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Dyson [10] defined the rank of a partition to be its largest part minus its number of parts, and the partition rank function is defined by N m, n : # partitions of n with rank equal to m . p q “ t u For example, N 7, 2 2, because precisely 2 of the 15 partitions of n 7 have rank equal to 2; these arep 2 ´ 2q“2 1, and 3 1 1 1 1. “ Partition´ rank functions` ` ` have a rich` history` in` the` areas of combinatorics, q-hypergeometric series, number theory and modular forms. As one particularly notable example, Dyson con- jectured that the rank could be used to combinatorially explain Ramanujan’s famous parti- tion congruences modulo 5 and 7; this conjecture was later proved by Atkin and Swinnerton- Dyer [2]. It is well-known that the associated two variable generating function for N m, n may be expressed as a q-hypergeometric series p q n2 8 8 8 q (1.3) N m, n wmqn : R w; q , wq q w 1q q 1 m n 0 p q “ n 0 ; n ´ ; n “ p q ÿ“´8 ÿ“ ÿ“ p q p q noting here that N m, 0 δm0, where δij is the Kronecker delta function. Specializations inp the q“w-variable of the rank generating function have been of particular interest in the area of modular forms. For example, when w 1, we have that “ 8 1 n 24 1 (1.4) R1 1; q 1 p n q q η´ τ p q“ ` n 1 p q “ p q ÿ“ thus recovering (1.2), which shows that the generating function for p n is (essentially1) the 1 p q reciprocal of a weight 2 modular form. If instead we let w 1, then “´ n2 8 q (1.5) R 1; q : f q . 1 q q 2 p´ q“ n 0 ; n “ p q ÿ“ p´ q The function f q is not a modular form, but one of Ramanujan’s original third order mock theta functions.p q Mock theta functions, and more generally mock modular forms and harmonic Maass forms have been major areas of study. In particular, understanding how Ramanujan’s mock theta functions fit into the theory of modular forms was a question that persisted from Ramanu- jan’s death in 1920 until the groundbreaking 2002 thesis of Zwegers [20]: we now know that

1Here and throughout, as is standard in this subject for simplicity’s sake, we may slightly abuse termi- nology and refer to a function as a modular form or other modular object when in reality it must first be multiplied by a suitable power of q to transform appropriately. 2 Ramanujan’s mock theta functions, a finite list of curious q-hypergeometric functions in- cluding f q , exhibit suitable modular transformation properties after they are completed by the additionp q of certain nonholomorphic functions. In particular, Ramanujan’s mock theta functions are examples of mock modular forms, the holomorphic parts of harmonic Maass forms. Briefly speaking, harmonic Maass forms, originally defined by Bruiner and Funke [8], are nonholomorphic generalizations of ordinary modular forms that in addition to satisfying appropriate modular transformations, must be eigenfunctions of a certain weight k-Laplacian operator, and satisfy suitable growth conditions at cusps (see [4, 8, 16, 18] for more). Given that specializing R1 at w 1 yields two different modular objects, namely an ordinary modular form and a mock“ modular ˘ form as seen in (1.4) and (1.5), it is natural to ask about the modular properties of R1 at other values of w. Bringmann and Ono answered this question in [6], and used the theory of harmonic Maass forms to prove that upon specialization of the parameter w to complex roots of unity not equal to 1, the rank generating function R1 is also a . (See also [18] for related work.) Theorem ([6] Theorem 1.1). If 0 a c, then ă ă 1 πa 2 i a ℓc i sin ℓc 8 Θ ; ℓcρ 24 a ℓc c c q´ R1 ζc ; q dρ p q` ?3 τ i τ ρ ` ˘ ż´ ´` p ` ˘q 1 a is a harmonic Maass form of weight 2 on Γc.

2πia a c a Here, ζc : e is a c-th root of unity, Θ c ; ℓcτ is a certain weight 3 2 cusp form, ℓc : lcm 2c2, 24“, and Γ is a particular subgroup of SL Z . { “ c ` ˘2 Inp this paper,q as well as in prior work of two ofp theq authors [12], we study the related problem of understanding the modular properties of certain combinatorial q-hypergeometric series arising from objects called n-marked Durfee symbols, originally defined by Andrews in his notable work [1]. To understand n-marked Durfee symbols, we first describe Durfee symbols. For each partition, the Durfee symbol catalogs the size of its Durfee square, as well as the length of the columns to the right as well as the length of the rows beneath the Durfee square. For example, below we have the partitions of 4, followed by their Ferrers diagrams with any element belonging to their Durfee squares marked by a square ‚ , followed by their Durfee symbols. p q 4 3 1 2 2 2 1 1 1 1 1 1 ` ` ` ` ` `‚ ` ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚ ‚ ‚ ‚ 1 1 1 1 1 1 ‚ 1 2 1 1 1 1 1 ˆ ˙1 ˆ ˙1 ˆ ˙1 ˆ ˙1 ` ˘ Andrews defined the rank of a Durfee symbol to be the length of the partition in the top row, minus the length of the partition in the bottom row. Notice that this gives Dyson’s original rank of the associated partition. Andrews refined this idea by defining n-marked Durfee symbols, which use n copies of the integers. For example, the following is a 3-marked 3 Durfee symbol of 55, where αj, βj indicate the partitions in their respective columns. 3 2 1 43 43 32 32 22 21 α α α : 3 2 1 53 32 22 21 “ β β β ˆ ˙5 ˆ ˙5 Each n-marked Durfee symbol has n ranks, one defined for each column. Let len π denote the length of a partition π. Then the nth rank is defined to be len αn len βn p, andq each jth rank for 1 j n is defined by len αj len βj 1. Thus thep aboveq´ examplep q has 3rd rank equal to 1,ď 2ndă rank equal to 0, andp 1stq´ rankp equalq´ to 1. ´ Let Dn m1, m2,...,mn; r denote the number of n-marked Durfee symbols arising from p q partitions of r with ith rank equal to mi. In [1], Andrews showed that the n 1 - variable rank generating function for Durfee symbols may be expressed in terms ofp certain` q q-hypergeometric series, analogous to (1.3). To describe this, for n 2, define ě (1.6)

