Ramanujan s radial limits

Transcription

1 Contemporary Mathematics Ramanujan s radial limits Amanda Folsom, Ken Ono, and Roert C. Rhoades Astract. Ramanujan s famous deathed letter to G. H. Hardy concerns the asymptotic properties of modular forms and his so-called mock theta functions. For his mock theta function f(q), he asserts, as q approaches an even order k root of unity, that we have f(q) ( 1) k (1 q)(1 q 3 )(1 q 5 ) `1 q + q 4 O(1). We give two proofs of this claim y offering exact formulas for these limiting values. One formula is a specialization of a general result which relates Dyson s rank mock theta function and the Andrews-Garvan crank modular form. The second formula is ad hoc, and it relies on the elementary manipulation of q- series which are themselves mock modular forms. Both proofs show that the O(1) constants are not mysterious; they are values of special q-series which are finite sums at even order roots of unity. 1. Introduction and Statement of Results Ramanujan s enigmatic last letter to Hardy [7] gave tantalizing hints of his theory of mock theta functions. By work of Zwegers [31, 3], it is now known that Ramanujan s examples are essentially holomorphic parts of weight 1/ harmonic weak Maass forms (see [13] for their definition). This realization has resulted in many applications in cominatorics, numer theory, physics, and representation theory (for example, see [5, 9]). Here we revisit Ramanujan s original claims from this letter [7]. The letter egins y summarizing the asymptotic properties, near roots of unity, of Eulerian series which are modular forms. He then asks whether other Eulerian series with similar asymptotics are necessarily the sum of a modular form and a function which is O(1) at all roots of unity. He writes: The answer is it is not necessarily so. When it is not so I call the function Mock ϑ-function. I have not proved rigorously that it is not necessarily so. But I have constructed a numer of examples in which it is inconceivale to construct a ϑ- function to cut out the singularities of the original function. 000 Mathematics Suject Classification. 11F99, 11F37, 33D15. Key words and phrases. Mock theta functions, quantum modular forms. The authors thank the NSF and the Asa Griggs Candler Fund for their generous support. 1 c 0000 (copyright holder)

2 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES Ramanujan offers a specific example for the q-hypergeometric function (1.1) f(q) : 1 + q (1 + q) + q 4 (1 + q) (1 + q ) +..., which he claims (ut does not prove) is a mock theta function according to his imprecise definition. This function is convergent for q < 1 and those roots of unity q with odd order. For even order roots of unity, it has exponential singularities. For example, as q 1, we have f( 0.994) , f( 0.996) , f( 0.998) To cancel the exponential singularity at q 1, Ramanujan found the function (q), which is modular 1 up to multiplication y q 1 4, defined y (1.) (q) : (1 q)(1 q 3 )(1 q 5 ) (1 q + q 4 ). The exponential ehavior illustrated aove is canceled in the numerics elow. q f(q) + (q) It appears that lim q 1 (f(q) + (q)) 4. More generally, as q approaches an even order k root of unity radially within the unit disk, Ramanujan claimed that (1.3) f(q) ( 1) k (q) O(1). Ramanujan s point is that (q) is a near miss, a modular form which almost cuts out the exponential singularities of f(q). He asserts that (q) cuts out the exponential singularities of f(q) for half of the even order roots of unity, while (q) cuts out the exponential singularities for the remaining even order roots of unity. Of course, if f(q) is a mock theta function according to his definition, then there are no modular forms which exactly cut out its exponential singularitites. Remark. Claim (1.3) is intimately related to the prolem of determining the asymptotics of the coefficients of f(q). Andrews [1] and Dragonette [16] otained asymptotics for these coefficients, and Bringmann and the second author [11] later otained an exact formula for these coefficients. In a recent paper [0], the authors proved (1.3) y otaining a simple closed formula for the implied O(1) constants. Theorem 1.1 (Theorem 1.1 of [0]). If ζ is a primitive even order k root of unity, then, as q approaches ζ radially within the unit disk, we have that ( lim f(q) ( 1) k (q) ) k 1 4 (1 + ζ) (1 + ζ ) (1 + ζ n ) ζ n+1. q ζ Example. Since empty products equal 1, Theorem 1.1 confirms that lim (f(q) + (q)) 4. q 1 1 Here q 1 4 (q) is modular with respect to z where q : e πiz.

