# The f(q) mock theta function conjecture and partition ranks

@article{Bringmann2006TheFM, title={The f(q) mock theta function conjecture and partition ranks}, author={Kathrin Bringmann and Ken Ono}, journal={Inventiones mathematicae}, year={2006}, volume={165}, pages={243-266} }

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for Ne(n) (resp. No(n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known…

## 67 Citations

A Survey of Classical Mock Theta Functions

- Mathematics
- 2012

In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an…

THE COEFFICIENTS OF THE ω(q) MOCK THETA FUNCTION

- Mathematics
- 2008

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of…

Effective Congruences for Mock Theta Functions

- Mathematics
- 2013

Let M(q) =∑ c(n)q n be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: c(An + B) ≡ 0 (mod l j ) where A is a multiple of l and…

Weak Maaß forms and $({\mathfrak{g}},K)$-modules

- Mathematics
- 2011

Harmonic weak Maaß forms have recently been shown to have quite a few interesting arithmetic applications, including their connection to Ramanujan’s work on (mock) theta functions (Bringmann and Ono…

Maass forms and the mock theta function f(q)

- MathematicsMathematische Annalen
- 2019

Let $$f(q):=1+\sum _{n=1}^{\infty } \alpha (n)q^n$$f(q):=1+∑n=1∞α(n)qn be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form…

FROM SHEAVES ON P 2 TO A GENERALIZATION OF THE

- Mathematics
- 2010

Moduli spaces of stable coherent sheaves on a surface are of much interest for both mathematics and physics. Yoshioka computed generating functions of Poincare poly- nomials of such moduli spaces if…

Some Relations on Sixth Order Mock Theta Functions

- Mathematics
- 2013

During the last years of his life, Ramanujan defined 17 functions F(q), where | q | < 1. Ramanujan named them as mock theta functions, because as q radially approaches any point e 2∏iΓ (r rational),…

Properties of Lerch Sums and Ramanujan's Mock Theta Functions.

- Mathematics
- 2018

In this article we study properties of Ramanujan's mock theta functions that can be expressed in Lerch sums. We mainly show that each Lerch sum is actually the integral of a Jacobian theta function…

Asymptotic formulas related to the $$M_2$$-rank of partitions without repeated odd parts

- MathematicsThe Ramanujan Journal
- 2019

We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method…

## References

SHOWING 1-10 OF 28 REFERENCES

On the seventh order mock theta functions

- Mathematics
- 1988

In a recent paper [H], we proved the "Mock Theta Conjectures". These are identities, stated by Ramanujan in his "lost notebook" JR2, pp. 19-20], involving two of the 5th order mock 0-functions. In…

Dyson's crank of a partition

- Mathematics
- 1988

holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)…

Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors

- Mathematics
- 2002

Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The…

Partitions: At the Interface of q-Series and Modular Forms

- Mathematics
- 2003

In this paper we explore five topics from the theory of partitions: (1) the Rademacher conjecture, (2) the Herschel-Cayley-Sylvester formulas, (3) the asymptotic expansions of E.M. Wright, (4) the…

On two geometric theta lifts

- Mathematics
- 2002

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result…

Partitions with short sequences and mock theta functions.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 2005

The object now is to develop the subject intrinsically to both provide deeper understanding of the theory and application of partitions and reveal the surprising role of Ramanujan's mock theta functions.

Modular Forms of Half Integral Weight

- Mathematics
- 1973

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function
$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty…

The Selberg trace formula for PSL (2, IR)

- Mathematics, Environmental Science
- 1983

Development of the trace formula (version A).- Poincare series and the spectral decomposition of L2(? \H, ?).- Version B of the selberg trace formula.- Version C of the selberg trace formula.-…

On the number of divisors of quadratic polynomials

- Physics
- 1963

A switch for a suspended railway vehicle having two elastic wheels, comprises four rails forming three paths one of which branches off into two other paths. A gap is bounded between the four rails.…