Rnpx; qq :“ 2 q m1 m2 mn m1 mn´1 m1 mn´2 m1 p ` `¨¨¨` q `p `¨¨¨` q`p `¨¨¨` q`¨¨¨` , q qm1 qm1`¨¨¨`mn´1 x q q q x qm1 q q x qm1 mn´1 q q m1 0p 1 ; qm1 x1 ; p 2 ; qm2 1 x2 ; ¨¨¨p n `¨¨¨` ; qmn 1 x ; ą m1 ` m2 1 ` n m 1 m2,...,mÿ n 0 n ě ´ ¯ ´ ¯ ` ´ ¯ ` where x xn : x1, x2,...,xn . For n 1, the function R1 x; q is defined as the q- hypergeometric“ series“ p in (1.3). Inq what follows,“ for ease of notatiop qn, we may also write R1 x; q to denote R1 x; q , with the understanding that x : x. In [1], Andrews established thep followingq result, generalizingp q (1.3). “ Theorem ([1] Theorem 10). For n 1 we have that ě 8 8 m1 m2 mn r (1.7) Dn m1, m2,...,mn; r x x x q Rn x; q . p q 1 2 ¨ ¨ ¨ n “ p q m1,m2,...,mn r 0 ÿ “´8 ÿ“ When n 1, one recovers Dyson’s rank, that is, D1 m1; r N m1,r , so that (1.7) reduces to (1.3)“ in this case. The mock modularity of thep associatedq “ twop variableq generating function R1 x1; q was established in [6] as described in the Theorem above. When n 2, the p q “ modular properties of R2 1, 1; q were originally studied by Bringmann in [3], who showed that p q m 1 3m m 1 2 1 1 ´ q p ` q{ R2 1, 1; q : p´ q q q qm 2 p q “ ; m 0 1 p q8 ÿ‰ p ´ q is a quasimock theta function. In [5], Bringmann, Garvan, and Mahlburg showed more generally that Rn 1, 1,..., 1; q is a quasimock theta function for n 2. (See [3, 5] for precise details of thesep statements.)q ě In [12], two of the authors established the automorphic properties of Rn x; q , for more p q arbitrary parameters x x1, x2,...,xn , thereby treating families of n-marked Durfee rank “p q functions with additional singularities beyond those of Rn 1, 1,..., 1; q . We point out that the techniques of Andrews [1] and Bringmann [3] were notp directly applicableq in this setting due to the presence of such additional singularities. These singular combinatorial families are essentially mixed mock and quasimock modular forms. To precisely state a result from [12] along these lines, we first introduce some notation, which we also use for the remainder 4 of this paper. Namely, we consider functions evaluated at certain length n vectors ζn of roots of unity defined as follows (as in [12]). In what follows, we let n be a fixed integer satisfying n 2. Suppose for 1 j n, αr αs ě ď ď αj Z and βj N, where βj ∤ αj, βj ∤ 2αj, and that Z if 1 r s n. Let P P βr ˘ βs R ď ‰ ď α1 α2 αn n αn : , ,..., Q “ β1 β2 βn P ´ α1 α2 αn¯ n (1.8) ζn : ζ ,ζ ,...,ζ C . “ β1 β2 βn P Remark 1.1. We point out that the` dependence of˘ the vector ζn on n is reflected only in the length of the vector, and not (necessarily) in the roots of unity that comprise its components. In particular, the vector components may be chosen to be m-th roots of unity for different values of m.

Remark 1.2. The conditions stated above for ζn, as given in [12], do not require gcd αj, βj 1. Instead, they merely require that αj 1 Z. Without loss of generality, we willp assumeq“ βj R 2 here that gcd αj, βj 1. Then, requiring that βj ∤ 2αj is the same as saying βj 2. p q“ ‰ In [12], the authors proved that (under the hypotheses for ζn given above) the completed nonholomorphic function 1 24 (1.9) A ζn; q q´ Rn ζn; q A´ ζn; q p q“ p q` p q transforms like a modular form. Here the nonholomorphic part A´ is defined by p α n R j 3´ β , 2τ; τ 1 3αj αj j ´ (1.10) A´ ζn; q : ζ´ ζ´ , η τ 2βj 2βj ´ ¯ p q “ j 1p ´ q Πj: αn p q ÿ“ p q R where 3 is defined in (2.4), and the constant Πj: is defined in [12, (4.2), with n j and k n]. Precisely, we have the following special case of a theorem established by twoÞÑ of the authorsÞÑ in [12].

Theorem ([12] Theorem 1.1). If n 2 is an integer, then A ζn; q is a nonholomorphic ě 1 p q modular form of weight 1 2 on Γn with character χ´ . { γ p Here, the subgroup Γn SL2 Z under which A ζn; q transforms is defined by Ď p q p q n 2 (1.11) Γn : Γ0 2βj p Γ1 2βj , “ j 1 X p q č“ ` ˘ and the Nebentypus character χγ is given in Lemma 2.1. 1.2. Quantum modular forms. In this paper, we study the quantum modular properties of the n 1 -variable rank generating function for n-marked Durfee symbols Rn x; q . Looselyp speaking,` q a quantum modular form is similar to a mock modular form in thatp q it exhibits a modular-like transformation with respect to the action of a suitable subgroup of SL2 Z ; however, the domain of a quantum modular form is not the upper half-plan H, but ratherp q the set of rationals Q or an appropriate subset. The formal definition of a quantum modular form was originally introduced by Zagier in [19] and has been slightly modified to allow for half- weights, subgroups of SL2 Z , etc. (see [4]). 5 p q 1 Definition 1.3. A weight k 2 Z quantum modular form is a complex-valued function f on a b P 1 Q, such that for all γ SL2 Z , the functions hγ : Q γ´ i C defined by “p c d q P p q z p 8q Ñ 1 k ax b hγ x : f x ε´ γ cx d ´ f ` p q “ p q´ p qp ` q cx d ˆ ` ˙ satisfy a “suitable” property of continuity or analyticity in a subset of R. Remark 1.4. The complex numbers ε γ , which satisfy ε γ 1, are such as those ap- pearing in the theory of half-integral weightp q modular forms.| p q| “ Remark 1.5. We may modify Definition 1.3 appropriately to allow transformations on subgroups of SL2 Z . We may also restrict the domains of the functions hγ to be suitable subsets of Q. p q The subject of quantum modular forms has been widely studied since the time of origin of the above definition. For example, quantum modular forms have been shown to be related to the diverse areas of Maass forms, Eichler , partial theta functions, colored Jones polynomials, meromorphic Jacobi forms, and vertex algebras, among other things (see [4] and references therein). In particular, the notion of a quantum modular form is now known to have direct connection to Ramanujan’s original definition of a mock theta function. Namely, in his last letter to Hardy, Ramanujan examined the asymptotic difference between mock theta and modular theta functions as q tends towards roots of unity ζ radially within the unit disk (equivalently, as τ approaches rational numbers vertically in the upper half plane, with q e2πiτ , τ H), and we now know that these radial limit differences are equal to special values“ of quantumP modular forms at rational numbers (see [4, 7, 13]).