3 RAMANUJAN S RADIAL LIMITS 3 Example. For k, Theorem 1.1 gives lim q i (f(q) (q)) 4i. The tale elow nicely illustrates this fact: q 0.99i 0.994i 0.996i f(q) i i i f(q) (q) i i i Remark. Zudilin [30] has recently otained an elementary proof of Theorem 1.1. Theorem 1.1 is a special case of a more general theorem, one which surprisingly relates two well-known q-series: Dyson s rank function R(w; q) and the Andrews- Garvan crank function C(w; q). These series play a prominent role in the study of integer partition congruences (for example, see [4, 6, 1, 17, 4]). To define these series, throughout we let (a; q) : (1 a)(1 aq)(1 aq ) and for n Z (a; q) n : (a; q) (aq n. ; q) Dyson s rank function is given y (1.4) R(w; q) N(m, n)w m q n : 1 + m Z n1 q n (wq; q) n (w 1 q; q) n. Here N(m, n) is the numer of partitions of n with rank m, where the rank of a partition is defined to e its largest part minus the numer of its parts. If w 1 is a root of unity, then it is known that R(w; q) is (up to a power of q) a mock theta function (i.e. the holomorphic part of a weight 1/ harmonic Maass form) (for example, see [1] or [9]). The Andrews-Garvan crank function is defined y (1.5) C(w; q) M(m, n)w m q n : m Z (q; q) (wq; q) (w 1 q; q). Here M(m, n) is the numer of partitions of n with crank m [4]. For roots of unity w, C(w; q) is (up to a power of q) a modular form. We also require the series U(w; q) [, 5, 14, 6], which arises in the study of unimodal sequences. This q-hypergeometric series is defined y (1.6) U(w; q) u(m, n)( w) m q n : (wq; q) n (w 1 q; q) n q n+1. m Z Here u(m, n) is the numer of strongly unimodal sequences of size n with rank m [14]. Theorem 1.1 is a special case of the following theorem which relates these three q-series. Throughout, we let ζ n : e πi/n. Theorem 1.. (Theorem 1. of [0]) Let 1 a < and 1 h < k e integers with gcd(a, ) gcd(h, k) 1 and k. If h is an integer satisfying hh 1 (mod k), then, as q approaches ζk h radially within the unit disk, we have that ( ) R (ζ a ; q) ζ a h k C (ζ a ; q) (1 ζ a )(1 ζ a ) U(ζ a ; ζk h ). lim q ζ h k

4 4 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES Four remarks. 1) There is an integer c(a,, h, k) such that the limit in Theorem 1. reduces to the finite sum c(a,,h,k) (1 ζ a )(1 ζ a ) (ζ a ζk h ; ζk h ) n (ζ a ζk h ; ζk h ) n ζ h(n+1) k. ) Theorem 1.1 is the a 1 and case of Theorem 1. ecause R( 1; q) f(q), comined with the elementary fact that C( 1; q) (q). 3) A variant of Theorem 1. holds when k. This is otained y modifying the proof to guarantee that the two resulting asymptotic expressions match. 4) Garvan [1] was the first to compare the asymptotics of the rank and crank generating functions. His oservations were made in the context of the moments for the rank and crank statistic. A precise form of these results has een otained y Bringmann and Mahlurg and the third author [9, 10]. Theorem 1., which in turn implies Theorem 1.1, relies on an identity of Choi [15] and Ramanujan (see Entry in [3]). This identity reduces the proof of Theorem 1. to the claim, upon appropriate specialization of variales, that a certain mixed mock modular form is asymptotic to a suitale multiple of the modular crank function. Here we offer a second proof of (1.3). This proof is ad hoc, and gives a different formula for the O(1) constants in (1.3). Theorem 1.3. If ζ is a primitive even order k root of unity, then, as q approaches ζ radially within the unit disk, we have ( f(q) ( 1) k (q) ) lim q ζ k 1 4 ( 1) n ζ n+1 (1 + ζ )(1 + ζ 4 ) (1 + ζ n ) if k 0 (mod ), k 1 + ( 1) n+1 ζ n+1 (1 + ζ)(1 + ζ 3 ) (1 + ζ n 1 ) if k 1 (mod ). Remark. We leave it as a challenge to find an elementary proof that the constants appearing in Theorem 1.3 match those appearing in Theorem 1.1. In Section we sketch the proof of Theorem 1., and in Section 3 we give the proof of Theorem 1.3. In the last section we conclude with a discussion of the oservations which played a role in the discovery of Theorem 1.3. Acknowledgements The authors thank Roert Lemke Oliver for computing the numerical examples in this paper, and they thank George Andrews, Bruce Berndt, Kathrin Bringmann, and Jeremy Lovejoy for helpful conversations. The complete proof is contained in [0].

