Note on the problem of Ramanujan’s radial limits
© Chen and Zhou; licensee Springer. 2014
Received: 7 May 2014
Accepted: 1 July 2014
Published: 23 July 2014
Ramanujan in his deathbed letter to GH Hardy concerned the asymptotic properties of modular forms and mock theta functions. For the mock theta function , he claimed that as q approaches an even order 2k root of unity ζ,
where . 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.
In his deathbed letter to Hardy, Ramanujan gave no definition of mock theta functions but just listed 17 examples and a qualitative description of the key properties that he had noticed. Since that time, many papers studying the 17 specific examples have been written by many famous mathematicians such as Watson, Selberg and Andrews . Due to the work of Zweger [2, 3], Bringmann and Ono [4, 5], Zagier  and others, we are able to recognize Ramanujan’s mock theta functions as holomorphic parts of certain harmonic weak Maass forms of weight , originally defined by Bruinier and Funke . This realization has resulted in many applications in combinatorics, number theory, physics and representation theory.
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. Here 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. He then asked whether others with similar asymptotic were necessary for the summation of a modular form and a function which is at all roots of unity. In fact, the recent work by Griffin et al.  has confirmed that there are no weakly holomorphic modular forms which exactly cut out the singularities of Ramanujan’s mock theta functions.
Claim (Ramanujan )
Throughout this paper, let . The function is convergent for and those roots of unity q with odd order. For the even order roots of unity, has exponential singularities. For example, , , .
Ramanujan’s last letter also inspired the problem of determining the asymptotic of the coefficients of mock theta functions such as . Andrews  and Dragonette  obtained asymptotic for coefficients of , then Bringmann and Ono  proved an exact formula for these coefficients. In the recent work, Folsom et al. [12, 13] provided two closed formulas for the implied constant .
Theorem 1.1 makes Ramanujan’s claim a special case of a more general result.
- (2)Since empty products equal 1, then Theorem 1.1 shows that(5)
Zudilin  has given an elementary proof of Theorem 1.1 by using Dyson’s rank function and the Andrews-Garvan crank function in his recent work.
In the meantime, Folsom et al.  proved a different form formula for the constant as follows.
Theorem 1.2 (Folsom-Ono-Rhoades Theorem 1.3 of )
Remark The authors left an open problem as a challenge for someone to find an elementary proof to show that the constants appearing in Theorem 1.1 match those appearing in Theorem 1.2. Interesting enough, Theorem 1.2 possesses a proof in  that makes use of q-series transformations only, while Theorem 1.1 is a particular instance of a much more general result whose proof uses a machinery of mock theta functions . The principal goal of this note is to give a new proof of the problem without using the relation of these two theorems.
2 Statement of results
As pointed out by many authors, Theorem 1.1 is a special case of a more general one, which surprisingly relates two well-known q-series: Dyson’s rank function and the Andrews-Garvan crank function . They play a prominent role in studying integer partition congruences.
Here and are two third-order mock theta functions in Ramanujan’s Lost Notebook.
where is the number of partitions of n with rank m.
The rank of a partition is defined to be its largest part minus the number of its parts. If is a root of unity, it is known that is (up to a power of q) a mock theta function which is the holomorphic part of a harmonic Maass form of weight .
where is the number of partitions of n with crank m.
where is the number of strongly unimodal sequences of size n with rank m.
Theorem 1.1 is a special case of the following theorem which relates these three q-series, here we define .
Theorem 2.1 (Folsom-Ono-Rhoades Theorem 1.2 of )
By taking , , and , so that , is a primitive even order 2k root of unity, Theorem 1.1 follows directly because of the fact that , and .
Based on the results above, in this note, we try to explain the relation between the two theorems, our results are as follows.
- (1)if , then(21)
- (2)if , then(22)
Obviously, we can get the following corollary from Theorem 2.2 directly.
- (1)if , then(23)
- (2)if , then(24)
- (1)One can use the uniqueness property of limits to obtain the result of Theorem 2.2 by combining with the relationship between these three functions(25)
In this note, we prove Theorem 2.2 by using the result related with the generating functions of convex compositions given by Andrews  and the online Encyclopedia of Integer Sequences  as well as Appell-Lerch sums, we get the desired results.
3 Proof of the theorem
Before we give the proof of the theorem, we would like to first introduce the recent work given by Andrews , which is about the theory of concave and convex compositions that linked the related generating functions to combinations of classical, false, or mock theta functions and other Appell-Lerch sums.
where , and c is called the central part and is ≥0. We denote the number of strictly convex compositions of n by . For example, .
It has been termed a second-order mock theta function by McIntosh .
to prove Theorem 1.2 and to show the new forms for and in (15), (16), respectively.
where is defined in (13) of Theorem 1.1.
The first claim in Theorem 2.2 is proved.
We have proved the second claim in Theorem 2.2.
On the other hand, we can prove the second claim in Theorem 2.2 in a new way which is related to the properties of Appell-Lerch sums.
The second claim in Theorem 2.2 is also completed.
We thank the editor and the referees for their valuable suggestions to improve the quality of this article. In addition, this article is supported by the Natural Science Foundation of China under Grant (11271283), the Natural Science Foundation of Shaanxi Province of China under Grant (2012JM1021), the Foundation of CMIRI of Shaanxi Province under Grant (2013JMR11) and the Key Subject Foundation under Grant (14SXZD002).
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