1.3. Results. On one hand, exploring the quantum modular properties of the rank gen- erating function for n-marked Durfee symbols Rn in (1.7) is a natural problem given that two of the authors have established automorphic properties of this function on H (see [12, Theorem 1.1] above), that Q is a natural boundary to H, and that there has been much progress made in understanding the relationship between quantum modular forms and mock modular forms recently [4]. Moreover, given that Rn is a vast generalization of the two vari- able rank generating function in (1.3) - both a combinatorial q-hypergeometric series and a mock modular form - understanding its automorphic properties in general is of interest. On the other hand, there is no reason to a priori expect Rn to converge on Q, let alone exhibit quantum modular properties there. Nevertheless, we establish quantum modular properties for the rank generating function for n-marked Durfee symbols Rn in this paper. For the remainder of this paper, we use the notation

Vn τ : V ζn; q , p q “ p q where V may refer to any one of the functions A, A´, Rn, etc. Moreover, we will write

1 24 (1.12) An τ q´ Rn ζn; q p q“ p p q for the holomorphic part of A; from [12, Theorem 1.1] above, we have that this function is 1 a mock modular form of weight 1 2 with character χ´ (see Lemma 2.1) for the group Γn { γ defined in (1.11). Here, we willp show that An is also a quantum modular form, under the 6 action of a subgroup Γζn Γn defined in (1.15), with quantum set Ď h Z,k N, gcd h, k 1, βj ∤ k 1 j n, h P P p q“ @ ď ď (1.13) Qζn : Q α α 1 , “ $k P ˇ j k j k 1 j n , ˇ β ´ β ą 6 @ ď ď & ˇ ˇ j „ j ˇ . ˇ ˇ ˇ where x is the closest% integerˇ toˇ x. ˇ - r s ˇ ˇ ˇ Remark 1.6. For x 1 Z, different sources define x to mean either x 1 or x 1 . The P 2 ` r s ´ 2 ` 2 definition of Qζn involving is well-defined for either of these conventions in the case of x 1 Z, as x x 1 . r¨s P 2 ` | ´r s| “ 2 To define the exact subgroup under which An transforms as a quantum automorphic object, we let

2 6 lcm β1,...,βn if 3 ∤ βj for all 1 j n, (1.14) ℓ ℓ ζn : r p qs2 ď ď “ p q “ 2 lcm β1,...,βn if 3 βj for some 1 j n, # r p qs | ď ď 1 0 1 1 and let Sℓ : ℓ 1 , T : 0 1 . We then define the group generated by these two matrices as “ p q “ p q

(1.15) Γζn : Sℓ, T . “x y x We now state our first main result, which proves that An x , and hence e Rn ζn; e x p q p´ 24 q p p qq is a quantum modular form on Qζn with respect to Γζn . Here and throughout we let e x : e2πix. p q “

a b Theorem 1.7. Let n 2. For all γ Γζn , and x Qζn , ě “p c d q P P 1 2 Hn,γ x : An x χγ cx d ´ An γx p q “ p q´ p ` q p q c is defined, and extends to an analytic function in x on R ´ . In particular, for the matrix ´t d u Sℓ,

3 αj 3αj g 1 1 αj 1 3ρ n i , ?3 ζ2β ζ2β 8 3 2 β 2 p j ´ j q 1 ˘ ` ´ j ` p q Hn,Sℓ x ζ6˘ dρ p q“ 2 : » 1 fi j 1 Πj αn ℓ i ρ x ÿ“ p q ÿ˘ ż ´ p ` q – n a3αj αj fl ζ2´β ζ2´β 1 α j j 2 ℓ j p ´ q ℓx 1 ´ ζ24´ E1 ,ℓ; x , ` : α p ` q βj j 1 Πj n ˆ ˙ ÿ“ p q where the weight 3 2 theta functions ga,b are defined in (2.5), and E1 is defined in (4.3). {

Remark 1.8. As mentioned above, the constants Πj: are defined in [12, (4.2)]. With the exception of replacing n j and k n, we have preserved the notation for these constants from [12]. ÞÑ ÞÑ Remark 1.9. Our results apply to any n 2, as the quantum modular properties in the case n 1 readily follow from existing results.ě Namely, proceeding as in the proof of Theorem 3.2,“ one may determine a suitable quantum set for the normalized rank generating function 7 in [6, Theorem 1.1]. Using [6, Theorem 1.1], a short calculation shows that the error to modularity (with respect to the nontrivial generator of Γc) is a multiple of i a 8 Θ ; ℓcρ p c q dρ x i τ ρ ż ´ p ` q for some x Q. When viewed as a functiona of τ in a subset of R, this integral is analytic (e.g., see [15,P 19]). One could also establish the quantum properties of a non-normalized version of R1 by rewriting it in terms of the Appell-Lerch sum A3, and proceeding as in the proof of Theorem

1.7. In this case, R1 ζ1; q (where ζ1 e α1 β1 ) converges on the quantum set Qζ1 , where this set is defined byp lettingq n 1 in“ (1.13).p { q The interested reader may also“ wish to consult [9] for general results on quantum properties associated to mock modular forms. Remark 1.10. In a forthcoming joint work [11], we extend Theorem 1.7 to hold for the more general vectors of roots of unity considered in [12], i.e., those with repeated entries. Allowing repeated roots of unity introduces additional singularities, and the modular completion of Rn is significantly more complicated. This precludes us from proving the more general case in the same way as the restricted case we address here.