5 RAMANUJAN S RADIAL LIMITS 5. Sketch of the proof of Theorem 1. The proof of Theorem 1. requires the theory of modular units and mock modular forms. In particular, we employ the modular properties of Dedekind s η-function η(z) : q 1 4 (1 q n ), n1 and certain Klein forms t (r,s) (z) t (N) (r,s)(z) [3]. We also require the Appell-Lerch function which played an important role in the work of Zwegers [3] on Ramanujan s mock theta functions. For q e πiz, z H, and u, v C \ (Zz + Z), this function is defined y µ(u, v; z) : eπiu ϑ(v; z) n Z Here the Jacoi theta function is defined y ( 1) n q n(n+1) e πinv 1 e πiu q n. (.1) ϑ(v; z) : i n Z ( 1) n q 1 (n+ 1 ) e πiv(n+ 1 ) iq 1 8 e πiv (1 q n )(1 e πiv q n 1 )(1 e πiv q n ). n1 The µ-function satisfies the following eautiful ilateral series identity. Theorem.1 (see p.67 of [3]). Let q e πiz, where z H. For suitale complex numers α e πiu and β e πiv, we have (αβ) n q n + q n (α 1 ; q) n (β 1 ; q) n (αq; q) n (βq; q) n n1 iq 1 8 (1 α)(βα 1 ( ) 1 qα 1 ; q ) ( β 1 ; q ) µ(u, v; z). Remark. Theorem.1 is also otained in a eautiful paper y Choi [15]. To make use of this identity, we study the function A(u, v; z) : ϑ(v; z)µ(u, v; z) which was previously studied y Zwegers [33] and the first author and Bringmann [8]. We employ the completed function Â(u, v; z), defined y Zwegers as (.) where R(v; z) : n Z Â(u, v; z) : A(u, v; z) + i ϑ(v; z)r(u v; z), { ( sgn n + 1 ) (( E n Im(v) ) )} Im(z) Im(z) and for w C we have ( 1) n q 1 (n+ 1 ) e πiv(n+ 1 ), E(w) : w 0 e πu du. Using the transformation properties [3] of the functions µ and ϑ, we have, for integers m, n, r, s and γ ( a c d ) SL (Z), that

6 6 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES (.3)  (u + mz + n, v + rz + s; z) ( 1) m+n e πiu(m r) e πivm q m mr Â(u, v; z), ( ) u  cz + d, v cz + d ; γz ( u +uv) πic (.4) (cz + d)e (cz+d) Â(u, v; z). Sketch of the Proof of Theorem 1.. The proof makes use of the modular transformation properties descried aove. We consider Choi s identity with α ζ a and β ζ a, (hence u a, v a πi ), and q replaced y e k (h+iz), and we define (.5) m(a, ; u) : ie πiu 4 (1 ζ a )ζ a (ζ a e πiu ; e πiu ) (ζ a ; e πiu ). To prove Theorem 1., noting that the function U(ζ a; ζh k ) is a finite convergent sum when k, it suffices to prove that upon appropriate specialization of variales, the mixed mock modular form m µ is asymptotic to a suitale multiple of the modular crank generating function C. To e precise, let k, gcd(a, ) 1, gcd(h, k) 1, where a,, h, k are positive integers. As z 0 +, we must prove that (.6) m (a, ; 1k (h + iz) ) µ ( a, a ; 1k (h + iz) ) ζ a h k C (ζ a ; 1k ) (h + iz). Aove and in what follows, we let z R +, and let z 0 +. This corresponds to the radial limit q e πi k (h+iz) ζk h from within the unit disk. The claim (.6) is otained y comparing separate asymptotic results for the crank function and the mixed mock modular form. To descrie this, we let (.7) q : e πi k (h+iz), q 1 : e πi k (h + i z ). For the mixed mock modular m µ, we otain the following asymptotics. Let k, gcd(a, ) 1, gcd(h, k) 1, where a,, h, k are positive integers, and let and h e positive integers such that k and hh 1 (mod k). For z R +, as z 0 +, we estalished in Theorem 3. of [0] that there is an α > 1/4 for which m (a, ; 1k ) (h + iz) µ ( a, a ; 1k ) (h + iz) ( ) 1 i (ψ(γ)) 1 q 1 4 q ( 1) a ζ ah a z ζ a 1 (1 + O(q α 1 ζ ah 1 )). ζ 3a kh Here, γ γ(h, k) SL (Z), and ψ(γ) is a 4th root of unity, oth of which are defined in [0]. Under the same hypotheses, we show in Proposition 3.3 of [0] that the modularity of Dedekind s eta-function and the Klein forms implies, for the crank function, that C (ζ a ; 1k ) (.8) (h + iz) ( ) 1 i (ψ(γ)) 1 q 1 4 q ( 1) a ζ ah a z ζ a kh ζ a 1 (1 + O(q β 1 ζ ah 1 )),

