ON A MODULARITY CONJECTURE OF ANDREWS, DIXIT, SCHULTZ, AND YEE FOR A VARIATION OF RAMANUJAN S ω(q)

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1 ON A MODULARITY CONJECTURE OF ANDREWS, DIXIT, SCHULTZ, AND YEE FOR A VARIATION OF RAMANUJAN S ωq KATHRIN BRINGMANN, CHRIS JENNINGS-SHAFFER, AND KARL MAHLBURG Abstract We analyze the mock modular behavior of P ωq, a partition function introduced by Andrews, Dixit, Schultz, and Yee This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is ωq We find that the modular completion of P ωq is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space Introduction and statement of results Andrews introduced the smallest parts partition function sptn in [], which enumerates the partitions of n weighted by the multiplicity of their smallest parts In the following years, a large volume of subsequent work has established its importance as a rich source for study, with many interesting examples of combinatorial and algebraic results [, 8], connections to modular forms [], as well as generalized families of spt-functions [9, ] A particularly striking feature of the smallest parts function is found in the automorphic properties of its generating function [5] Indeed, the generating function provides a natural example of a mock theta function, in the modern sense; ie, the holomorphic part of a harmonic Maass form [8, ] The classical notion of a mock theta function was based on Ramanujan s last letter to Hardy [], and these functions are now understood in the framework of real-analytic modular forms thanks to Zwegers seminal PhD thesis [] In particular, in Theorem of [], Andrews showed that sptnq n = n q; q n nq n q n + q; q where throughout the paper we use the standard q-factorial notation a n = a; q n := n j=0 n aq j n q nn+ + q n q n, for n N 0 { } The key feature of is that up to rational powers of q it expresses the generating function as a derivative of a linear combination of theta functions and Appell-Lerch sums, which are the main components of Zwegers work [] see Section below In a recent paper [], Andrews, Dixit, and Yee considered the question of constructing smallest parts partition functions directly from mock theta functions, and proved new results arising from The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union s Seventh Framework Programme FP/007-0 / ERC Grant agreement n 50 - AQSER

2 the classical mock theta functions ωq, νq, and φq Originally ω is defined as q nn+ ωq := q; q ; n+ in Theorem of [], they proved the new representation P ω q := qωq = n q n q n q n+ ; q n q n+ ; q Basic combinatorial arguments show that the series is the generating function for p ω n, which enumerates the partitions of n in which the odd parts are less than twice the smallest part Furthermore, this naturally gives a corresponding smallest parts function, which is a weighted count of each partition by the multiplicity of its smallest part: spt ω nq n := q n q n n n q n+ ; q n q n+ ; q We note that this spt-function first appeared in the literature as spt + o n in [9], where Patkowski studied two Bailey pair identities that are directly related to Andrews original smallest parts function It also appears as spt C n in [], where many examples of spt-functions were derived from Slater s extensive list of partition identities arising from Bailey pairs [] One of the main results arising from these various studies of spt ω n states that its generating function has a representation in terms of Appell-Lerch sums Specifically, Theorem in [9] and Lemma 6 in [] provide two different proofs of the identity spt ω nq n = n q ; q n nq n q n + q ; q n n + q n q nn+ q n As discussed further below, this is essentially a mock modular form of weight ; note that this also yields a Hecke-type indefinite theta series representation by expanding the denominators using geometric series In [], Andrews, Dixit, Schultz, and Yee considered an overpartition analog of p ω n and spt ω n In particular, they defined p ω n to be the number of overpartitions of n such that all odd parts are less than twice the smallest part, and in which the smallest part is always overlined The generating function in this case is P ω q := p ω nq n = q n+ ; q n q n+ ; q q n q n+ ; q n n n q n+ ; q q n, with corresponding smallest parts function spt ω nq n = q n+ ; q n q n+ ; q q n n n q n 5 q n+ ; q n q n+ ; q The generating function in 5 previously appeared in [6], where several additional families of spt-functions arising from Bailey pairs were studied systematically Indeed, one of the functions defined in [6] is spt G nq n := q n q n n n q n+ ; q, n qn+ ; q q n+ ; q and a short calculation shows that this is termwise equivalent to 5

