Please use this identifier to cite or link to this item: `https://hdl.handle.net/10356/137118`
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dc.contributor.authorHong, Nankunen_US
dc.date.accessioned2020-02-26T04:47:07Z-
dc.date.available2020-02-26T04:47:07Z-
dc.date.issued2020-
dc.identifier.citationHong, N. (2020). On the ranks of partitions modulo certain integers. Doctoral thesis, Nanyang Technological University, Singapore.en_US
dc.identifier.urihttps://hdl.handle.net/10356/137118-
dc.description.abstractThis thesis focuses on the rank of partition functions, identities related to generating functions of ranks modulo different integers and Ramanujan's convolution sum. Most results in Chapters 2 and 4 are reproduced from [14] and [10], respectively. Ramanujan had three famous congruences for the partition function modulo 5, 7 and 11. F. J. Dyson defined the the rank of partitions and conjectured that ranks could provide combinatorial explanations for the cases of 5 and 7. A. O. L. Atkin and H. P. F. Swinnerton-Dyer proved his conjecture using generating functions for the rank difference modulo 5 and 7. From Theorem 8.16 in F. G. Garvan's paper [20], we know that results on dissections of the rank modulo m are equivalent to results on rank difference results modulo m, which inspired us to find a 3-dissection of ranks modulo 9 in Chapter 2. We also give an identity involving generating functions of ranks modulo 3 and 9. Ramanujan recorded several entries which are related to generating functions of the rank modulo different integers. Finding analogous identities is the motivation of Chapter 3. We give some identities, some of which are obtained by using Ramanujan's entries. In Ramanujan's paper [36], he proved a formula for convolutions of sum of divisors functions. In Chapter 4, we find formulas for convolutions of the sum of divisor functions twisted by the Dirichlet character, which are analogous to Ramanujan's.en_US
dc.language.isoenen_US
dc.publisherNanyang Technological Universityen_US
dc.subjectScience::Mathematics::Number theoryen_US
dc.titleOn the ranks of partitions modulo certain integersen_US
dc.typeThesis-Doctor of Philosophyen_US
dc.contributor.supervisorChan Song Hengen_US
dc.contributor.schoolSchool of Physical and Mathematical Sciencesen_US
dc.description.degreeDoctor of Philosophyen_US
dc.contributor.supervisoremailchansh@ntu.edu.sgen_US
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Appears in Collections:SPMS Theses
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