Recurrence formulae for the coefficients of mock theta functions of order 5 and 7

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Research Collection

Student Paper

Recurrence formulae for the coefficients of mock theta functions of order 5 and 7

Author(s):

Depouilly, Baptiste Publication Date:

2021

Permanent Link:

https://doi.org/10.3929/ethz-b-000477696

Rights / License:

In Copyright - Non-Commercial Use Permitted

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Recurrence formulae for the coefficients of mock theta functions of order 5 and 7

Baptiste Depouilly

ABSTRACT. We compute recurrence formulae for the coefficients of Ramanujan’s mock theta functions of order 7 and 5. Our computations rely on a method developed by J.H. Bruinier and M. Schwagenscheidt, who obtained an equation for such coefficients by evaluating regularized theta lifts of harmonic Maass forms in two different ways.

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1 INTRODUCTION

1 Introduction

The story of mock theta functions begins in the twilight of Ramanujan’s life and his friendship with Hardy. In his last letter to Hardy - sent three months before his death -, a sick Ramanujan announces to his friend the discovery of a new class of functions. He lists 17 of them, organized by order: four of order 3, ten of order 5 and three of order 7. Ramanujan goes on to explain that these have the same type of expansion as modular forms at every rational point but that no single one of them has the expansion of a modular form at all rational points (cf. [Zag09]).

In accordance with his field of predilection, Ramanujan wrote the definition of his 17 functions and the identities between them in the form of hypergeometric q-series. A mere 82 years later, in 2002, Sander Zwegers will publish his thesis and give the first intrinsic characterization of mock theta functions. This breakthrough facilitates greatly the proof of identities between functions of this class, as it is now sufficient to compute some of their invariant to conclude that two given mock theta functions are equal whereas previous proofs relied on a great deal of analytic ingenuity.

Since then, the theory of mock theta functions has been further developed by mathematicians, such as Kathrin Bringmann and Ken Ono, and this previously mysterious class of function has found various applications. For example, the theory of mock theta functions allowed Bringmann and Ono [BO10] to compute the quantities N(r, t;n), counting the number of partitions of n whose rank in congruent to r mod t.

The aim of this paper is to compute recurrence formulae for the coefficients of mock theta functions of order 7 and 5. The starting point of our computations is an equation proven by J.H. Bruinier and M.

Schwagenscheidt in their paper [BS20] and based on the evaluation of regularized theta lifts of harmonic Maass forms. Following [GM12], we define (a;q)∞=Q∞

n=0(1−aqⁿ) and take F₀(q) =

∞

X

n=0

qⁿ² (qⁿ⁺¹;q)n

, (1)

F₁(q) =

∞

X

n=0

q⁽ⁿ⁺¹⁾²

(qⁿ⁺¹;q)_n+1, (2)

F₂(q) =

∞

X

n=0

qⁿ⁽ⁿ⁺¹⁾

(qⁿ⁺¹;q)_n+1 (3)

as definitions for the three order 7 mock theta functions given by Ramanujan in his last letter to Hardy.

We write the first few terms of the Fourier series of these three functions F₀(q) = 1 +q+q³+q⁴+q⁵+ 2q⁷+q⁸+· · ·

F₁(q) =q+q²+q³+ 2q⁴+q⁵+ 2q⁶+ 2q⁷+ 2q⁸+· · ·

F₂(q) = 1 +q+ 2q²+q³+ 2q⁴+ 2q⁵+ 3q⁶+ 2q⁷+ 3q⁸+· · ·, These expansions can be found athttps://oeis.org/search?q=mock+theta+7th+order.