3 Theorem 5 of [] shows that, just like and, the overpartition analog also has a representation in terms of Appell-Lerch sums and hence well-understood modularity properties, namely spt ω nq n = n q ; q q ; q n nq n q q n + ; q n q nn+ q ; q q n n However, there is a significant difference between P ω q and P ω q, as the modularity properties of P ω q come directly from the mock theta function ωq in Indeed, Andrews, Dixit, Schultz, and Yee commented on the difficulty of relating P ω q to modular forms in [], and raised the question of whether this is even possible as Problem The main result of the present paper provides an affirmative answer to the question of Andrews, Dixit, Schultz, and Yee on the modularity properties of P ω Theorem The function P ω q + ητ, where ητ := q ητ n qn q := e πiτ throughout is Dedekind s η-function, can be completed to P ω τ, which transforms like a weight modular form Remark We recall that a mock modular form f of weight k has a modular completion f with the property that ξ k f := iv k τ fτ 6 is a weakly holomorphic modular form of weight k the shadow of f In contrast, here ξ P ω is not a weakly holomorphic modular form However, L P ω τ := iv τ P ω τ lies in the space v S Γ 0, χ H Γ, χ v M! Γ 0, χ M! Γ 0, χ ητ For this reason we call the function P ω q+ a higher depth mock modular form, as L P ητ ω is in a relatively simple space Although we use the term somewhat informally in this article, we say that higher depth mock modular forms are automorphic functions that are characterized and inductively defined by the key property that their images under lowering operators essentially lie in lower depth spaces For example, the classical mock modular forms have depth, since the Maass lowering operator essentially yields a classical modular form Similarly, the main result of this paper shows that P ω q + ητ is a mock modular form of depth For the precise ητ details of the modular transformation properties of P ω, see Theorem The remainder of the paper is organized as follows In Section we give a review of harmonic Maass forms, classical and mock Jacobi forms, and state two q-series identities which we require below In Section we rewrite P ω as an indefinite theta function in the form of a triple sum In Section we use the results of Section to express P ω in terms of known mock modular forms, from which Theorem follows after a series of calculations Acknowledgments The authors thank the referees for their careful reading of the paper and many helpful comments Preliminaries Harmonic Maass forms In this section we recall basic facts of harmonic Maass forms, first introduced by Bruinier and Funke in [8] We begin with their definition

4 Definition For k Z, a weight k harmonic Maass form for a congruence subgroup Γ SL Z contained in Γ 0 if k + Z is any smooth function f : H C satisfying the following conditions: For all c d Γ, we have aτ + b f = cτ + d { cτ + d k fτ if k Z, c d ε k d cτ + d k fτ if k + Z Here c d is the generalized Legendre symbol in the sense of Shimura [0] and ε d := if d mod and ε d := i if d mod We have k f = 0, where k is the weight k hyperbolic Laplacian τ = u + iv, u, v R throughout k := v u + v There exists a polynomial P f τ C[q ] such that fτ P f τ = Oe εv + ikv u + i v as v for some ε > 0 Analogous conditions are required at all cusps Remark Similarly one can define harmonic Maass forms of weight k on Γ with multiplier Denote the space of such harmonic Maass forms of weight k on Γ by H k Γ Every f H k Γ has an expansion of the form fτ = f + τ + f τ with the holomorphic part having a q-expansion f + τ = n Q,n c + f nqn and the non-holomorphic part having an expansion of the form f τ = c f nγ k, πnvq n n Q,n>0 Here Γs, v is the incomplete gamma function defined, for v > 0, as the integral Γs, v := v t s e t dt The function f + is called a mock modular form The non-holomorphic part of a harmonic Mass form may be written as a non-holomorphic Eichler integral of a classical modular form, the socalled shadow of the mock modular form Indeed, if f is a harmonic Maass form of weight k, then the shadow of its holomorphic part can be recovered as ξ k f, where ξ k is definied in 6 This operator maps harmonic Maass forms of weight k to weakly holomorphic modular forms of dual weight k and conjugated multipliers It is related to the Maass lowering operator L := iv τ, via ξ k = v k L Harmonic Maass forms for which f = 0 are weakly holomorphic modular forms and we denote the corresponding space by M k! Γ Additionally we denote the space of cusp forms by S k Γ For forms with multiplier, we use the notation H k Γ, χ, M k! Γ, χ, and S kγ, χ We note that if fτ transforms like a modular form of weight k with multiplier χ c d for a subgroup Γ of SL Z, then f τv k transforms like a modular form of weight k with multiplier χ a b c d on Γ := { c d : a b c d Γ} Note that Γ = Γ for the congruence subgroups Γ 0 N

