On three theorems of Folsom, Ono and Rhoades

In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to Dyson's rank (mock theta) function and the Andrews--Garvan crank (modular) function---the match with explicit formulae for implied $O(1)$ constants. In this note we give another elementary proof of Ramanujan's original claim and outline some heuristics which may be useful for obtaining a new proof of the general Folsom--Ono--Rhoades theorem.


Ramanujan's claim
In his deathbed letter to G. H. Hardy, Ramanujan gave a vague definition of a mock modular function. It mainly referred to a specific asymptotic behaviour of such a function at roots of unity, and Ramanujan singled out the following illustrative example. The parameter q below is always assumed to be inside the unit disc.
Claim (Ramanujan [3]). As q approaches an even root of unity of order 2k, the difference f (q) − (−1) k b(q) is absolutely bounded.
In order to discuss and analyse the claim we introduce the standard q-Pochhammer notation (a; q) n := (1 − aq j ) for n = 0, 1, . . . , ∞; the above functions can be given then as follows: Recently Folsom, Ono and Rhoades proved, in two different ways, that Ramanujan's claim can be significantly refined. Namely, they showed that the difference f (q) − (−1) k b(q) has a limit as q approaches the corresponding even root.
Theorem 1 (Folsom, Ono and Rhoades [6,7]). If ζ is a primitive even order 2k root of unity, then, as q approaches ζ radially within the unit disc, where Theorem 2 (Folsom, Ono and Rhoades [7]). If ζ is a primitive even order 2k root of unity, then, as q approaches ζ radially within the unit disc, where Note that the sums on the right-hand sides in (1) and (2) terminate at the even root of unity.
It is not at all obvious that there are no modular forms which exactly cut out the singularities of a mock theta function, in particular, of Ramanujan's f (q): the presence on the right-hand side of (1) (or (2)) of a nonzero term, which depends on the root of unity, is essential. This is proven Griffin, Ono and Rolen in the 2013 paper [5].
The reader intrigued by Ramanujan's mock theta functions is referred to the inspiring expositions of Ono [9] and Zagier [11] on development of the subject.
Interestingly enough, Theorem 2 possesses an elementary proof [7] that makes use of q-series transformations only, while Theorem 1 is a particular instance of a much more general result (Theorem 3 stated below) whose proof uses a modern machinery of mock theta functions [6]. The principal goal of this note is to produce a simpler proof (challenged in [7]) of Theorem 1, a proof that Ramanujan had all ingredients for. All ingredients except possibly time.

Partition generating functions
As pointed out by many authors, the relation between Dyson's rank generating function R(w; q) := ∞ n=0 q n 2 (wq; q) n · (w −1 q; q) n and the series which is related to count of strongly unimodal sequences, has been already given by Ramanujan [2, Entry 3.4.7, p. 67]: The left-hand side of the series is nothing but a limiting case of bilateral 2 ψ 2 -series, and the above equality (3) follows from a general transformation due to W. N. Bailey. Though the Andrews-Garvan crank function is not immediately linked to R(w; q) and U(w; q), its similarity with the right-hand side of (3) becomes apparent from the expression [1, Entry 12.2.2, p. 264] A surprising fact is that the asymptotics of C(w; q) is related, in a simple way, to the asymptotics of (3) when w is chosen to be a root of unity and q approaches another root of unity radially. This remarkable relation is proven in the recent work of Folsom, Ono and Rhoades. The notation ζ m below is used for the root of unity e 2πi/m . Theorem 3 (Folsom, Ono and Rhoades [6,7]). Let 1 ≤ a < b and 1 ≤ h < m be integers with gcd(a, b) = gcd(h, m) = 1 and b | m. If h ′ is an integer satisfying hh ′ ≡ 1 (mod m), then, as q approaches ζ h m radially within the unit disc, we have Taking a = 1, b = 2 and m = 2k, so that ζ a b = −1 and ζ = ζ h m is a primitive even order 2k root of unity, we arrive at Theorem 1, because in this case f (q) = R(−1; q), b(q) = C(−1; q) and u(q) = U(−1; q).

Proof of Theorem 1
Proof of Theorem 1. Since f (q) = R(−1; q), b(q) = C(−1; q) and u(q) = U(−1; q), formulas (3) and (4) imply The right-hand sides can be further transformed: the first formula is [1, eq. (12.2.7), p. 264] with c = 1 and q replaced with −q, while the second one is [1, eq. (12.2.6), p. 264] with c = 1. It remains to notice that the pre-factor (−q; q) ∞ vanishes at any even root of unity, while the sums on the right-hand sides in (6) have finite limits as q approaches an order 2k root of unity, when k is even and odd, respectively.
The right-hand side in (6a) admits a different presentation [1, eq. (12.5.1), p. 280] that leads to a formula that can be used instead of (6a) in the proof. The 'literal' analogue of the latter sum for the right-hand side in (6b) is instead a partial theta-function [2, eq. (6.3.5), p. 120], In the odd k case the radial asymptotics can be also controlled with a help of the different identity [2, eq. (3.6.14), p. 79] An analogue of this identity for the even k case, with a proof similar to the one given in [2, p. 79], seems to be inadequate for the purposes: It is worth mentioning that the two identities (6) were used in [8] to give a short proof of another result about mock theta functions from the paper [4].

Asymptotics related to Theorem 3
It would be interesting to extend the identities of the previous section to also prove general Theorem 3. Here we briefly discuss some related asymptotics.
In light of (3), Theorem 3 means in particular that the quotient of as q approaches ζ h m radially. This fact follows from an elementary argument reproduced below, though knowledge of the limiting behaviour of the quotient, of course, does not imply that the difference of the two tends to 0.
To see this fact, write q = ζ h m r, where r → 1 − , and split the Appell-Lerch sums (including the one in (4)) into m subsums according to the residue n (mod m): As r → 1 − , each double sum involves a single collapsing subsum that corresponds to the residue c = c 0 for which ζ −a b ζ hc 0 m = ζ −am/b+hc 0 m = 1, so that c 0 ≡ h ′ am/b (mod m). This results in the root of unity for the limit of the quotient as r → 1 − .

Algebraic independence of q-zeta values
The elementary technique of Section 4 about asymptotic behaviour at roots of unity was used by Pupyrev in [10] to establish some (functional) linear and algebraic independence results for the so-called q-zeta values For even s, these q-series are related, in a simple way, to the classical Eisenstein series. In particular, P (q) := 1 − 24ζ q (2) (a quasi-modular form), and Q(q) := 1 + 240ζ q (4) and R(q) := 1 − 504ζ q (6) (modular forms of weight 4 and 6) are algebraically independent over C(q), while all other even q-zeta values are expressible as polynomials in ζ q (4) and ζ q (6).