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1 INTRODUCTION

The formula towards which we will work for mock theta functions of order 7 equates a finite combination of coefficients of F₀,F₁ and F₂ to a combination of well-understood divisor sums. Namely, we will prove that

Proposition 1.1. For any m∈N, X

r∈Z r²≡1 (168)

−84 r

raF0

m+1−r² 168

− X

r∈Z r²≡25 (168)

−84 r

raF1

m+25−r² 168

− X

r∈Z r²≡121 (168)

−84 r

raF₂

m+−r²−47 168

=−48σ₁(m) +X

d|m

4 sgnm

d −42d

minm d,42d

−4 sgn 2m

d −21d min

2m d,21d

−4 sgn 3m

d −14d min

3m d,14d

+ 4 sgn 6m

d −7d min

6m d,7d

, whereaFi(n)denotes the order-ncoefficient ofF_iandσ1(m) =P

d|ndis the usual sum-of-divisors function.

Remark. The equation presented in Proposition 1.1 and the ones stated in Proposition 4.1 and Proposi- tion 4.2 were all numerically checked in Sage (https://www.sagemath.org/).

Example. Note that all the sums in Proposition 1.1 are finite and that only a few coefficients of each mock theta function appear for each increment of m. For example, we have

• For m= 1:

−13a_F₀(0) +aF₀(1)−5aF₁(1)−11aF₂(0) =−24σ₁(1)−4;

• form= 2:

−13a_F₀(1) +aF0(2)−19aF1(0)−5aF1(2)−17aF2(0)−11aF2(1) =−24σ₁(2) + 26;

• form= 3:

−13a_F₀(2) +aF0(3)−23aF1(0)−19aF1(1)−5aF1(3)−17aF2(1)−11aF2(2) =−24σ₁(3) + 34.

We additionally note that similar recurrence formulae have been obtained for the coefficients of the order 3 mock theta functions

f(q) = 1 +

∞

X

n=1

qⁿ²

(1 +q)²(1 +q²)²· · ·(1 +qⁿ)² and ω(q) =

∞

X

n=0

q²ⁿ²⁺²ⁿ

(1−q)²(1−q³)²· · ·(1−q²ⁿ⁺¹)² by Imamoglu-Raum-Richter [IRR14] using the method of holomorphic projection. A few years after, Bruinier-Schwagenscheidt used the method they developed, and that will be used in this paper, to find results of the same nature for f and ω.

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2 PRELIMINARIES

We will in addition use our method to obtain two formulae for the coefficients of eight order 5 mock theta functions. These results are collected in Proposition 4.1 and Proposition 4.2. Moreover, this method can also be employed to derive similar formulae for mock theta functions of other orders. For example, in [KK20], Klein-Kupka have found formulae for mock theta functions of order 6 and 10 that can be used as a basis for the computation of similar results.

2 Preliminaries

In this section we will introduce the objects that will be the focus of the paper and we hope to help the reader gain some familiarity with them. However, if you already have the basics covered, feel free to start directly with Section 3.

2.1 Lattices

Throughout the paper, we will work on an latticeL endowed with a symmetric Z-valued bilinear form Q. In other words we will consider a finitely generated free module over Zand a pairing

Q(·,·) :L×L7→Z

that induces a linear map from L to Z when fixing either of the two components. We will assume that the quadratic space (L, Q) is regular, i.e. if for some x ∈ L we have Q(x, y) = 0,∀y ∈ L, then x = 0.

We can then diagonalize the form Q and denote λ₁, . . . , λ_R 6= 0 the values on the diagonal of Q. The signature of the quadratic space (L, Q) is defined to be the pair (n+, n−), describing the number of positive and negative λ_i’s, respectively. We define the dual lattice L^∗ to be the set of linear combination of the lattice generators over the field Q, i.e. the set of x ∈ L⊗Q, such that Q(x, y) ∈ Z,∀y ∈ L. Note that L is a submodule of L^∗ by definition and therefore that L^∗/L defines an abelian group. It is called the determinant group and turns out to be finite.

2.2 The metaplectic group and the Weil representation

Let us start with establishing the notation that will appear throughout the paper. We will denoteHthe complex upper-half plane and τ =u+iv, u, v∈R, the standard complex variable onH. Additionally, we will note e(z) :=e^2iπz for z∈C and q :=e^2iπτ. For any α ∈C, we denote e^α^log(z) by z^α, where log(·) is the principal branch of the logarithm.