5 Jacobi forms and Zwegers µ-function In this section, we recall certain automorphic forms that we encounter in this paper We start with classical Jacobi forms, following Eichler and Zagier s seminal work [0] Definition A holomorphic Jacobi form of weight k and index m k, m N on a subgroup Γ SL Z of finite index is a holomorphic function ϕ : C H C which, for all c d Γ and λ, µ Z, satisfies ϕ z cτ+d ; aτ+b cτ+d = cτ + d k e πimcz cτ+d ϕz; τ, ϕz + λτ + µ; τ = e πimλτ+λz ϕz; τ, ϕz; τ has a Fourier expansion of the form n,r cn, rqn ζ r ζ := e πiz throughout with cn, r = 0 unless n r /m Jacobi forms with multipliers and of half integral weight, meromorphic Jacobi forms, and weak Jacobi forms are defined similarly with obvious modifications made Remark When specializing Jacobi forms to torsion points ie, z Q + Qτ one obtains classical modular forms with respect to some congruence subgroup of SL Z A special Jacobi form used in this paper is Jacobi s theta function, defined by ϑz = ϑz; τ :=, n +Z e πinτ+πinz+ where here and throughout, we may omit the dependence of various functions on the variable τ if the context is clear This function is odd and well known to satisfy the following transformation law [7] Lemma For λ, µ Z and c d SL Z, we have that ϑ ϑz + λτ + µ; τ = λ+µ q λ ζ λ ϑz; τ, z cτ + d ; aτ + b = ψ cτ + d πicz e cτ+d ϑz; τ, cτ + d c d where ψ is the multiplier of η, ie, η aτ+b cτ+d = ψ c d d c ψ = c d c πi d e e πi a+dc bdc c ac d +db c+ cτ + d ητ More explicitly, we have if c is odd, if c is even Zwegers mock Jacobi forms do not quite satisfy the elliptic and modular transformations given in and in the above definition of holomorphic Jacobi forms, but instead must be completed by adding a certain non-holomorphic function in order to satisfy suitable transformation laws To describe the simplest case, we define for z, z C \ Z + Zτ and τ H the function and its completion µz, z = µz, z ; τ := eπiz ϑz ; τ n e πinz q n +n e πiz q n µz, z = µz, z ; τ := µz, z ; τ + i Rz z ; τ Here the non-holomorphic part is given by τ = u + iv, z = x + iy the even function Rz = Rz; τ := sgnn E n + y v n q n ζ n v n +Z 5

6 with w Ew := e πt dt 0 For real arguments w, this can be expressed in terms of the incomplete gamma function, as Ew = sgnw Γ π, πw We also note that for certain values of z, τ Rz; τ is in fact weakly holomorphic We have the following transformation laws Lemma Zwegers [] We have the symmetry relations µz, z = µz, z, µ z, z = µz, z For c d SL Z and r, r, s, s Z, we have z µ cτ + d, z cτ + d ; aτ + b = ψ cτ + d e πicz z cτ+d µz cτ + d c d, z ; τ, µ z + r τ + s, z + r τ + s = r +s +r +s e πir r τ+πir r z z µz, z Moreover R satisfies the elliptic shifts For the function µ, we have Rz + = Rz, Rz + τ = ζq Rz + ζ q 8 µz, z = µz, z, µz + τ, z = e πiz z +πiτ µz, z ie πiz z + πiτ We note that when restricting z and z to torsion points, µ gives rise to a harmonic Maass form of weight see Corollary 85 in [6] The corresponding shadow is obtained from Raτ + b = i e πa v n+ n + ae πin τ πinaτ+b τ v n +Z Thus the shadow of µz, z ; τ, for z j specialized to torsion points, may be written in terms of the weight / unary theta function a, b Q ne πinb q n n a+z There is a similar identity for derivatives of R To be more precise, we compute [ ] τ z Rz = e πa v z=aτ+b n + πaa + n e πinτ πinaτ+b v v n +Z Combinatorial results In this section and the next, we will additionally use the notation a,, a k n := a ; q n a k ; q n for n N 0 { } In our initial identities for P ω, we need two standard q-series identities The first one is a finite version of the Jacobi triple product identity [, p 9 ex ] Lemma For n N 0, we have that ζ, ζ q n q n = n j= n 6 j ζ j q jj q n j q n+j

7 Secondly, we require Heine s transformation [5, p 59 eq III] Lemma Suppose q, ζ <, and c < b Then we have a, b n ζ n c b =, bζ c, q n c, ζ abζ c, b n c b n bζ, q n Representation in terms of indefinite theta functions In this section we prove a fundamental identity for P ω that provides a representation in terms of indefinite theta functions In fact, we find a new representation for a -parameter generalization of P ω Define P ω ζ; q := q ζ, ζ q q; q n ζ, ζ q n q; q n q n q n ; this also appeared in [6] It is not hard to see, comparing termwise to, that P ω q = P ω ; q We prove a double series representation for P ω ζ; q that ultimately helps us identify the modularity properties of P ω q Theorem We have P ω ζ; q = ζ, ζ q, q j j+ i n ζ j ζ q j q jj+ Proof: To prove this identity, we isolate the coefficient of ζ j in + iq j+n+ P ωζ; q := ζ, ζ q P ω ζ; q, +nj+ transform this coefficient with standard q-series techniques, and then sum over j For convenience we use the notation that [ζ j ]F ζ is the coefficient of ζ j in a series F ζ Noting that P ωζ; q is symmetric in ζ and ζ q, we have [ζ j ]P ωζ; q = q j [ζ j ]P ωζ; q, and so we only need to determine the coefficients of the non-negative powers of ζ By Lemma, we have that P ωζ; q = q q; q n q; q n qn n j= n j ζ j q jj q n j q n+j From the above we see that the calculation of the coefficients of ζ slightly differs depending on whether j or j = 0 For j, we have that [ ζ j ] P ωζ; q = j q jj q q; q = j q jj+ q q; q j q; q q j = j q jj+ q q; q j q; q n qn = j q q n=j n j q n+j q; q iq j+, iq j+ iq j+, iq j+ q; q q j q j+, q 7 jj q n qn q j+, q n q; q n+j qn+j q n q n+j + iq j+ i n q nj+ + iq j+ +n