2. Preliminaries 2.1. Modular, mock modular and Jacobi forms. A special ordinary modular form we require is Dedekind’s η-function, defined in (1.1). This function is well known to satisfy the following transformation law [17].

a b Lemma 2.1. For γ c d SL2 Z , we have that “p q P p q 1 2 η γτ χγ cτ d η τ , p q“ p ` q p q where b e 24 , if c 0,d 1, χγ : 1 a d “ “ “ ? i ω´ e ` , if c 0, # `´ ˘ d,c 24c ą 1 with ωd,c : e s d,c . Here the Dedekind` sum˘s m, t is given for integers m and t by “ p 2 p qq p q j mj s m, t : , t t p q “ j mod t ÿ ˆˆ ˙˙ˆˆ ˙˙ where x : x txu 1 2 if x R Z, and x : 0 if x Z. pp qq “ ´ ´ { P z pp qq “ P The following gives a useful expression for χγ (see [14, Ch. 4, Thm. 2]): d 1 2 c e 24 a d c bd c 1 3c if c 1 mod 2 , (2.1) χγ pp ` q ´ p ´ q´ q ” p q “ |c | e 1 a d c bd c2 1 3d 3 3cd if d 1 mod 2 , " ` d ˘ `24 pp ` q ´ p ´ q` ´˘ ´ q ” p q α where β is the generalized` ˘ ` Legendre symbol. ˘ We require two additional functions, namely the Jacobi theta function ϑ u; τ , an ordinary ` ˘ Jacobi form, and a nonholomorphic modular-like function R u; τ used byp Zwegersq in [20]. In what follows, we will also need certain transformation propertiesp q of these functions. 8 Proposition 2.2. For u C and τ H, define P P 2 1 πiν τ 2πiν u 2 (2.2) ϑ u; τ : e ` p ` q. p q “ 1 ν Z Pÿ2 ` Then ϑ satisfies (1) ϑ u 1; τ ϑ u; τ , p ` q“´ p q πiτ 2πiu (2) ϑ u τ; τ e´ ´ ϑ u; τ , p ` q“´ p q

πiτ 4 πiu 8 2πimτ 2πiu 2πiτ m 1 2πiu 2πimτ (3) ϑ u; τ ie { e´ 1 e 1 e e p ´ q 1 e´ e . p q“´ m 1p ´ qp ´ qp ´ q ź“ The nonholomorphic function R u; τ is defined in [20] by p q Im u ν 1 πiν2τ 2πiνu R u; τ : sgn ν E ν p q 2 Im τ 1 ´ 2 e´ ´ , p q “ 1 p q´ ` Im τ p q p´ q ν 2 Z " ˆˆ ˙ ˙* Pÿ` p q a where z πt2 E z : 2 e´ dt. p q “ ż0 The function R transforms like a (nonholomorphic) mock Jacobi form as follows. Proposition 2.3 (Propositions 1.9 and 1.10, [20]). The function R satsifies the following transformation properties: (1) R u 1; τ R u; τ , p ` q“´ p q 2πiu πiτ πiu πiτ 4 (2) R u; τ e´ ´ R u τ; τ 2e´ ´ { , p q` p ` q“ (3) R u; τ R u; τ , p q“ p´ q πi (4) R u; τ 1 e´ 4 R u; τ , p ` q“ p q 1 πiu2 τ u 1 (5) ? iτ e { R τ ; τ R u; τ h u; τ , where the Mordell integral is defined by ´ ´ ` p q“ p q 2 ` ˘ eπiτt 2πut (2.3) h u; τ : ´ dt. p q “ cosh πt żR Using the functions ϑ and R, Zwegers defined the nonholomorphic function 2 i 2πiju (2.4) R3 u, v; τ : e ϑ v jτ 1;3τ R 3u v jτ 1;3τ p q “2 j 0 p ` ` q p ´ ´ ´ q ÿ“ i 2 e2πijuϑ v jτ;3τ R 3u v jτ;3τ , “2 j 0 p ` q p ´ ´ q ÿ“ where the equality of the two expressions in (2.4) is justified by Proposition 2.2 and Propo- sition 2.3. This function is used to complete the level three Appell function (see [21] or 9 [4]) n 3n n 1 2 2πinv 3πiu 1 q p ` q{ e A3 u, v; τ : e p´ q , 1 e2πiuqn p q “ n Z ÿP ´ where u, v C, as P A3 u, v; τ : A3 u, v; τ R3 u, v; τ . p q “ p q` p q This completed function transforms like a (non-holmorphic) Jacobi form, and in particular satisfies the following ellipticp transformation.

Theorem 2.4 ([21, Theorem 2.2]). For n1, n2, m1, m2 Z, the completed level 3 Appell P function A3 satisfies 2 n1 m1 2πi u 3n1 n2 vn1 3n1 2 n1n2 A3 u n1τ m1, v n2τ m2; τ 1 ` e p p ´ q´ qq { ´ A3 u, v; τ . p `p ` ` ` q“p´ q p q The following relationship between the Appell series A3 and the combinatorial series Rn is provedp in [12]. p

Proposition ([12, Proposition 4.2]). Under the hypotheses given above on ζn, we have that α n j A3 β , 2τ; τ 1 3αj αj j ´ Rn ζn; q ζ´ ζ´ . q 2βj 2βj ´ ¯ p q“ j 1 ´ Πj: αn p q8 ÿ“ ´ ¯ p q We also note that α n j A3 β , 2τ; τ 1 3αj αj j ´ An τ ζ´ ζ´ . η τ 2βj 2βj ´ ¯ p q“ j 1p ´ q Πj: αn p q ÿ“ p p q p In addition to working with the Appell sum A3, we also make use of additional properties of the functions h and R. In particular, Zwegers also showed how under certain hypotheses, these functions can be written in terms of integralsp involving the weight 3 2 modular forms { ga,b τ , defined for a, b R and τ H by p q P P πiν2τ 2πiνb (2.5) ga,b τ : νe ` . p q “ ν a Z Pÿ` We will make use of the following results.

Proposition 2.5 ([20, Proposition 1.15 (1), (2), (4), (5)]). The function ga,b satisfies:

(1) ga 1,b τ ga,b τ , ` p q“ p q 2πia (2) ga,b 1 τ e ga,b τ , ` p q“ p q πia a 1 (3) ga,b τ 1 e´ p ` qga,a b 1 τ , p ` q“ ` ` 2 p q 1 2πiab 3 (4) ga,b ie iτ 2 gb, a τ . p´ τ q“ p´ q ´ p q Theorem 2.6 ([20, Theorem 1.16 (2)]). Let τ H. For a, b 1 , 1 , we have P P p´ 2 2 q i 1 1 2 8 ga ,b ρ a τ 1 ` 2 ` 2 p q h aτ b; τ e 2 a b 2 dρ. p ´ q“´ ´ p ` q 0 i ρ τ ´ 10 ¯ ż ´ p ` q a 3. The quantum set We call a subset S Q a quantum set for a function F with respect to the group G Ď Ď SL2 Z if both F x and F Mx exist (are non-singular) for all x S and M G. p q p q p q P P In this section, we will show that Qζn as defined in (1.13) is a quantum set for An with

respect to the group Γζn . Recall that Qζn is defined as

h Z,k N, gcd h, k 1, βj ∤ k 1 j n, h P P p q“ @ ď ď Qζn : Q α α 1 , “ $k P ˇ j k j k 1 j n , ˇ β ´ β ą 6 @ ď ď & ˇ ˇ j „ j ˇ . ˇ ˇ ˇ where x is the closest integerˇ ˇ to x (see Remarkˇ 1.6). r s % ˇ ˇ ˇ - 1 Moreover, recall that the “holomorphic part” we consider (see §1.3) is An τ q´ 24 Rn ζn; q . p q“ p q To show that Qζn is a quantum set for An τ , we must first show that the the multi-sum h h p q defining Rn ζn; ζk converges for k Qζn . In what follows, as in the definition of Qζn , we take h Z, pk N suchq that gcd h, kP 1. P P p q“ h We start by addressing the restriction that for Qζn , βj ∤ k for all 1 j n. k P ď ď h h Lemma 3.1. For k Q, all summands of Rn ζn; ζk are finite if and only if βj ∤ k for all 1 j n. P p q ď ď h h Proof. Examining the multi-sum Rn ζn; ζk , we see that all terms are a power of ζk divided p αqj ζ˘ ζhm m by a product of factors of the form 1 βj k for some integer 1. Therefore, to have ´ αj ě each summand be finite, it is enough to ensure that 1 ζ˘ ζhm 0 for all m 1 and for ´ βj k ‰ ě all 1 j n. For ease of notation in this proof, we will omit the subscripts for αj and βj. ď ďα hm If 1 ζ˘ ζ 0 for some m N, we have that ´ β k “ P α hm Z. ˘β ` k P α hm Let K lcm β,k ββ1 kk1. Then Z is the same as αβ1 hmk1 KZ. “ p q “ “ ˘ β ` k R ˘ ` R Since K kk1, if k1 ∤ αβ1, this is always true and we do not have a singularity. “ ββ1 However, since K ββ1 kk1, if k1 αβ1, then k αβ1. This implies that β αk and that β k since gcd α, β “1. “ | | | | p q“ Therefore, if β ∤ k, it is always the case that k1 ∤ αβ1, so for all m N, P α hm Z. ˘β ` k R 

h h Now that we have shown that all summands in Rn ζn; ζk are finite for k Qζn , we will show that the sum converges. p q P h h Theorem 3.2. For ζn as in (1.8), if k Qζn , then Rn ζn; ζk converges and can be evaluated as a finite sum. In particular, we haveP that: p q

n 1 (3.1) R ζ ; ζh n n k k k 1 p q“ j 1 1 1 xj 1 xj´ ´ ź“ ´ pp ´ qp ´ 11qq 2 h m1 m2 mn m1 mn´1 m1 mn´2 m1 ζk rp ` `¨¨¨` q `p `¨¨¨` q`p `¨¨¨` q`¨¨¨` s hm ˆ ζh ζ 1 0 m1 k h h k h hm1 h k h x1ζ ; ζ 1 ; ζ x2ζ ; ζ 2 1 ; ζ ă2 ď k k m x1 k k k m x2 k 0 m ÿ,...,mn k m1 ` ď ă p q p q m2 1 ´ ¯ ˆ ˙ ` 1 , hpm1`m2q hpm1`¨¨¨`mn´1q ˆ h m1 m2 h ζk h h m1 mn´1 h ζk h x3ζk p ` q; ζk m3 1 ; ζk xnζk p `¨¨¨` q; ζk mn 1 ; ζk ` x3 ` xn p q m¨¨¨p3 1 q mn 1 ˆ ˙ ` ˆ ˙ ` where ζn x1, x2,...,xn . “p q h h Proof of Theorem 3.2. We start by taking k Qζn , and write ζ ζk . For ease of notation, P 2πiα“j βj we will use xj to denote the j-th component in ζn, so xj e { . Further, for clarity of argument, we will carry out the proof in the case of n 2,“ with comments throughout about how the proof follows for n 2. We have that “ ą 2 m1 m2 m1 ζp ` q ` R2 x1, x2 ; ζ 1 m1 1 m1 pp q q“ x1ζ; ζ m1 x1´ ζ; ζ m1 x2ζ ; ζ m2 1 x2´ ζ ; ζ m2 1 m1 0 ` ` m2ą0 p q p q p q p q ÿě 1 (3.2) “ xk M1 x k M1 xk M2 x k M2 M1,M2 0 1 1 1 1´ 1 2 1 2´ ě p ´ q p ´ q p ´ q p ´ q ÿ 2 ζ s1 s2 s1 (3.3) p ` q ` , 1 s1 1 s1 ˆ x1ζ; ζ s1 x1´ ζ; ζ s1 x2ζ ; ζ s2 1 x2´ ζ ; ζ s2 1 0 s1 k ` ` 0ăÿs2ďk p q p q p q p q ď ă where we have let mj sj Mjk for 0 s1 k, 0 s2 k, and Mj N0, and have used the fact that “ ` ă ď ď ă P r k M r xζ ; ζ s Mk 1 x xζ ; ζ s, p q ` “p ´ q p q which holds for any M,r,s N0. (We note that for n 2, we proceed as above, additionally P ą taking 0 sj k 1 for j 2.) The second sum in (3.3) is a finite sum, as desired. For the first sumď in (3.2)ď we´ noticeą that we in fact have the product of two geometric series, each of the form Mj 1 . k k 1 x 1 x´ Mj 0 ˜ j j ¸ ÿě p ´ qp ´ q 2παj By definition, we have xj cos θj i sin θj where θj . Therefore, this sum converges “ ` “ βj if and only if

k k 1 1 x 1 x´ 2 2 cos kθj 1 cos kθj . | ´ j || ´ j |“ ´ p qą ðñ p qă 2 1 π For cos kθj 2 , it must be that kθj r 2πM where π r π, r 6 , and M Z. This isp equivalentq ă to saying “ ` ´ ă ď | | ą P α α 1 j k j k 1 j n, β ´ β ą 6 @ ď ď ˇ j „ j ˇ ˇ ˇ h as in the definition of Q n inˇ (1.13). Therefore,ˇ we see that for Q n , R x , x ; ζ ζ ˇ ˇ k ζ 2 1 2 converges to the claimed expression in (3.1). P pp q q 12 We note that by Abel’s theorem, having shown convergence of R2 x1, x2 ; ζ , we have pp q q that R2 x1, x2 ; q converges to R2 x1, x2 ; ζ as q ζ radially within the unit disc. pp q q pp q q Ñ As noted, the above argument extends to n 2. Letting mj sj Mjk with 0 s1 k ą “ ` ă ď and 0 sj k for j 2, rewriting as in (3.1), and then summing the resulting geometric ď ă ě series gives the desired exact formula for Rn ζn; ζ .  p q