Let us denote {e_i}^r_i=1 the generators of the group algebra C[L^∗/L]. We recall that SL₂(Z) admits the double cover Mp₂(Z), described as follows: the elements of Mp₂(R) are pairs of the form (M, φ(τ)),where M = ^{a b}_{c d}

∈ SL₂(R), and φ: H 7→ C is holomorphic such that φ(τ)² = cτ +d. Since for any complex number z there are two choices of square root, this is in fact a two-sheeted covering of SL₂(R). We then

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2.2 The metaplectic group and the Weil representation 2 PRELIMINARIES

denote Mp₂(Z) the inverse image of SL₂(Z) with respect to the covering map π: Mp₂(R) 7→ SL2(R)

(M, φ(τ)) 7→ M The group Mp₂(Z) is generated by the elements

S˜:= ⁰_{1 0}⁻¹ ,√

τ T˜:= ((^{1 1}_{0 1}),1).

We recall that there is a unitary representation ρL of Mp₂(Z) on the group algebra C[L^∗/L], defined by the action of ˜S and ˜T as follows:

ρL( ˜T)eγ:=e(q(γ))eγ

ρL( ˜S)eγ:=

√ iⁿ⁻⁻ⁿ⁺ p|L^∗/L|

X

δ∈L^∗/L

e(−(γ, δ))e_δ.

The representation will be referred to as the Weil representation. We denote ρ^∗_L the dual representation of ρ_L. For (M, φ) ∈Mp₂(Z), if we think of ρ^∗_L(M, φ) as a matrix with entries in C, then ρ^∗_L(M, φ) is the complex conjugate of ρ_L(M, φ).

Definition 2.1 (Petersson slash operator). For k∈ ¹₂Z, (M, φ) ∈ Mp₂(Z) and a map f :H7→ C[L^∗/L], we define the Petersson slash operator as follows:

f |_k (M, φ)(τ) :=φ(τ)^−kρ_L(M, φ)⁻¹f(M τ).

This defines an operation of Mp₂(Z) on functions fromH toC[L^∗/L].

Note that iff is invariant under this operation with respect to ˜T := ((^{1 1}_{0 1}),1), we have X

γ∈L^∗/L

f_γ(τ + 1)e_γ= X

γ∈L^∗/L

f_γ(τ)ρ_L( ˜T)e_γ

= X

γ∈L^∗/L

fγ(τ)e(q(γ))eγ. Hence f_γ(τ)e(−q(γ)τ) is periodic for allγ and f admits a Fourier extension

f(τ) = X

γ∈L^∗/L

X

n∈Z−q(γ)

a_f(γ, n, v)e(nτ)eγ. We can define the analogous operation for ρ^∗_L:

f 7→f |^∗_k(M, φ), f |^∗_k(M, φ)(τ) :=φ(τ)^−kρ^∗_L(M, φ)⁻¹f(M τ).

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2.3 Harmonic Maass forms 2 PRELIMINARIES

Definition 2.2. A modular form of weight k ∈ ¹₂Z with respect to the representation ρ_L is a map f :H7→C[L^∗/L] such that

(1) f |_k(M, φ)≡f for all (M, φ)∈Mp₂(Z);

(2) f is holomorphic on H; (3) f is holomorphic at ∞.

The space of modular forms of weight kwith respect to ρL and Mp₂(Z) is denotedM_k,L. If instead of the third condition f has a pole at∞, we say that f is a weakly holomorphic modular form of weight k with respect to ρL and Mp₂(Z). The space of weakly holomorphic modular forms of weight k with respect to ρL and Mp₂(Z) is denotedM_k,L^! .

Remark. The third condition ensures that f admits a Fourier expansion on the upper half-plane such that f(τ) = X

γ∈L^∗/L

X

n∈Z−q(γ) n≥0

a_f(γ, n)e(nτ)eγ.