8 = j q jj+ q i n q nj+, + iq j+ +n where in the penultimate equality we apply Lemma with ζ = q, a = iq j+, b = iq j+, and c = q j+ For j = 0, we instead have that [ ζ 0 ] P ωζ; q = q q; q n q; q n qn q n = i n q n q + iq q +n q; q = q q; q q; q n qn q q n q; q Here the final equality follows by observing that the sum is exactly the j = 0 case from Summing over j then gives that P ωζ; q = q i n q n + iq +n q q; q + q However, we note that P ω; q = 0 and so Thus q j i n q n + iq +n q q; q = q j ζ j + q j ζ j j q jj+ + q j j q jj+ Pωζ; q = ζ j + q j ζ j q j j q jj+ q j i n q nj+ + iq j+ +n i n q nj+ + iq j+ +n i n q nj+ + iq j+ +n Factoring gives the claim Theorem immediately leads to the following indefinite theta series representation of P ω q Corollary We have P ω q = q n,j,l 0 + n,j,l<0 Proof: Taking ζ in Theorem and using that gives that P ω q = q j q j j+n+l q jj+ +nj+lj+nl+n+l ζ j lim ζ ζ = j j q j j i n q jj+ j + iq j+n+ +nj+ Note that throughout the following calculations we frequently also include the j = 0 term if it gives a more convenient representation 8

9 Expanding the geometric series then gives q P ω q = j n,l 0 j j i n l q j q jj+ +nj+ +lj+n+ Changing n, l l, n, one sees that the contribution from n l + mod vanishes Splitting the sum according to the parity on n and l and then changing j, n, l j, n, l in the second sum gives q P ω q = j j+n+l q j q jj+ +nj+lj+nl+n+l This yields the claim Let j,n,l 0 + j,n,l = j,n,l 0 j j+n+l q j q jj +nj+lj n l+nl + j,n,l<0 j j+n+l q j q jj+ +nj+lj+n+l+nl Proof of Theorem i P ω τ := π ητ 6 F 0; τ + e where recall q = e πiτ and ζ = e πiz π πi ητ ητ ητ F 0; τ, Fz; τ := q 8 ζ ϑz; τ µ z, τ + ; τ In this section, we prove that this function is a non-holomorphic modular form for { Γ := SL c d Z : c, c d } b mod However, first we introduce a few auxiliary functions and determine their modular properties As we see below, these functions are related to P ω q and P ω τ Set G z, z, z ; τ := q 8 ζ ζ ζ k>0 l, + k 0 l,n<0 k q kk+ +kl+kn+ln ζ k ζ l ζ n, where throughout this section we write ζ j := e πiz j for the Jacobi parameters z j It is not hard to see that recall that we drop τ-dependencies whenever they are clear from the context Gz, z, z = i α β F z, z + α, z + β, with 0 α,β F z, z, z ; τ := q 8 ζ ζ ζ k>0 l, + k 0 l,n<0 9 k q kk+ +kl+kn+ln ζ k ζ l ζ n

10 It is shown in Theorem of [7] that F z, z, z = iϑz µ z, z µ z, z η ϑz + z ϑz ϑz µ z, z + z Below we plug in z = 0, and it is important to note that F has a removable singularity at this value, even though the individual terms in may have poles Indeed, this follows from the evaluations ϑ 0 = πη, lim z z µz, z = 0 Recalling, we define the modular completion of this function F by πiϑz F z, z, z ; τ := iϑz ; τ µz, z ; τ µz, z ; τ ητ ϑz + z ; τ ϑz ; τϑz ; τ µz, z + z ; τ With Lemmas and, we find the following Jacobi transformation laws hold Lemma For c d SL Z and n j, m j, α, β Z, we have z F cτ + d, z cτ + d, z cτ + d ; aτ + b cτ + d = ψ cτ + d πic e cτ+d z z z +z z +z z F z, z c d, z ; τ, F z + n τ + m, z + n τ + m, z + n τ + m F q n = n +m +n +m +n +m ζ n n n ζ n n ζ n n z + n τ + m, z + α + n τ + m, z + β + n τ + m + n + n n ζ = n +α+β+n +α+n +β+m +m +m ζ n n n q n + n + n n n n n F z, z + α, z + β The modular completion of G is now naturally defined as Ĝ z, z, z ; τ := i α β F 0 α,β n n n n F z, z, z, n n ζ n z, z + α, z + β ; τ We see below that modular completions introduce additional terms that can be either holomorphic or non-holomorphic, which need to be identified separately By applying 5 and then 6, we find modular transformations for Ĝ We also use 6 to obtain elliptic transformations for Ĝ The proofs are nothing more than lengthy, but straightforward calculations reducing exponents and roots of unity, and as such are omitted Lemma For c d Γ, we have z Ĝ cτ + d, z cτ + d, z cτ + d ; aτ + b cτ + d = ψ cτ + d πic cτ+d e c d z z z +z z +z z For m, n Z, m, n, m, n Z, and n n n mod, we have Ĝz + n τ + m, z + n τ + m, z + n τ + m 0 Ĝ z, z, z ; τ 8