7 RAMANUJAN S RADIAL LIMITS 7 for some β > 1/4. Comining these asymptotics gives (.6), which in turn implies Theorem 1.. It is important to explain the automorphic reasons which underlie the asymptotic relationship of Theorem 1.. The rank generating function R(w; q) and the crank generating function C(w; q) are Jacoi forms with the same weight and multiplier. This coincidence explains why their radial asymptotics are closely related. However, they are not of the same index, and this difference accounts for the ( 1) k in Theorem 1.1, and the ζ a h k in Theorem 1.. Of course, these facts alone do not directly lead to Theorem 1.. To make this step requires the ilateral series of Theorem.1, namely, n Z q n (wq) n (w 1 q) n R(w; q) + (1 w)(1 w 1 ) q n+1 (wq) n (w 1 q) n. It turns out that this series is related to a mixed-mock Jacoi form which has asymptotics resemling that of crank generating function. Although such coincidences are mysterious, it is not uncommon that such a ilateral series possesses etter modular properties than either half of the series (See [] for more dealing with ilateral series and mock theta functions). 3. Proof of Theorem 1.3 Here we prove Theorem 1.3, a second formula for the O(1) constants in (1.3). Proof of Theorem 1.3. In his deathed letter, Ramanujan defined four third order mock theta functions. Three of them are f(q), which we saw aove, and φ(q) :1 + q 1 + q + q 4 (1 + q )(1 + q 4 ) + q n ( q ; q ) n ψ(q) : q 1 q + q 4 (1 q)(1 q 3 ) + q 9 (1 q)(1 q 3 )(1 q 5 ) + q n (q; q. ) n In his final letter, Ramanujan stated the following relation etween these three functions (3.1) φ( q) f(q) f(q) + 4ψ( q) (q), where (q) is defined in (1.). These relations were proved y Watson [8]. Later, Fine [19] gave elegant proofs which rely on his detailed study of the asic hypergeometric series (3.) F (a, ; t) F (a, ; t, q) : (3.3) From (3.1), we have f(q) (q) 4ψ( q). (aq; q) n (q; q) n t n. Note that if q ζ and ζ is a k root of unity with k even, then oth functions on the left hand side of (3.3) must have singularities, while the function on the right n1

8 8 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES hand side of (3.3) is a constant. Likewise, if q ζ and ζ is a k root of unity with k odd, then we have (3.4) f(q) + (q) φ( q). The function on the right hand side of (3.4) tends to a finite numer, while the functions on the left hand side of (3.4) each have singularities. Therefore, to estalish the theorem it is enough to show that (3.5) ψ(q) q n+1 ( q ; q ) n and φ(q) 1 + q n+1 ( 1) n (q; q ) n. Both of these identities follow as special cases of Fine [19] (7.31), which is F (/t, 0; t) 1 1 t ( ) n q n +n. (tq; q) n To prove the first claim in (3.5), we take q 1 and t q 1 to otain 1 1 q 1 q n +n (q 3 ; q) n q n ( q; q)n. Sending q q and multiplying y q gives the first claim. To prove the second claim in (3.5), take q 1 and t q to otain q q n +n ( q ; q) n ( q) n (q 1 ; q)n. Sending q q and multiplying y q, then adding 1, gives the second claim of (3.5). 4. Discussion related to Theorem 1.3 The remainder of the paper will explain the discovery of Theorem 1.3. Recall that ( lim f( e t ) + ( e t ) ) 4U( 1; 1) 4. t 0 + However, the calculations of [0] yield the stronger asymptotic relation Computation gives f( e t ) + ( e t ) 4U( 1; e t ) as t 0 +. (4.1) U( 1; e t ) 1 t + 7 t! 17t3 3! t4 4! 35831t5 5! t6 6! The coefficients of this t-series can e given in closed form. Theorem 4.1. We have f( e t ) + ( e t ) a n n! ( t)n as t 0 +,