2.3 Harmonic Maass forms

Definition 2.3. Fork∈R, we define the weight khyperbolic Laplacian on Has

∆k:=−v² ∂²

∂u² + ∂²

∂v²

+ikv ∂

∂u +i ∂

∂v

=−4v² ∂

∂τ

∂

∂τ + 2ikv ∂

∂τ.

The method used in Bruinier-Schwagenscheidt [BS20] to obtain the equation on which we base our computations regards a special class of functions defined by Bruinier and Funke in [BF04].

Definition 2.4. A harmonic Maass form of weightk∈ ¹₂Zfor the representation ρLis a smooth function f :H7→C[L^∗/L] such that

(1) f(M·τ) =φ(τ)^2kρL(M, φ)f(τ) for any (M, φ)∈Mp₂(Z);

(2) ∆_kf ≡0;

(3) there exists a polynomial P_f(τ)∈C[q⁻¹] such that

f(τ)−P_f(τ) =O(e^−v), for some >0, wheneverv= Im(τ)→ ∞.

We denote H_k,L the space of harmonic Maass forms of weight kwith respect to ρ_L and Mp₂(Z).

Definition 2.5. We now define the antilinear differential operator ξ_k= 2iv^k ∂

∂τ,

mappingH_k,L toM_2−k,L^! −, whereL⁻denotes the lattice (L,−Q).We denoteH_k,L^hol andH_k,L^cuspthe subsets of Hk,Lthat are mapped to the subspacesM2−k,L⁻, of modular forms, andS2−k,L⁻, of cusp forms, respectively.

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2.4 Unary theta series and Atkin-Lehner involutions 2 PRELIMINARIES

Fork≤0, anyf ∈H_k,L^hol can be decomposed as a sumf =f⁺+f⁻such that f⁺is holomorphic andf⁻ isn’t. Their respective Fourier expansions are given by

f⁺(τ) = X

γ∈L^∗/L

X

n∈Q n−∞

a⁺_f(γ, n)qⁿe_γ (4)

f⁻(τ) = X

γ∈L^∗/L





a⁻_f(γ,0)v^1−k+X

n∈Q n<0

a⁻_f(γ, n)Γ(1−k,4π|n|v)qⁿ





e_γ, (5)

where a^±_f(γ, n)∈C,q =e^2iπτ, and Γ(s, x) is the incomplete Gamma-function Γ(s, x) :=

Z ∞ x

e^−tt^s−1dt.

2.4 Unary theta series and Atkin-Lehner involutions

Let N ∈ N. The lattice (Z, N x²) has level 4N, its discriminant group is isomorphic to Z/2NZ and is endwowed with the quadratic Q/Z-valued quadratic form x 7→ _4N^x². The weight-3/2 unary theta series associated to this lattice are defined to be

Θ3

2,N(τ) := X

r(2N)

X

n∈Z n≡2r(2N)

nqⁿ²^/4Ne_r.

Using Poisson’s summation formula and a direct computation of the Fourier transform, Borcherds proved in [Bor98], Theorem 4.1, that Θ³

2,N is a modular form of weight 3/2 for the Weil representation associated to (Z, N x²).

Definition 2.6. For an exact divisorc||N, we define the Atkin-Lehner involution σcofZ/2NZto be the map such that

σc(x)≡ −x(2c) σc(x)≡x(2N/c).

We define an action of the set of Atkin-Lehner involutions on the vector-valued modular forms for the Weil representation associated to (Z, N x²) as follows:



 X

r∈Z/2NZ

fr(τ)er





σc

:= X

r∈Z/2NZ

f_σ_c_(r)(τ)er.

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2.5 Theta lifts for Lorentzian lattices and recurrence formulae 2 PRELIMINARIES

2.5 Theta lifts for Lorentzian lattices and recurrence formulae

Definition 2.7 (Siegel theta function). LetL denote an even lattice of signature (b⁺, b⁻) with respect to the quadratic form Q. We define Gr(L) to be the manifold of b⁺-dimensional positive definite subspaces of V := L⊗R. For z ∈Gr(L), we letz^⊥ be its orthogonal complement so that V =z⊕z^⊥. For x ∈V, let x_z be its orthogonal projection toz and x_z^⊥ its orthogonal projection to z^⊥. Then, the Siegel theta function attached to L is defined by

Θ_L(τ, z) =v^b⁻^/2 X

λ∈L^∗

e(τ q(λ_z) +τ q(λ_z⊥))e_L+λ, (τ =u+iv∈H, z∈Gr(L)).