To complete the argument that Qζn is a quantum set for Rn ζn; ζ with respect to Γζn , ph q 2πiγ k h it remains to be seen that Rn ζn; ξ converges, where ξ e p q for k Qζn and γ Γζn , defined in (1.15). For the easep of theq reader, we recall from“ (1.14) and (1.15)P that P 1 1 1 0 Γζn : , , “ 0 1 ℓ 1 Bˆ ˙ ˆ ˙F where 2 6 lcm β1,...,βk if j, 3 βj ℓ ℓβ : r p qs2 @ ­ | “ “ 2 lcm β1,...,βk if j, 3 βj. # r p qs D | The convergence of Rn ζn; ξ is a direct consequence of the following lemma. p q

Lemma 3.3. The set Qζn is closed under the action of Γζn .

Proof. Since Γζn is given as a set with two generators, it is enough to show that Qζn is closed under action of each of those generators. h 1 1 h h k Let k Qζn . Then 0 1 k `k . Note that gcd h k,k gcd h, k 1 and we already P p q “ p ` q“ p q“1 1 h know that k satisfies the conditions in the definition of Qζn . Therefore, 0 1 k Qζn . Under the action of 1 0 , we have p q P p ℓ 1 q 1 0 h h . ℓ 1 k “ hℓ k ˆ ˙ ` We first note that gcd h, hℓ k gcd h, k 1, and βj ∤ hℓ k as βj ℓ and βj ∤ k. It remains to check that p ` q “ p q “ p ` q | α α 1 j hℓ k j hℓ k 1 j n. β p ` q´ β p ` q ą 6 @ ď ď ˇ j „ j ˇ ˇ ˇ We have that ˇ ˇ ˇ ˇ α α α hℓ α α hℓ α j hℓ k j hℓ k j j k j j k β p ` q´ β p ` q “ β ` β ´ β ` β ˇ j „ j ˇ ˇ j j „ j j ˇ ˇ ˇ ˇα α 1 ˇ (3.4) ˇ ˇ ˇ j k j k , ˇ ˇ ˇ “ ˇ β ´ β ą 6 ˇ ˇ j „ j ˇ ˇ αj ℓ ˇ where we can simplify as in (3.4) since, by definitionˇ of ℓ, ˇ Z. Thus, Qζn is closed under ˇ βj ˇ P  the action of Γζn .

4. Proof of Theorem 1.7

We now prove Theorem 1.7. Our first goal is to establish that Hn,γ is analytic in x on c a b R ´d for all x Qζn and γ c d Γζn . As shown in Section 3, we have that An x ´ t u P “ p q P 13 p q and An γx are defined for all x Qζn and γ Γζn . Note that it suffices to consider only p q P P the generators Sℓ and T of Γζn , since 1 2 Hn,γγ1 τ Hn,γ1 τ χγ1 Cτ D ´ Hn,γ γ1τ p q“ p q` p ` q p q a b A B for γ and γ1 . “p c d q “p C D q First, consider γ T . Then by definition, χT ζ24, and so Hn,T x An x ζ24An x 1 . “ 2πix “ p q“ p q´ p ` q When we map x x 1, q e remains invariant. Then since the definition of Rn x in (1.6) can be expressedÞÑ ` as a“ series only involving integer powers of q, it is also invariant.p q Thus ´2πipx`1q 24 1 An x 1 e Rn x ζ´ An x , p ` q“ p q“ 24 p q and so Hn,T x 0. p q“ ℓ We now consider the case γ Sℓ. In this case using (2.1) we calculate that χSℓ ζ24´ . Thus, “ “ 1 ℓ 2 Hn,Sℓ x An x ζ24´ ℓx 1 ´ An Sℓx . p q“ p q´ p ` q p 1q ℓ 2 From the modularity of An we have that An x ζ24´ ℓx 1 ´ An Sℓx . Thus (1.9) and (1.12) give that p q “ p ` q p q 1 p p ℓ 2 p (4.1) Hn,S x A´ x ζ´ ℓx 1 ´ A´ Sℓx , ℓ p q“´ n p q` 24 p ` q n p q where An´ is defined in (1.10). Using the Jacobi triple product identity from Proposition 2.2 item (3), we can simplify the 2 1 theta functions to get that ϑ 2τ;3τ iq´ 3 η τ , ϑ τ;3τ iq´ 6 η τ , and ϑ 0;3τ 0. Thus, p´ q“ p q p´ q“ p q p q“ 1 α 1 2 α δ 3α j 3 j 2 j R3 , 2τ; τ q´ η τ e δ q R 2 δ τ;3τ . β ´ “´2 p q β β `p ´ q ˆ j ˙ δ 0 ˆ j ˙ ˆ j ˙ ÿ“ Using Proposition 2.3 item (2), we can rewrite

3αj 3αj 5 3αj 1 3αj R 2τ;3τ 2e q 8 e q 2 R τ;3τ , β ` “ 2β ´ β β ´ ˆ j ˙ ˆ j ˙ ˆ j ˙ ˆ j ˙ so that