9 RAMANUJAN S RADIAL LIMITS 9 where the a n are given y a n ( 1) n a++cn + ( 1) n n! a!()!c! a+n n! a!()! ( ) a ( ) 3 5 E a+ ( ) a ( ) 3 1 E a+, where E n are the Euler numers, and summation is taken over a,, c N 0. This closed form is not enlightening. For instance, it is not clear from this formula that the a n are all positive. The significance of this result lies in (4.1). A search in the Online Encyclopedia of Integer Sequences [7] shows that these numers are also the values of G(e t ) as t 0 + where (4.) G(q) : 1 + ( 1) n (q; q ) n. n1 This is a function which, as defined, exists only for q a root of unity. The asymptotic relationship etween U( 1; e t ) and G(e t ) led us to guess that there might e a relation at other roots of unity. We computed the following tale with ζ k e πi/k. k U( 1; ζ k ) G(ζ k ) i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i From this tale and additional computation it is clear that for odd roots of unity ζ we have U( 1; ζ) G(ζ). Remark. We would like an elementary proof of this claim.

10 10 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES Next, we wondered if it was possile to extend the function G to converge in the domain q < 1. The ovious thing to try is to write G(q) 1 + ( 1) n+1 (q; q ) n ( 1) n+1 (q; q ) n ( 1) n+1 q n+1 (q; q ) n n1 ( 1) n (q; q ) n + ( 1) n q n+1 (q; q ) n. This implies that G(q) 1 + ( 1) n q n+1 (q; q ) n 1 + q q 3 + q 4 + q 5 q 6 q 7 + q 9 q 11 q 14 q 15 + q q 17 + A search in the Online Encyclopedia of Integer Sequences [7] shows that this q-series matches that of the mock theta function φ(q). A literature search for relations etween f(q) and φ(q) turned up the result f(q) + (q) φ( q), which led to Theorem Proof of Theorem 4.1. Here we prove Theorem 4.1. Proof of Theorem 4.1. From [] page 18 with q e t we have ( π π f( q) + t exp 4t t ) (4.3) f( e π t ) 4 4t π e t 4 e cosh ( 5 3 tx tx) + cosh ( 1 tx) dx. cosh (3tx) Notice that f( e π t ) 1 + O 0 (e π t ) as t 0 +. Next, using the fact that (e πiτ ) e πiτ/1 η 3 (τ)/η (τ), together with the modular transformation law for the Dedekind η-function, a straightforward calculation shows that ( π π ( e t ) t exp 4t t ) (1 + O(e π t )) 4 as t 0 +. Thus, using these facts, to prove Theorem 4.1 it suffices to analyze the asymptotic as t 0 + of the right hand side of (4.3). A simple change of variales gives 4t π 0 e cosh ( 5 3 tx tx) + cosh ( 1 tx) cosh (3tx) π 3t R e π w 6t dx cosh ( 5 6 πw) + cosh ( 1 6 πw) cosh (πw) Recall that the Mordell integral as defined y Zwegers [3] for z C, τ H, is given y e πiτx πzx (4.4) h(z; τ) : R cosh (πx) dx cosh (πzw) πiτw e R cosh (πw) dw. dw.

