Rank of partition and mock theta conjecture

order 3 Ramanujan mock theta function

• 3rd order mock theta functions \[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
• coefficients 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244

 

Andrews-Dragonette

• [Dragonette1952] and [Andrews1966]
• concerns the question of partitions with even rank and odd rank
• rank of partition =  largest part - number of parts 9의 분할인 {7,1,1}의 경우, rank=7-3=4 9의 분할인 {4,3,1,1}의 경우, rank=4-4=0
• \(N_e(n), N_o(n)\) number of partition with even rank and odd rank
• \(p(n)=N_e(n)+N_o(n)\)
• \(\alpha(n)=N_e(n)-N_o(n)\)
• this is in fact the coefficient of the 3rd order mock theta functions

\[f(q) = \sum_{n\ge 0} \alpha(n)q^n\]

• thus we need modularity of f(q) to get exact formula for \(\alpha(n)\) as \(p(n)\) was obtained by the circle method

 

 

harmonic Maass form of weight 1/2

• Zweger's completion

 

 

 

 

generalization

• crank

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

related items

• 3rd order mock theta functions
• Dyson rank generating function 

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxSzZNSVVMNFZIUzg/edit
• http://oeis.org/A000025

expositions

• Puzzle Solved: Ramanujan's Mock Theta Conjectures
• Partitions : at the interface of q-series and modular forms Andrews, George E., 2003

articles

• The f(q) mock theta function conjecture and partition ranks Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006
• George E. Andrews and F. G. Garvan, Ramanujan's “Lost” Notebook VI: The mock theta conjectures 1989
• Hickerson, Dean, A proof of the mock theta conjectures (1988), Inventiones Mathematicae 94 (3): 639–660, doi:10.1007/BF01394279, MR969247, ISSN 0020-9910
• [Andrews1966]On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
  • Andrews, George E. (1966), American Journal of Mathematics 88: 454–490
• [Dragonette1952]Some asymptotic formulae for the mock theta series of Ramanujan
  • Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
• Watson, G. N. (1937), "The Mock Theta Functions (2)", Proc. London Math. Soc. s2-42: 274–304, doi:10.1112/plms/s2-42.1.274
• Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", J. London Math. Soc. 11: 55–80, doi:10.1112/jlms/s1-11.1.55