics for overpartitions, originally introduced by Bringmann and Lovejoy. Utilizing the Hardy-Ramanujan circle method, we derive asymptotic formulas for the moments of these two cranks, as well as for the symmetrized moments proposed by Jennings-Shaffer. Building on these, we employ the probabilistic method of moments to prove that both two cranks asymptotically follow a logistic distribution when appropriately normalized. Consequently, our results recover the asymptotic formulas for the positive moments first obtained by Zapata Rolon using Wright's circle method.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 737-760"},"PeriodicalIF":0.7000,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotics and limiting distributions of several overpartition statistics\",\"authors\":\"Helen W.J. Zhang, Ying Zhong\",\"doi\":\"10.1016/j.jnt.2025.09.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper primarily is dedicated to studying the asymptotics and limiting distributions of several statistics in overpartitions. As a preliminary result, we use asymptotic methods to prove that the number of distinct parts and distinct integers in overpartitions is asymptotically normal, extending the results of Corteel and Hitczenko. Furthermore, we investigate the asymptotic and distributional properties of two types of crank statistics for overpartitions, originally introduced by Bringmann and Lovejoy. Utilizing the Hardy-Ramanujan circle method, we derive asymptotic formulas for the moments of these two cranks, as well as for the symmetrized moments proposed by Jennings-Shaffer. Building on these, we employ the probabilistic method of moments to prove that both two cranks asymptotically follow a logistic distribution when appropriately normalized. Consequently, our results recover the asymptotic formulas for the positive moments first obtained by Zapata Rolon using Wright's circle method.</div></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"280 \",\"pages\":\"Pages 737-760\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-10-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X25002598\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X25002598","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Asymptotics and limiting distributions of several overpartition statistics
This paper primarily is dedicated to studying the asymptotics and limiting distributions of several statistics in overpartitions. As a preliminary result, we use asymptotic methods to prove that the number of distinct parts and distinct integers in overpartitions is asymptotically normal, extending the results of Corteel and Hitczenko. Furthermore, we investigate the asymptotic and distributional properties of two types of crank statistics for overpartitions, originally introduced by Bringmann and Lovejoy. Utilizing the Hardy-Ramanujan circle method, we derive asymptotic formulas for the moments of these two cranks, as well as for the symmetrized moments proposed by Jennings-Shaffer. Building on these, we employ the probabilistic method of moments to prove that both two cranks asymptotically follow a logistic distribution when appropriately normalized. Consequently, our results recover the asymptotic formulas for the positive moments first obtained by Zapata Rolon using Wright's circle method.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.