11 n ζ = n + n + n +m + m + m ζ n n n n n ζ n q n + n 8 + n 8 n n n n Ĝz, z, z 9 Additionally, we require the following shifted G-functions Hz; τ := q ζg z, τ +, τ + ; τ, Ĥz; τ := q ζ Ĝ z, τ +, τ + ; τ Using 8 and 9, we determine the modular transformation of Ĥ in the following lemma Again, we omit the proof Lemma For c d Γ, z Ĥ cτ + d ; aτ + b = ψ exp πi cτ + d c d Before finally returning to P ω, we additionally define the functions ab + cd cτ + d exp πicz Ĥ z; τ cτ + d 0 f τ := v η τ, f τ := F 0; τ ητ, f τ := ητ ητ, f τ := v η τ 5 η τ η τ and the multipliers a χ c a χ c b d := ψ c d b := d ψ ψ c d, χ := c d, χ c d := c d {i c 8 d if 8 c, ie πic 6 if c, ψ ψ c d 5 c d ψ c d We note that v f τ is a weight cusp form, f τ is a weight weakly holomorphic modular form, and v f τ is a weight weakly holomorphic modular form, on Γ 0 with multipliers χ, χ, and χ, respectively Theorem The function P ω has the following properties i We have, for γ = c d Γ, aτ + b P ω = e πic 8 cτ + d Pω τ cτ + d ii The holomorphic part of Pω by which we mean the part that can be expressed as a q-series anqn is P ω q + ητ ητ iii The Maass lowering operator acts as L Pω = e πi f f e πi π π f f Furthermore, the function f is a harmonic Maass form of weight with multiplier χ on Γ and shadow e πi πητ

12 Proof: We first relate P ω to a Jacobi-derivative of the indefinite theta-function Gz, z, z defined in Using the representation from Corollary, we obtain P ω q = iq [ 8 ζ G q z, τ +, τ + ] [ ζ ζ G z, τ +, τ + ] ζ=q By the definition of Hz; τ, can be rewritten as [ i ] [ P ω q = ητ ζ Hz ζ ] ζ Hz ζ=q and the modular completion of, which we show below is P ω τ, becomes [ i ] [ ητ ζ Ĥz ζ ] ζ Ĥz ζ=q Below we determine its holomorphic part Indeed, we see below that if z is fixed, τ Ĥz; τ Hz; τ may be a mix of holomorphic and non-holomorphic functions We now regroup the terms from, first noting that, for a function f : C C, we have [ζ ] fz = [ ] πi z fz = [ ] [ ] fz + τ = fz + τ πi z ζ=q z=τ Thus the second term in can be expanded as [ ] q ζ Ĥz + τ = q Ĥτ z=0 [ ] q ζ Ĥz + τ We can now combine terms that satisfy the same modular transformations For this, we set Ĥ τ := Ĥ τ + Ĥτ, Ĥ τ := Ĥτ Ĥτ, where Ĥ τ := Ĥ0; τ, Ĥ τ := q Ĥτ; τ, [ ] [ ] Ĥ τ := Ĥz; τ, Ĥ τ := q ζ Ĥz + τ; τ We see below that Ĥ and Ĥ, respectively, satisfy modular transformations of weights and 5 In a similar fashion, we define the functions H τ := H0; τ + q Hτ; τ, [ ] [ ] H τ := Hz; τ + q ζ Hz + τ; τ We note that in this notation becomes P ω = i H η + H, and its modular completion is given by i η Ĥ + Ĥ Again, we must show that this modular completion is in fact P ω We next prove modularity of Ĥ and Ĥ Firstly, from Lemma, we have for c d Γ aτ + b Ĥ = ψ e πi ab+ cd cτ + d Ĥ τ, 5 cτ + d c d aτ + b Ĥ = ψ e πi ab+ cd 5 cτ + d Ĥ τ 6 cτ + d c d