11 RAMANUJAN S RADIAL LIMITS 11 By Proposition 1. (5) of [3] we have h(z; τ) 1 e πiz τ iτ R e πi R Thus, for rational 0 < A < 1 we have e π w cosh(aπw) 6t (4.5) 6t cosh(πw) dw π e 3 A t 0 cosh ( πw z τ w τ cosh (πw) R ) dw. cos(6atw) 6tw e cosh(πw) dw. Letting A 5 6, and then A 1 6 in (4.5), and adding, we find that 4t π e t 4 e cosh ( 5 3 tx tx) + cosh ( 1 tx) g(w; t) dx cosh (3tx) cosh(πw) dw, where (4.6) g(w; t) :e 6tw ( e t cos(5tw) + cos(tw) ) + (1 1w )t + ( ) 1 19w + 36w 4 t + O(t 3 ). We make use of the well known identity (see [18] for example) w n cosh(πw) dw ( 1)n E n n, R where E n are the Euler numers. The Taylor expansion of g(w; t) and these constants give the asymptotic as stated in Theorem 4.1. References [1] G. E. Andrews, On the theorems of Watson and Dragonette for Ramanujan s mock theta functions, Amer. J. Math. 88 (1966), [] G. E., Andrews, Concave and convex compositions, Ramanujan J., to appear. [3] G. E. Andrews and B. C. Berndt, Ramanujan s Lost Noteook. Part II. Springer, New York, 009. [4] G. E. Andrews and F. Garvan, Dyson s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), [5] G. E. Andrews, R. C. Rhoades, S. Zwegers, Modularity of the concave composition generating function, Alg. Num. Th., to appear. [6] A. O. L. Atkin and H.P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 4 (1954), [7] B. C. Berndt and R. A. Rankin, Ramanujan: Letters and commentary, Amer. Math. Soc., Providence, [8] K. Bringmann and A. Folsom, On the asymptotic ehavior of Kac-Wakimoto characters, Proc. Amer. Math. Soc., to appear. [9] K. Bringmann and K. Mahlurg, Inequalities etween crank and rank moments, Proc. Amer. Math. Soc. 137 (009), [10] K. Bringmann, K. Mahlurg, R. C. Rhoades, Taylor Coefficients of Mock-Jacoi Forms and Moments of Partition Statistics, to appear Math. Proc. Cam. Phil. Soc. [11] K. Bringmann and K. Ono, The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165 (006), [1] K. Bringmann and K. Ono, Dyson s ranks and Maass forms, Ann. of Math. 171 (010), [13] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 15 (004), [14] J. Bryson, K. Ono, S. Pitman, and R. C. Rhoades, Unimodal sequences and quantum and mock modular forms, Proc. Natl. Acad. Sci. USA, 109, No. 40 (01), pages [15] Y.-S. Choi, The asic ilateral hypergeometric series and the mock theta functions, Ramanujan J. 4 (011),

12 1 AMANDA FOLSOM, KEN ONO, AND ROBERT C. RHOADES [16] L. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. Amer. Math. Soc. 7 (195), [17] F. Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), [18] A. Erdelyi, W. Magnus, F. Oerhettinger, and F. Tricomi, Higher transcendental functions, Vol. III, ased on notes left y Harry Bateman, reprint of the 1955 original, Roert E. Krieger Pulishing Co., Inc., Melourne, Fla., [19] N. J., Fine, Basic hypergeometric series and applications, Math. Surveys and Monographs, no. 7, Amer. Math. Soc., Providence, [0] A. Folsom K. Ono, and R. C. Rhoades, Mock theta functions and quantum modular forms, Forum of Mathematics, Pi, 1 (013), e. [1] F. Garvan, Congruences for Andrews smallest parts partition function and new congruences for Dyson s rank, Int. J. Numer Theory 6 (010), 1 9. [] B. Gordon and R. J. McIntosh, A survey of the classical mock theta functions, Partitions, q-series, and modular forms, Dev. Math. 3, Springer, New York, 01, [3] D. S. Kuert and S. Lang, Modular units, Springer-Verlag, Berlin, [4] K. Mahlurg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA 105 (005), [5] K. Ono, Unearthing the visions of a master: harmonic Maass forms and numer theory, Proc. 008 Harvard-MIT Current Developments in Mathematics Conf., (009), Somerville, Ma., [6] R. C. Rhoades, Asymptotics for the numer of strongly unimodal sequences, Int. Math. Res. Notices, to appear. [7] N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, [8] G. N. Watson, The final prolem: An account of the mock theta functions, J. London Math. Soc. () (1936), [9] D. Zagier, Ramanujan s mock theta functions and their applications [d aprés Zwegers and Bringmann-Ono], Sém. Bouraki (007/008), Astérisque, No. 36, Exp. No. 986, vii-viii, (010), [30] W. Zudilin, On three theorems of Folsom, Ono, and Rhoades, Proc. Amer. Math. Soc., accepted for pulication. [31] S. Zwegers, Mock ϑ-functions and real analytic modular forms, q-series with applications to cominatorics, numer theory, and physics (Ed. B. C. Berndt and K. Ono), Contemp. Math. 91, Amer. Math. Soc., (001), [3] S. Zwegers, Mock theta functions, Ph.D. Thesis (Advisor: D. Zagier), Universiteit Utrecht, (00). [33] S. Zwegers, Multivariale Appell functions, preprint. Department of Mathematics, Yale University, New Haven, CT address: amanda.folsom@yale.edu Department of Mathematics, Emory University, Atlanta, GA address: ono@mathcs.emory.edu Department of Mathematics, Stanford University, Stanford, CA address: rhoades@math.stanford.edu