This note is to remark the large supremum norms of some Hecke-eigenforms of large squarefree levels and nebentypus of small conductors,based on the recent prominent progress by Soundararajan on the extreme central val...This note is to remark the large supremum norms of some Hecke-eigenforms of large squarefree levels and nebentypus of small conductors,based on the recent prominent progress by Soundararajan on the extreme central values of L-functions.展开更多
The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)mo...The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).展开更多
基金supported by the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No.HKU 7032/06P)
文摘This note is to remark the large supremum norms of some Hecke-eigenforms of large squarefree levels and nebentypus of small conductors,based on the recent prominent progress by Soundararajan on the extreme central values of L-functions.
基金Supported in part by the National Key Research and Development Program of China(No.2021YFA1000700)Natural Science Basic Research Program of Shaanxi(Nos.2023-JC-QN-0024,2023-JC-YB-077)+1 种基金Foundation of Shaanxi Educational Committee(No.2023-JC-YB-013)Shaanxi Fundamental Science Research Project for Mathematics and Physics(No.22JSQ010)。
基金Supported financially in part by the National Key Research and Development Program of China(No.2021YFA1000700)Natural Science Basic Research Program of Shaanxi(Nos.2023-JC-QN-0024,2023-JCYB-077)+1 种基金Foundation of Shaanxi Educational Committee(No.2023-JCYB-013)Shaanxi Fundamental Science Research Project for Mathematics and Physics(No.22JSQ010)
基金supported by National Basic Research Program of China (973 Project) (Grant No. 2011CB808003)the PCSIRT Project of the Ministry of EducationNational Natural Science Foundation of China (Grant No. 11231004)
文摘The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).
