Abstract
In his famous “deathbed” letter, Ramanujan “defined” the notion of a mock theta function and offered some examples of functions he believed satisfied his definition. Very recently, Griffin et al. established for the first time that Ramanujan’s mock theta functions actually satisfy his own definition. On the other hand, Zwegers’ 2002 doctoral thesis [Zwegers S (2002) Mock theta functions. PhD thesis (Univ Utrecht, Utrecht, The Netherlands)] showed that all of Ramanujan’s examples are holomorphic parts of harmonic Maass forms. This has led to an alternate definition of a mock theta function. This paper shows that Ramanujan’s definition of mock theta function is not equivalent to the modern definition.
In his famous “deathbed” letter (1), Ramanujan introduced the notion of a mock theta function. The following version of Ramanujan’s definition follows Andrews and Hickerson (2) and Zwegers (3).
Ramanujan’s Definition.
A mock theta function is a function f of the complex variable q, defined by a q-series of a particular type (Ramanujan calls this the Eulerian form), which converges for 
and satisfies the following conditions:
(1) infinitely many roots of unity are exponential singularities;
(2) for every root of unity ξ, there is a theta function
, such that the difference 
is bounded as 
radially; and(3) f is not the sum of two functions, one of which is a theta function and the other a function that is bounded radially toward all roots of unity.
Remark:
Ramanujan’s theta function 
, with 
, is more general than the notion of theta function considered here. As in refs. 3 and 4, we define a theta function as a weakly holomorphic modular form. Consequentially, these objects will have asymptotics that resemble those discussed by Ramanujan in his deathbed letter. Moreover, all of the examples constructed in his letter satisfy his definition with each 
a weakly holomorphic modular form [see the work of Griffin et al. (4) and the discussion therein].
Ramanujan gave 17 examples of functions he believed satisfied these properties. The most famous is
 |
Clearly 
when ξ is an odd order root of unity. (Here and throughout the remainder of the paper, all limits are radial limits.) In his letter, Ramanujan claimed that at all primitive, even 
roots of unity, say ξ,
 |
where 
is, up to a power of q, a weakly holomorphic modular form. Watson (1) proved this claim. Very recently (4), Griffin et al. proved that there is no weakly holomorphic modular form M such that 
is bounded radially toward all roots of unity. Until the results of that paper were announced, it was not known whether any of Ramanujan’s mock theta functions satisfied his own definition.
Despite the lack of a definition, Ramanujan’s mock theta functions were shown to possess many striking properties. For example, Ramanujan himself related certain sums of mock theta functions to modular forms. As an example, he claimed that
 |
where 
and 
are as above, and 
is one of his third-order mock theta functions. Identities of this flavor are often referred to as “mock theta conjectures” [see the survey of Gordon and McIntosh (5) or the work of Andrews and Garvan (6)]. Many examples of such identities proved themselves very difficult to establish. The most significant were proved in Hickerson’s works (7, 8).
Other striking properties are the Hecke-type series found by Andrews (9). As an example, consider Ramanujan’s fifth-order mock theta function
 |
Andrews proved that
 |
Note the resemblance to
 |
(see equation 5.15 of ref. 9, for example).
These hints of structure and many others led to Dyson’s (10, p 20) statement in 1987:
Somehow it should be possible to build them (the mock theta functions) into a coherent group-theoretic structure, analogous to the structure of modular forms which Hecke built around the old theta-functions of Jacobi. This remains a challenge for the future.
This group-theoretic structure was discovered by Zwegers. Zwegers’ 2001 PhD thesis (11) was a breakthrough in the study of the mock theta functions. As a result of his thesis, it is known that all of Ramanujan’s examples are essentially the holomorphic part of weight 
weak harmonic Maass forms whose nonholomorphic parts are period integrals of weight 
unary theta functions.
This has led to a huge number of results that are out of reach otherwise. Perhaps the most astonishing breakthroughs are the works of Bringmann and Ono (12, 13), which establish congruence properties and exact formulas for the coefficients of mock theta functions. Zwegers’ thesis also led to a simpler and more conceptual proof of the mock theta conjectures [see Folsom’s work (14)].
Zwegers’ construction results in an alternative definition of “mock theta function.” However, that definition has seemingly nothing to do with Ramanujan’s definition. Theorem 1.1 shows these definitions cannot be equivalent. A number of questions that will hopefully lead to a reconciliation of the two definitions are raised in 2. Questions and Remarks.
1. Modern Definition of Mock Theta Function
Following Zagier (15)‡, we offer the following definition of a mock theta function.
Modern Definition.
A mock theta function is a q-series 
such that there exists a rational number 
and a unary theta function 
of weight k, such that 
is a nonholomorphic modular form of weight 
, where
 |
with 
, the incomplete Gamma function and 
a constant that depends only on k. The function g is called the shadow.