1 αj δ 3αj e δ q 2 R 2 δ τ;3τ β β `p ´ q “ δ 0 ˆ j ˙ ˆ j ˙ ÿ“ 3αj 5 2αj 1 αj 3αj 2e q 8 e q 2 e R τ;3τ . 2βj ` βj ˘ ¯ βj βj ˘ ˆ ˙ ˆ ˙ ÿ˘ ˆ ˙ ˆ ˙ Thus we see that

n 3αj αj 1 ζ2´β ζ2´β 2α 1 α 3α j j j 6 j j (4.2) An´ τ p ´ qe q´ e R τ;3τ p q“´2 : α βj ˘ ¯ βj βj ˘ j 1 Πj k ˆ ˙ ˆ ˙ ˆ ˙ ÿ“ p q ÿ˘ n 3αj αj 1 ζ´ ζ´ 2βj 2βj 3αj q´ 24 p ´ qe . β ´ j 1 Πj: αk 2 j “ p q ˆ ˙ 14 ÿ Now to compute A´ Sℓτ we first define n p q 1 α 3α 6 Fα,β τ : q´ e R τ;3τ . p q “ ˘ ¯β β ˘ ÿ˘ ˆ ˙ ˆ ˙ Then by (4.1) and (4.2) we can write

n 3αj αj ζ´ ζ´ 1 1 2βj 2βj 2αj p ´ q ℓ 2 Hn,Sℓ τ e Fαj ,βj τ ζ24´ ℓτ 1 ´ Fαj ,βj Sℓτ p q“ 2 : α βj p q´ p ` q p q j 1 Πj k ˆ ˙ ÿ“ p q ” ı n 3αj αj ζ2´β ζ2´β 1 α j j 2 ℓ j p ´ q ℓτ 1 ´ ζ24´ E1 ,ℓ; τ , ` : α p ` q βj j 1 Πj k ˆ ˙ ÿ“ p q where

α 1 1 3 α S τ 3 α 2 ℓ 24 ℓ (4.3) E1 ,ℓ; τ : ℓτ 1 ζ q´ e e ´ e . β “p ` q 24 2 β ´ 24 2 β ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ 1 Thus in order to prove that Hn,Sℓ x is analytic on R ´ℓ it suffices to show that for each 1 j n, p q ´ t u ď ď 1 ℓ 2 Gα ,β τ : Fα ,β τ ζ´ ℓτ 1 ´ Fα ,β Sℓτ j j p q “ j j p q´ 24 p ` q j j p q 1 is analytic on R ´ . We establish this in Proposition 4.1 below. ´ t ℓ u Proposition 4.1. Fix 1 j n and set α, β : αj, βj . With notation and hypotheses as above, we have that ď ď p q “ p q

i g 1 1 1 α 3ρ 8 3 2 , 2 3 1 ˘ ` ´ β Gα,β τ ?3 e p qdρ, p q“ ¯ ¯6 1 ℓ i ρ τ ÿ˘ ˆ ˙ ż ´ p ` q 1 which is analytic on R ´ . a ´ ℓ ( 3α 1 1 Proof. Fix 1 j n and set α, β : αj, βj . Define m : Z, r , so that ď ď p q “ p q “ β P P p´ 2 2 q 3α m r. We note that r 1 since β 2. Using Proposition” ı 2.3 (1), we have that β “ ` ‰˘ 2 ‰ 1 r m 6 m (4.4) Fα,β τ q´ e ¯ e ¯ 1 R τ r;3τ . p q“ ˘ 3 3 p´ q p˘ ` q ÿ˘ ´ ¯ ´ ¯ 1 1 r 1 Letting τℓ : ℓ we have Sℓτ ´ . Using Proposition 2.3 (5) with u τℓ and “ ´ τ ´ “ τℓ “ 3 ¯ 3 τ τℓ we see that ÞÑ 3 1 3 (4.5) R r ; ´ ¯ τ τ “ ˆ ℓ ℓ ˙ 2 iτℓ 1 rτℓ 1 3 rτℓ 1 τℓ rτℓ 1 τℓ ´ e h ; R ; . 3 ¨ ´2 3 ¯ 3 τ 3 ¯ 3 3 ´ 3 ¯ 3 3 c ˜ ˆ ˙ ˆ ℓ ˙¸ „ ˆ ˙ ˆ ˙ 15 rτℓ 1 τℓ ℓ r 1 1 Using Proposition 2.3 parts (1) and (4) we see that R 3 3 ; 3 ζ24R ´3τ 3 ; ´3τ . Then using Proposition 2.3 (5) with u τ r and τ 3τ we¯ obtain“ that ¯ “¯ ´ ÞÑ ` ˘ ` ˘ 2 rτℓ 1 τℓ τ r R ; ζℓ i 3τ e ´p¯ ´ q h τ r;3τ R τ r;3τ , 3 ¯ 3 3 “ 24 ´ p q ¨ 6τ r p¯ ´ q´ p¯ ´ qs ˆ ˙ ˜ ¸ a which together with (4.4) and (4.5) gives

Fα,β Sℓτ p q“ 2 1 r m iτℓ 1 rτℓ 1 3 e e ¯ e ¯ 1 m ´ e 6τℓ ˘ 3 3 p´ q c 3 ¨ ˜´2 3 ¯ 3 τℓ ¸ ¨ ˆ ˙ ÿ˘ ´ ¯ ´ ¯ ˆ ˙ ˆ ˙ 2 rτℓ 1 τℓ τ r h ; ζℓ i 3τ e ´p¯ ´ q h τ r;3τ R τ r;3τ . 3 ¯ 3 3 ´ 24 ´ p q ¨ 6τ r p¯ ´ q´ p¯ ´ qs « ˆ ˙ ˜ ¸ ff a 2 By the definition of r and ℓ we have that r ℓ Z. Simplifying thus gives that 6 P 2 m m r iτℓ rτℓ 1 τℓ Fα,β Sℓτ 1 e ¯ e ´ h ; p q“ ˘p´ q 3 6τ c 3 3 ¯ 3 3 ÿ˘ ´ ¯ ˆ ˙ ˆ ˙ m m r 1 ℓ 1 1 e ¯ e ¯ q´ 6 ζ ℓτ 1 2 h τ r;3τ ´ ˘p´ q 3 3 24p ` q ¨ p¯ ´ q ÿ˘ ´ ¯ ´ ¯ m m r 1 ℓ 1 1 e ¯ e ¯ q´ 6 ζ ℓτ 1 2 R τ r;3τ , ` ˘p´ q 3 3 24p ` q ¨ p¯ ´ q ÿ˘ ´ ¯ ´ ¯ and so using Proposition 2.3 (3) and the fact that h u; τ h u; τ which comes directly from the definition of h in (2.3), we see that p q “ p´ q