13 The other two functions from also satisfy modular transformations, which we see by rewriting them A direct calculation, with 7 and 6, yields Ĥz + τ = q ζ i α+β F z, τ + + α, τ + + β 7 0 α,β One can verify the transformation formulas for q ζ F z, τ + + α, τ + + β to find that the function q ζ Ĥz + τ satisfies the same modular transformation as in Lemma Thus the analogue of 5 also holds for Ĥ, and 6 holds for Ĥ Using, the modular completion of F requires the additional terms in F F, which equal R z, z, z ; τ := ϑz ; τµz, z ; τrz z ; τ ϑz ; τrz z ; τµz, z ; τ i ϑz ; τrz z ; τrz z ; τ iητ ϑz + z ; τ ϑz ; τϑz ; τ Rz z z ; τ 8 We use this to help identify the excess terms in the modular completion Ĥ In particular, plugging into the definition of Ĥ as well as 7, we have Ĥ H = i α β q R 0, τ + + α, τ + + β + q R τ, τ + + α, τ + + β 0 α,β 9 Recalling, we compute that R 0, z, z = i Rz η ϑz + Rz ϑz ϑz + z ϑz ϑz Rz + z 0 Similarly, a calculation with Lemmas and yields R τ, z, z = q ζ ζ R 0, z, z + iq 8 ζ ζ 0 α,β η ζ ϑz + ζ ϑz ϑz + z ϑz ϑz We now determine whether Ĥ H adds any additional holomorphic terms to P ω The first term of combined with 0 contributes the following to 9 i α β q R 0, τ + + α, τ + + β + α+β q R 0, τ + + α, τ + + β = q R 0, τ +, τ + R 0, τ +, τ + = iq η R τ + ϑ τ + R τ + ϑ τ + Noting that ϑ τ + = q ητ ητ, τ ϑ + τ = iϑ +, ϑ ϑ τ + ϑ τ + R τ + ϑ τ + = ϑ τ +, + ϑ τ + ϑ τ + R τ + τ + = e πi q 8 ητ ητ,

14 R τ + = iq τ 8, R + τ = ir + e πi q 8, R τ + = R τ +, we obtain that the previous expression equals iητ ητ ητ + 8iq 8 ητ ητ ητ The second and third terms from contribute after switching the roles of α and β in the third term e πi q 8 η 0 α,β α i β ϑ τ + + β = 0 For the final term of, we note that ϑτ + + α+β = 0 if α β mod Thus the corresponding sum simplifies to α+β ϑ τ + q 8 η + α+β 0 α,β ϑ τ + + α = q ϑ τ + + β 8 η ϑ τ + ϑ τ + + ϑ τ + ϑ τ + = q 8 η ϑ τ + ϑ τ + = iq 8 ητ ητ ητ, where we use to rewrite ϑτ + and ϑ τ + in terms of eta quotients Combining with, we then obtain Ĥ τ H τ = iητ ητ ητ iq 8 ητ ητ ητ 5 We next consider Ĥ H, and note that Ĥ H = i α β q ζr z, τ + + α, τ + + β 0 α,β q R z + τ, τ + + α, τ + + β ] 6 We again need to determine the holomorphic components of these terms for our specific choices of z j We first evaluate the shifted term Using 8 and Lemmas and, and then calculating the derivative directly, we find that [ ] R z + τ, z, z [ ] = q ζ ζ ζr z, z, z [ ζ ζ q 8 ζ ϑzµz, z + i ζ ζ q 8 i ζ ζ q 8 [ [ ] [ ζ ζ q ] 8 ζ ϑzµz, z ζ ϑzrz z ζ ϑzrz z ] ] + i ζ ζ q 8 iq ζ ζ i ζ ζ q 8 [ ζ ϑzrz z [ ] ϑz [ ] ζ ϑzrz z ]

15 + iq ζ ζ [ ] ϑz iq 8 ζ ζ η ϑz + z ϑz ϑz After cancellation, we are left with the following terms: [ ] [ q ζ ζ ζr z, z, z ζ ζ q ] 8 ζ ϑzµz, z [ ζ ζ q ] 8 ζ ϑzµz, z iq ζ ζ [ ] ϑz iq 8 ζ ζ η ϑz + z ϑz ϑz 7 We now evaluate the contribution of each term of 7 to 6 Recalling, we find that the fourth term of 7 sums to iη α+β = 0 For convenience, we define fz j ; τ := 0 α,β [ ] ζ ϑz; τµz, zj ; τ = ζ j n n + q nn+ ζ j q n By rewriting the second and third terms in terms of fz j, we find that they also cancel completely, as e πi q τ 8 i α β α i β f + + α τ + β i α f + + β 0 α,β = e πi q τ 8 i α β f + + α = 0 0 α,β The fifth term of 7 does make a contribution to Ĥ H, which simplifies as q η i α β i ϑ τ + q 8 e πi τ + + α e πi τ + + β + α+β 0 α,β ϑ τ + + α ϑ τ + + β = iq 8 ητ ητ ητ, where we have the same cancellations as in Finally, the first term of 7 combines with R z, τ + + α, τ + + β from 6 to give q ζ i α β i α+β R z, τ + + α, τ + + β = iq [ 0 α,β, ζr z, τ +, τ + ], 8 where in the final equality we use the symmetry R z, z, z = R z, z, z To evaluate the above, recall that [ ] ζr z, z, z 5