As remarked in Dabholkar, et al.‡, the condition that the shadow be a unary theta function forces the weight k to be either 
or 
. All of Ramanujan’s examples have 
.
There are two particularly elegant examples of mock theta functions with shadow proportional to weight 
unary theta functions. The work of Bringmann and Lovejoy (8, 16) studies the series
 |
The coefficients of this series are related to the rank of an overpartition. They prove that this is a mock theta function with shadow proportional to the unary theta function 
. This series is particularly interesting because of its relation with the class number generating function of Zagier (18), which is also a mock theta function with the same shadow.
The second elegant example of a mock theta function with shadow proportional to a weight 
unary theta function is the generating function for the number of smallest parts of a partition, denoted spt. The value 
is defined as the number of appearances of the smallest parts in the partitions of n. Andrews (19) showed that
 |
Moreover, it was proved by Bringmann (20) that 
is essentially a mock modular form with shadow
 |
where 
is the Kronecker symbol.
Our theorem makes use of a different mock theta function with shadow proportional to 
.
Theorem 1.1.
Ramanujan’s definition of a mock theta function is not equivalent to the modern definition of a mock theta function.
Proof:
Define the two q-series
 |
 |
where 
.
Remark:
In the notation of ref. 21, 
and 
.
We show that either 
is a mock theta function according to the modern definition, but not Ramanujan’s definition, or 
is a mock theta function according to Ramanujan’s definition, but not the modern definition.
First, it follows from theorem 1.2 (1) of ref. 21 that 
is a mock theta function according to the modern definition with shadow proportional to 
.
Assume that 
is mock theta function according to Ramanujan’s definition as well. In this case, we prove that 
is a mock theta function according to Ramanujan’s definition, but that it is not a mock theta function according to the modern definition. It was proved by Andrews (22) that
 |
The right-hand side of this identity is 
for all roots of unity q, because the sum is finite. Therefore, 
and 
have all of the same singularities at roots of unity. Because 
is a mock theta function according to Ramanujan’s definition for every root of unity ξ, there exists a theta function 
such that 
is bounded as 
radially. Therefore, 
is also bound as 
radially. Moreover, because the exponential singularities of 
and 
are identical and 
was assumed to be a mock theta function according to Ramanujan, it follows that there is no single theta function that “cuts out” all of the singularities of 
.
To complete the proof it suffices to prove that 
is not a mock theta function according to the modern definition. However, by theorem 1.2 of ref. 21, 
is a mock theta function with shadow proportional to 
. Therefore, 
is a mixed-mock modular form; that is, it is the product of a mock theta function and a modular form. Because the product of two harmonic Maass forms is a harmonic Maass form only when both forms being multiplied are actually modular forms or one of them is a constant, it is clear that 
is not the holomorphic part of a harmonic Maass form and thus not a mock theta function according to the modern definition.
2. Questions and Remarks
The work of Griffin et al. (4) as well as this text leaves many questions to be answered.
2.1.
The proof of our main theorem uses a mixed-mock theta function (section 7.3 of Dabholkar, et al.‡). Are all mixed-mock theta functions mock theta functions according to Ramanujan’s definition?
2.2.
The proof of Theorem 1.1 uses an example of a weight 
mock theta function. Does there exist a weight 
counter example to the equivalence of Ramanujan’s definition and the modern definition?
2.3.
Let F be a mock theta function according to Ramanujan’s definition. Let 
be the set of nonzero weakly holomorphic modular forms from condition 2 of Ramanujan’s definition. Define the Gordon–McIntosh (5) universal mock theta functions by
 |
where 
.
Since 
where 
is Dyson’s partition rank generating function, it is clear that 
is finite. It should be true that for any root of unity ζ and integers A and B with 
and 
, the sets 
and 
are finite. It would be interesting to determine these sets in general.
Remark:
It should be noted that for some roots of unity the theta functions occasionally need to be modified by a power of q to get the radial limits to match.
2.4.
Is 
, given above, a mock theta function according to Ramanujan’s definition? The results of ref. 4 show that there is no weakly holomorphic modular form M such that 
is bounded radially toward all roots of unity. Is there a set of weakly holomorphic modular forms 
, where ξ runs over all roots of unity, such that 
is bounded as 
radially?
2.5.
There exists a modular form 
such that
 |
where 
is one of Ramanujan’s fifth-order mock theta functions (ref. 5, p. 104). Moreover, 
and 
have singularities at disjoint sets of roots of unity. As a result, in the notation above, the set 
contains only one nontrivial modular form. This appears to be typical of the fifth-order mock theta functions of Ramanujan.
Footnotes
The author declares no conflict of interest.
†This Direct Submission article had a prearranged editor.
‡Dabholkar A, Murthy S, Zagier D, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, preprint.
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