1 m r 6 m Gα,β τ q´ 1 e ¯ e ¯ h τ r;3τ p q“ ˘p´ q 3 3 p˘ ` q ÿ˘ ´ ¯ ´ ¯ 2 m m r ℓ i rτℓ 1 τℓ 1 e ¯ e ζ24´ h ; . ´ ˘p´ q 3 6τ c3τ ¨ 3 ¯ 3 3 ÿ˘ ´ ¯ ˆ ˙ ˆ ˙ 1 We now use Theorem 2.6 to convert the h functions into integrals. Letting a ˘3 , b r, and τ 3τ gives that “ “´ ÞÑ i 1 1 1 1 r 8 g , r z dz 6 1 ˘ 3 ` 2 2 ´ h τ r;3τ q ζ6¯ e ˘ p q . p˘ ` q“´ 3 0 i z 3τ ´ ¯ ż ´ p ` q 1 τℓ Letting a r, b ˘ , and τ gives that a “ “ 3 ÞÑ 3 2 i g 1 1 1 z dz rτℓ 1 τℓ r r r 8 r 2 , 3 2 h ; e ´ e ¯ e ´ ` ˘ ` p q . 3 ¯ 3 3 “´ 6τ 3 2 τℓ ˆ ˙ ˆ ˙ 0 i z ´ ¯ ´ ¯ ż ´ ` 3 Thus b ` ˘

i 1 1 1 m 8 g , r z dz 1 m ˘ 3 ` 2 2 ´ Gα,β τ ζ6¯ 1 e ¯ p q p q“´ ˘ p´ q 3 0 i z 3τ ż ÿ˘ ´ ¯ 16´ p ` q a i 1 1 1 m r r i 8 gr , z dz ℓ m ` 2 ˘ 3 ` 2 ζ24´ 1 e ¯ e ¯ e ´ p q . ` ˘ p´ q 3 3 2 c3τ 0 i z τℓ ÿ˘ ´ ¯ ´ ¯ ´ ¯ ż ´ ` 3 1 b ` ˘ By a simple change of variables (let z ℓ ) we can write “ 3 ´ z ℓ 1 i g 1 1 1 z dz 0 g 1 1 1 dz 8 r 2 , 3 2 r 2 , 3 2 3 z (4.6) ` ˘ ` p q ? 3τ ` 3 ˘ ` ´ . 3 0 τℓ “´ ´ 2 i z ℓ z i `z 3τ˘ ż ´ ` 3 ż ´ p ` q b a Moreover, using Proposition` 2.5 we˘ can convert

ℓ 1 ℓ 1 (4.7) gr 1 , 1 1 ζ24 gr 1 , 1 1 ´ ` 2 ˘ 3 ` 2 3 ´ z “ ¨ ´ 2 ˘ 3 ` 2 z ˆ ˙ ˆ ˙ 1 1 r r 3 ℓ 2 ζ24e e ¯ e ˘ e z g 1 1 , 1 r z . “´ 8 6 3 2 ¨ ˘ 3 ` 2 2 ´ p q ˆ ˙ ˆ ˙ ´ ¯ ´ ¯ Thus by (4.6) and (4.7) we have that

i 1 1 1 m 8 g , r z dz 1 m ˘ 3 ` 2 2 ´ Gα,β τ ζ6¯ 1 e ¯ p q p q“´ ˘ p´ q 3 0 i z 3τ ÿ˘ ´ ¯ ż ´ p ` q 0 g 1 1 1 z dz m a 3 2 , 2 r 1 m ˘ ` ´ ζ6¯ 1 e ¯ p q ´ ˘ p´ q 3 3 ℓ i z 3τ ÿ˘ ´ ¯ ż ´ p ` q i g 1 1 1 z dz m 8 3 a2 , 2 r 1 m ˘ ` ´ (4.8) ζ6¯ 1 e ¯ p q . “´ ˘ p´ q 3 3 ℓ i z 3τ ÿ˘ ´ ¯ ż ´ p ` q To complete the proof, one can deduce from Propositiona 2.5 (2) that for m Z, P

ga,b τ e ma ga,b m τ . p q“ p q ´ p q Applying this to (4.8) with a direct calculation gives us

i g 1 1 1 α 3z 8 3 2 , 2 3 1 ˘ ` ´ β Gα,β τ ?3 e p qdz, p q“ ¯ ¯6 1 ℓ i z τ ÿ˘ ˆ ˙ ż ´ p ` q 1 a which is analytic on R ´ as desired.  ´ t ℓ u

5. Conclusion

We have proven that when we restrict to vectors ζn which contain distinct roots of unity, 1 the mock modular form q´ 24 Rn ζn; q is also a quantum modular form. To consider the p q more general case where we allow roots of unity in ζn to repeat, the situation is significantly more complicated. In this setting, as shown in [12], the nonholomorphic completion of 1 q´ 24 Rn ζn; q is not modular, but is instead a sum of two (nonholomorphic) modular forms of differentp weights.q We will address this more general case in a forthcoming paper [11]. 17 References [1] G. E. Andrews Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37–73. [2] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 66 (1954), 84–106. [3] K. Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math. J., 144 (2008), 195-233. [4] K. Bringmann, A. Folsom, K. Ono, and L. Rolen, Harmonic Maass forms and mock modular forms: theory and applications, American Mathematical Society Colloquium Publications. American Math- ematical Society, Providence, RI, to appear. [5] K. Bringmann, F. Garvan, and K. Mahlburg, Partition statistics and quasiweak Maass forms, Int. Math. Res. Notices, 1 (2009), 63–97. [6] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math., 171 (2010), 419-449. [7] K. Bringmann and L. Rolen, Radial limits of mock theta functions, Res. Math. Sci. 2 (2015), Art. 17, 18 pp. [8] J. Bruiner and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45-90. [9] D. Choi, S. Lim, and R.C. Rhoades, Mock modular forms and quantum modular forms, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2337–2349. [10] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [11] A. Folsom, M-J. Jang, S. Kimport, and H. Swisher, Quantum modular forms and singular combina- torial series with repeated roots of unity, in preparation. [12] A. Folsom and S. Kimport Mock modular forms and singular combinatorial series, Acta Arith. 159 (2013), 257–297. [13] A. Folsom, K. Ono, and R.C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27 pp. [14] M. I. Knopp, Modular functions in analytic number theory, Markham Publishing Co., Chicago, Ill., 1970. [15] R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1) (1999) 93–107. [16] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, (2008), 347-454, Int. Press, Somerville, MA, (2009). [17] H. Rademacher, Topics in analytic number theory, Die Grundlehren der math. Wiss., Band 169, Springer-Verlag, Berlin, (1973). [18] D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono- Bringmann), S´eminaire Bourbaki Vol. 2007/2008, Ast´erisque 326 (2009), Exp. No. 986, vii-viii, 143-164 (2010). [19] D. Zagier, Quantum modular forms, Quanta of maths, 659-675, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010. [20] S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002. [21] S. Zwegers, Multivariable Appell functions, Preprint.

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