Definition 2.8 (Regularized theta lift). Let L be a Lorentzian lattice, i.e. an even lattice of signature (1, n) with n≥1. Let k= ¹⁻ⁿ₂ and f ∈H_k,L^cusp be a harmonic Maas form of weight k with respect to the dual Weil representation associated to L. Following Borcherds, forz∈Gr(L), the regularized theta lift of f is defined by

Φ(f, z) = lim

T→∞

Z

F_T

hf(τ),ΘL(τ, z)iv^kdudv

v² , (6)

where F_T denotes the standard fundamental domain of SL₂(Z) cut off at height T and u, respectively v, is the real, respectively imaginary, part of τ.

From here on, we will let L be an even isotropic lattice of signature (1,1) in the quadratic space (Q², Q(x, y) =xy).

Definition 2.9. Following [BS20], for a fixed isotropic vector l ∈ L⊗Q, we define the modified theta function

Θ^∗_L(τ, z) :=v^3/2 X

λ∈L^∗

λ, l_z

|l_z| λ, l_z⊥

|l_z⊥|

e(Q(λ_z)τ+Q(λ_z⊥)τ)e_λ+L, where τ =u+iv∈H, z∈Gr(L) and |λ|=p

|Q(λ)|forλ∈L⊗R.

Form ∈N≥0, let us denote jm the unique modular function for SL2(Z) whose Fourier expansion starts with q^−m +O(q). To exhibit the recurrence formulae for the coefficients of order 5 and order 7 mock modular forms, we will apply an equation obtained by Bruinier and Schwagenscheidt [BS20] to vector of mock theta functions. Namely, they proved that

Proposition 2.10 ([BS20], Equation 4.8). Letting (y₁, y₂)∈Z² be coprime such thaty₁y₂=N, we have 8√

y1y2

X

λ1,λ2∈N λ1λ2=N

sgn(λ2y1−λ1y2) min(λ1y2, λ2y1)

=X

n∈Z

am(−n)X

r∈Z

ra⁺_˜

Θ3 2,N

σy1(r), n− r² 4N

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where σc for c||dis the corresponding Atkin-Lehner involution onZ/2dZ,am(−n)is the −n-th coefficient of 2j_m+ 48σ₁(m) and Θ˜³

2,N is a ξ_1/2-preimage of Θ³

2,N.

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2.5 Theta lifts for Lorentzian lattices and recurrence formulae 2 PRELIMINARIES

Remark. Note that the equation 4.8 found in [BS20] differs from the one given above. Indeed, the factor 1/√

y₁y₂ that appears in the reference should be√

y₁y₂ and this typo was fixed in (7).

To obtain the above equation, the authors of [BS20] evaluated the regularized theta lift of harmonic Maass forms in two different ways.

First, the authors used the fact that every weakly holomorphic modular form can be written as a linear combination of Maass-Poincar´e series, as proved by Bruinier in [Bru02]. The authors then computed the regularized theta lift of an arbitrary Maass-Poincar´e series and used this result to obtain Fourier expansion for Φ(f, z).

On the other hand, the authors evaluated the regularized theta lift Φ(f, z) at a special point z = ω ∈Gr(L), directly from the definition, using a splitting of the modified Siegel theta function Θ^∗_L. More precisely, for a special pointω ∈Gr(L), i.e. a line in Gr(L) that can be generated by an element of L⊗Q, one can split L⊗Qasω⊕ω^⊥ and define

P :=L∩ω, M :=L∩ω^⊥.

