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Distribution of a Certain Partition Function Modulo Powers of Primes

Distribution of a Certain Partition Function Modulo Powers of Primes
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摘要 In this paper, we study a certain partition function a（n） defined by ∑n≥0 a（n）qn ：= ∏n=1（1- qn）-1（1 -2n）-1. We prove that given a positive integer j 〉 1 and a prime m _〉 5, there are infinitely many congruences of the type a（An ＋ B） ≡ 0 （rood m3）. This work is inspired by Ono＇s ground breaking result in the study of the distribution of the partition function p（n）. In this paper, we study a certain partition function a（n） defined by ∑n≥0 a（n）qn ：= ∏n=1（1- qn）-1（1 -2n）-1. We prove that given a positive integer j 〉 1 and a prime m _〉 5, there are infinitely many congruences of the type a（An ＋ B） ≡ 0 （rood m3）. This work is inspired by Ono＇s ground breaking result in the study of the distribution of the partition function p（n）.
作者 Hei-Chi CHAN
机构地区 Department of Mathematical Sciences
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2011年第4期625-634,共10页 数学学报（英文版）
关键词 Distribution of partition function Ono＇s theorem Ramanujan＇s congruences Distribution of partition function, Ono＇s theorem, Ramanujan＇s congruences
分类号 O211.3 [理学—概率论与数理统计] O156 [理学—基础数学]
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  • 1Chan, H. C.: Ramanujan's cubic continued fraction and a generalization of his "most beautiful identity". Int. J. Number Theory, accepted.
  • 2Andrews, C. E.: The Theory of Partitions, Encycl. Math. and Its Appl., Vol. 2, C.-C. Rota ed., Addison- Wesley, Reading, 1976 (Reissued: Cambridge University Press, Cambridge 1998).
  • 3Berndt, B. C.: Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, 2004.
  • 4Chu, W., Di Claudio, L.: Classical Partition Identities and Basic Hypergeometric Series, Universita deg;o Stido di Lecce, Lecce, 2004.
  • 5Hirschhorn, M. D.: An identity of Ramanujan, and applications. In: q-Series from a Contemporary Per- spective, Contemporary Mathematics, 254, American Mathematical Society, Providence, RI 2000, 229-234.
  • 6Hirschhorn, M. D.: Ramanujan's "most beautiful identity". Austral. Math. Soc. Gaz., 31, 259-262 (2005).
  • 7Chan, H. C.: Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function. Int. J. Number Theory, accepted.
  • 8Watson, G. N.: Ramanujans vermutung fiber zerfallungsanzahlen. J. Reine Angew. Math., 179, 97-128 (1938).
  • 9Berndt, B. C., Ono, K.: Ramanujan's Unpublished Manuscript on the Partition and Tau-functions, The Andrews Festschrift (D. Foata and G. N. Han ed.), Springer-Verlag, Berlin, 2001, 39-110.
  • 10Ono, K.: Distribution of the partition function modulo m. Ann. of Math., 151, 293-307 (2000).
Acta Mathematica Sinica,English Series
2011年 第4期

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