Note on the problem of Ramanujan’s radial limits

Bin Chen,Haigang Zhou

doi:10.1186/1687-1847-2014-191

Bin Chen, Haigang Zhou

Open Access

https://doi.org/10.1186/1687-1847-2014-191

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Abstract

Ramanujan in his deathbed letter to GH Hardy concerned the asymptotic properties of modular forms and mock theta functions. For the mock theta function f(q), he claimed that as q approaches an even order 2k root of unity ζ, lim q → ζ ( f ( q ) − ( − 1 ) k ( 1 − q ) ( 1 − q 3 ) ( 1 − q 5 ) ⋯ ( 1 − 2 q + 2 q 4 − ⋯ ) ) =O(1), where (1−q)(1− q 3 )(1− q 5 )⋯(1−2q+2 q 4 −⋯)= ∏ n = 1 ∞ 1 − q n ( 1 + q n ) 2 . Recently, Folsom, Ono and Rhoades have proved two closed formulas for the implied constant and formulated an open problem which is related to their two theorems. In this note, we give a new proof on the problem of the two theorems by using some results about the generating functions of convex compositions given by GE Andrews and Appell-Lerch sums.MSC:11F37, 11F03, 11F99.

Highlights

1 Introduction In his deathbed letter to Hardy, Ramanujan gave no definition of mock theta functions but just listed examples and a qualitative description of the key properties that he had noticed
While the theory of the weak Maass forms has led to a flood of applications in many disparate areas of mathematics, it is still not the case that we fully understand the deeper framework surrounding the contents of Ramanujan’s last letter to Hardy
We revisit Ramanujan’s original claims from his deathbed letter [ ], which begins by summarizing the asymptotic properties, near roots of unity, of the Eulerian series which were modular forms

Summary

Introduction

In his deathbed letter to Hardy, Ramanujan gave no definition of mock theta functions but just listed examples and a qualitative description of the key properties that he had noticed. For the mock theta function f (q), he claimed that as q approaches an even order 2k root of unity ζ , lim (f (q) – (–1)k(1 – q)(1 – q3)(1 – q5) · · · (1 – 2q + 2q4 – · · · )) = O(1), q→ζ where (1 – q)(1 – q3)(1 – q5) · · · (1 – 2q + 2q4 – · · · ) =

Results

Conclusion