16 = [ ζϑzµz, z Rz z ζϑzµz, z Rz z i ζϑzrz z Rz z ] iη ϑ z + z ϑz ϑz ζrz z z 9 Plugging in our specific values of z and z, we begin with the fourth term of 9 From the fact that ϑ vanishes at Z + Zτ, this term vanishes Next we consider the third term of 9 when plugging into 8 Noting that we need to differentiate ϑ see and using our specific values of z, z, along with the fact that R is even, this becomes iq τ η R + τ R + 0 Finally, the first and second terms of 9 result in iq τ f + τ R + τ + f + τ R + τ + g + [ ζ R z τ ] τ + g + [ ζ R z τ ], where gz; τ := ζ n q nn+ ζq n We now determine the holomorphic terms in 0 and We start with those that arise from not differentiating R Noting that sgn n { 0 if n 0, sgnn = if n = 0, we obtain R τ + = sgn n E n + v n e πin q nn+ n +Z = e πi q 8 e πi q 8 i n sgn n E n v q n πi = e q 8 + N τ, where in the last line we use that the sum over even integers n vanishes, and where we set N τ := e πi q 8 n sgnn + E n + v q n+ Now N is purely non-holomorphic, by which we mean that the coefficients of its q-expansion are incomplete Gamma functions with arguments in v cf Moreover, gives that τ R + = in τ + e πi q 8 We conclude that the holomorphic contribution of 0 to 6 is iη 6

17 Similarly, the holomorphic components from the first two terms in simplify as e πi q τ 8 if + τ + f + = iq n n + q nn+ 8 + q n+ = iq 8 n n + q nn+ = iη We rewrite the third term of as τ g + [ τ = ig + [ iζ R z τ ] + e πi q 8 ζ R z τ ] The third and fourth terms including the outside factor now become [ q ζ R z τ ] τ g + τ ig + To compute the derivative of R, we write ζ R z τ = e πi πi q 8 + e q 8 Recalling, we then find that [ ζ R z τ ] = e πi q 8 = e πi q 8 i n n n n sgnn E sgnn E n + y v i n q n ζ n v n v q n e πi + q 8 i n e πin τ vπ sgnn E n v q n + e πi q 8 π v n e πin τ since for both summands the sums over odd integers n vanish Thus is purely non-holomorphic We conclude that Ĥ H has the holomorphic part iητ + iq 8 ητ ητ ητ i η Ĥ + Ĥ, we can now find the holomorphic part by combining the Since is exactly above calculations above In particular, we use the decomposition Ĥ + Ĥ = Ĥ H + Ĥ H + H + H Recalling, and using 5 and, the holomorphic part of Ĥ + Ĥ equals iητ ητ ητ iq 8 ητ ητ ητ + iητ + iq 8 ητ ητ ητ iητ Pω q = iητ P ω q + ητ ητ 7

18 Having found the relation between Ĥ + Ĥ and P ω, we next connect it to P ω We start by showing that Ĥ = 0 By 7, a direct calculation gives that Ĥ = q F 0, τ +, τ + F 0, τ +, τ + We remember that F z, z, z has a removable singularity at z = 0, and so by applying Lemmas and, we obtain that F 0, τ +, τ + = F 0, τ + ; τ + From this we directly conclude that Ĥ = 0 We next rewrite Ĥ As a first simplification, we find Ĥz q ζ Ĥz + τ = q ζ 0 α,β = iq ζ F i α β i α+β F z, τ + + α, τ + + β z, τ +, τ + + F z, τ +, τ + = iq ζ F z, τ +, τ +, noting that F z, z, z = F z, z, z Thus we have that [ Ĥ = iq ζ F z, τ +, τ + We then compute F z, τ +, τ + = iϑz µ z, τ + µ z, τ + η ϑ τ + ϑ τ + ] lim w 0 ϑw + τ + µz, w + τ + By and Lemmas and, and further applying, we obtain [ Ĥ = iq iϑz µ z, τ + µ z, τ + ] η 6 ϑ τ + ϑz [ { }] = i FzF z η6 q ϑz ϑ τ +, 5 where F is defined in In order to evaluate this derivative, we calculate the Laurent series of the interior expression and show that it has a removable singularity First, implies that ϑz = πη z ϑ 0 π η 6 z + O z The Taylor expansion of the bracketed terms in 5 is F0 + η6 q ϑ τ + F0F 0 F 0 z + O z 6 8