Then,P is a positive definite one-dimensional subset ofLwhereasM is a negative definite one-dimensional subset ofL. Lety= (y1, y2)∈Lbe such thaty1, y2>0 and such that its norm with respect to the quadratic form Q(x, y) =xy is minimal among vectors ofLthat generate ωoverQ. We also definey^⊥= (y^⊥₁, y₂^⊥) to be the smallest multiple of (y₁,−y₂) that generatesω^⊥ overQ. We deduce that,

P '(Z, y₁y₂x²), M '(Z, y₁^⊥y^⊥₂x²).

Hence

Θ^∗_P =C_yΘ_3/2,y₁_y₂, Θ^∗_M− =C_y⊥Θ_3/2,−y⊥ 1y^⊥₂,

whereC_y,C_y⊥ respectively, are constants that only depend ony,y^⊥ respectively. Now, atω, the modified Siegel theta function associated to Lsplits as

Θ^∗_P_⊕M(τ, ω) =



 X

λ1∈P^∗

λ₁, l_ω

|l_ω|

e(Q(λ₁)τ)e_λ₁



⊗v^3/2



 X

λ2∈M^∗

λ₂, l_ω^⊥

|l_ω⊥|

e(Q(λ₂)τ)e_λ₂



, (8) by orthogonality of P and M, using the same notation as in Definition 2.9.

Taking (y₁, y₂) as in Proposition 2.10, we obtain

Θ^∗_P_⊕M(τ, ω) =C_yΘ_3/2,N(τ)⊗v^3/2C_y^⊥Θ_3/2,−y^⊥

1y^⊥₂ (τ). (9)

Bruinier-Schwagenscheidt concluded this evaluation by introducing the splitting into the integral defining the theta lift and then using Stokes theorem.

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3 ORDER 7 MOCK THETA FUNCTIONS

Zwegers’ thesis now provides a bridge between the machinery developed above and Ramanujan’s mock thetas. Indeed, he showed in [Zwe02] that mock theta functions can be completed to harmonic Maass forms.

Combining the two evaluations computed in [BS20] and specializing to lattices and harmonic Maass forms of interest, Bruinier-Schwagenscheidt finally obtained the recurrence formula stated in Proposition 2.10.

3 Order 7 mock theta functions

The goal of this section will be to use Bruinier’s and Schwagenscheidt’s formula, stated in Proposi- tion 2.10, to obtain a recursive formula for the coefficients of Ramanujan’s three order 7 mock modular forms, namely F₀,F₁ and F₂ as defined in equations (1)–(3). Our main result will be Proposition 1.1.

Before starting with our computations we need a couple more definitions and we will state a result proven by Zwegers that we will use throughout the paper. First, for all a, b∈R and τ ∈Hdefine

R_a,b(τ) := X

ν∈a+Z

sgn(ν)β(2ν²y)e^−πiν²^τ−2πiνb,

where β(x) =R∞

x u⁻¹²e^−πudu, x∈R≥0.Additionally, let g_a,b be the unary theta series ga,b(τ) = X

ν∈a+Z

νe^iπν²^τ+2iπνb.

Then, Zwegers ([Zwe02], Proposition 4.2) proved that

∂

∂τR_a,b(τ) =−i 1

√2yga,−b(−τ). (10) We can now get started with the computation of the desired recurrence formula. We follow [Zwe02] and arrange the three functions as a vector-valued mock modular form in the following way:

F₇(τ) :=







q⁻¹⁶⁸¹ F₀(q) q¹⁶⁸⁴⁷F₂(q) q⁻¹⁶⁸²⁵F₁(q)





.

Alike him, we set

G₇(τ) =





 G_7,0 G7,1

G7,2





(τ) :=−







ζ₈₄⁻¹³R¹³

42,−¹

2 +ζ84R⁴¹

42,−¹

2

ζ₈₄²⁹R¹¹

42,−⁵

2 +ζ₈₄⁻⁴¹R²⁵

42,−⁵

2

ζ₂₈⁵ R23

42,−³₂ +ζ₂₈⁻⁹R37 42,−³₂





(21τ).

