• Corpus ID: 64693773

# THE f(q) MOCK THETA FUNCTION CONJECTURE AND PARTITION RANKS

@inproceedings{Ono2005THEFM,
title={THE f(q) MOCK THETA FUNCTION CONJECTURE AND PARTITION RANKS},
author={Ken Ono},
year={2005}
}
• K. Ono
• Published 2005
• Mathematics
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…
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## References

SHOWING 1-10 OF 23 REFERENCES

### 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

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

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

### Modular forms of weight 1/2

• Mathematics
• 1980
We will construct θ-series of weight 1/2 and 3/2 for some congruence subgroups of SL(2,ℤ) by taking appropriate coefficients of the representation $$\mathop{R}\limits^{\sim }$$ of SL(2;ℝ),

### Partitions with short sequences and mock theta functions.

• G. Andrews
• Mathematics
Proceedings 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

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

### Tenth order mock theta functions in Ramanujan's Lost Notebook

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously, the author proved

### The Selberg trace formula for PSL (2, IR)

• D. Hejhal
• 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.-