19 We now show that the constant term of this expansion vanishes Indeed, by definition Fz = q 8 ζ ϑzµ z, τ + + i q 8 ζ ϑzr z τ, 7 and then implies that F0 = iη q 8 ϑ τ + 8 Recalling, and plugging 6 back in to 5, we conclude that Ĥ τ = π ητ F 0 e πi ητ ητ ητ F 0 = iητ Pω τ We are now in a position to prove the stated properties of Pω In particular, we see that i follows from 6 and 8, along with the fact that 6 ψ e πi ab+ cd πic = e 8 c d on Γ, and ii follows from Thus we are left to show iii For this we use and 7 to compute that noting that ϑ must differentiated with respect to ζ τ F 0 = πiq 8 ητ τ R τ + = Noting that the sum over even integers n vanishes, we obtain πie πi ητ v n n + e πin+τ = πe πi ητ v πi πe i n ne πinτ ητ η τ 9 v Next we have [ τ F 0 = q i 8 ζ ϑzr z z τ τ ] z=0 Since ϑ is odd, we again need to differentiate ϑ in 7 exactly once, so that by using, [ τ F 0 = πiq 8 ητ ζ R z τ z τ ] 0 z=0 The derivatives can be simplified and rewritten as [ ζ R z τ z τ ] = πi z=0 τ R τ + [ ] τ z Rz z= τ We now apply both and, finding that 0 equals πητ v e πinτ = πητ v η τ 5 η τ η τ We have now shown that the action of L is as claimed We are left to prove that F 0 is a harmonic Maass form of weight η whose shadow is as claimed The transformation follows by a lengthy calculation with Lemmas and The claim about the shadow follows from 9 by multiplying by iv ητ and conjugating 9

20 References [] G Andrews, The number of smallest parts in the partitions of n, J Reine Angew Math 6 008, [] G Andrews The theory of partitions Addison-Wesley Publishing Co, Reading, Mass-London-Amsterdam, 976 Encyclopedia of Mathematics and its Applications, Vol [] G Andrews, A Dixit, D Schultz, and A Yee, Overpartitions related to the mock theta function ωq, preprint arxiv:6005 [] G Andrews, A Dixit, and A Yee, Partitions associated with the Ramanujan/Watson mock theta functions ωq, νq and φq, Res Number Theory, 05 [5] K Bringmann, On the explicit construction of higher deformations of partition statistics, Duke Math Journal 008, 95- [6] K Bringmann, A Folsom, Ken Ono, and L Rolen, Harmonic Maass forms and mock modular forms: theory and applications, AMS Colloquium Series, to appear [7] K Bringmann, L Rolen, and S Zwegers, On the modularity of certain functions from Gromov-Witten theory of elliptic orbifolds, Royal Society Open Science 05, pages 500 [8] J Bruinier and J Funke, On two geometric theta lifts, Duke Math J 5 00, 5 90 [9] A Dixit, and A Yee, Generalized higher order spt-functions, Ramanujan J 0, no -, 9 [0] M Eichler and D Zagier, The theory of Jacobi forms, Progress in Mathematics, 55 Birkhäuser Boston, Boston, MA, 985 [] A Folsom, and K Ono, The spt-function of Andrews, Proc Natl Acad Sci USA , no 5, [] F Garvan, Congruences for Andrews smallest parts partition function and new congruences for Dyson s rank, Int J Number Theory 6 00, no, 8 09 [] F Garvan, Higher order spt-functions, Adv Math 8 0, no, 65 [] F Garvan, and C Jennings-Shaffer, Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E, Adv Math 99 06, [5] G Gasper and M Rahman Basic hypergeometric series, volume 96 of Encyclopedia of Mathematics and its Applications Cambridge University Press, Cambridge, second edition, 00 [6] C Jennings-Shaffer, Some smallest part functions from variations of Bailey s Lemma, preprint arxiv:50605 [7] M Knopp, Modular functions in analytic number theory, Markham Publishing Co, Chicago, 970 [8] K Ono, Congruences for the Andrews spt function, Proc Natl Acad Sci USA 08 0, no, 7 76 [9] A Patkowski, Another smallest part function related to Andrews spt function, Acta Arith 68 05, no, 0 05 [0] G Shimura, On modular forms of half integral weight, Annals Math 97 97, 0-8 [] L Slater, Further identities of the Rogers-Ramanujan type, Proc London Math Soc 5, 95, 7 67 [] G Watson, The final problem: An account of the mock theta functions, J London Math Soc 96, [] D Zagier, Ramanujan s mock theta functions and their applications [d aprés Zwegers and Bringmann-Ono] Astérisque 6 009, Soc Math de France, 6 [] S Zwegers, Mock theta functions, PhD Thesis, Universiteit Utrecht, 00 Mathematical Institute, University of Cologne, Weyertal 86-90, 509 Cologne, Germany address: kbringma@mathuni-koelnde Mathematical Institute, University of Cologne, Weyertal 86-90, 509 Cologne, Germany address: jennichr@mathoregonstateedu Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA address: mahlburg@matlsuedu 0