He then proved the following result:

(12)

Proposition 3.1 ([Zwe02], Proposition 4.5). The function

H7 :=F7−G7. (11)

transforms like a vector-valued (real-analytic) modular form of weight 1/2and is annihilated by the weight- 1/2 Laplacian ∆_1/2. More precisely, we have

H7(τ+ 1) =







ζ₁₆₈⁻¹ 0 0 0 ζ₁₆₈⁴⁷ 0 0 0 ζ₁₆₈⁻²⁵





H7(τ)

H₇

−1 τ

=√

−iτ 2

√7







sin^π₇ sin^3π₇ sin^2π₇ sin^3π₇ −sin^2π₇ sin^π₇ sin^2π₇ sin^π₇ −sin^3π₇





H₇(τ).

Moreover G7 is bounded wheneverτ ↓ξ, for ξ∈Q. In other words, H7 is a harmonic weak Maass form of weight 1/2.

We now compute ξ_1/2(H₇(τ)). Since F₇ is holomorphic, we have ξ_1/2(H₇(τ)) = ξ_1/2(−G₇(τ)). Using Equation (10), we first obtain that

∂

∂τ h

−

ζ₈₄⁻¹³R¹³

42,−¹

2 +ζ84R⁴¹

42,−¹

2

(21τ)

i

= 21i

√42v

ζ₈₄⁻¹³g¹³

42,¹₂ +ζ84g⁴¹

42,¹₂

(−21τ) We then deduce that

ξ¹

2

(−G_7,0(τ)) =√ 42h

ζ₈₄¹³g¹³

42,¹₂(−21τ) +ζ₈₄⁻¹g⁴¹

42,¹₂(−21τ)i .

We now compute the conjugate of g_a,b(τ). In general:

g_a,b(τ) =ga,−b(−τ). (12)

From this,

ξ¹

2(G_7,0)(τ) =√ 42h

ζ₈₄¹³g¹³

42,−¹

2 +ζ₈₄⁻¹g⁴¹

42,−¹

2

i (21τ).

Proceeding similarly with the other components of −G₇ we obtain:

Lemma 3.2.

ξ_1/2(H7(τ)) =

√ 42





 ζ₈₄¹³g¹³

42,−¹

2 +ζ₈₄⁻¹g⁴¹

42,−¹

2

ζ₈₄⁻²⁹g11

42,−⁵₂ +ζ₈₄⁴¹g25 42,−⁵₂

ζ₂₈⁻⁵g²³

42,−³

2 +ζ₂₈⁹ g³⁷

42,−³

2





(21τ). (13)

(13)

To use Bruinier’s and Schwagenscheidt’s result, we first need to find a harmonic Maass form for the dual Weil representation whose components can be written as linear combinations of components of H₇ and whose ξ_1/2-image can be written as a linear combination of unary theta functions. We use a result of Andersen, who proved that

Proposition 3.3 ([And17], Lemma 4). Noting H₇= (f₁, f₁₂₁, f₂₅)^T, the vector-valued function H˜7 =f1·(e1−e−1) +f1·(e41−e−41)− X

2≤r≤40 r²≡1,25,121 (168)

f_r² ·(er−e−r), (14)

transforms as a vector-valued modular form of weight 1/2 with respect to the dual Weil representation ρ^∗_L and Mp₂(Z).

We now claim that the following holds:

Proposition 3.4. We have that

ξ_1/2( ˜H7) = 1

√ 42

Θ³

2,42−Θ^σ3²

2,42+ Θ^σ3⁶

2,42−Θ^σ3³ 2,42

. (15)

Proof. Note that in our case all of the termsg_a,b appearing in the ξ_1/2-image of H₇ are of the form ga 42,₂^b

for some a, b∈Z such that (a,42) = 1. Note that by letting r:= 42nand N = 42, we arrive at ga

N,^b₂(21τ) = X

n∈₄₂^a+Z

nexp iπn²·21τ

e^2iπnb/2

= 1 N

X

r∈a+42Z

rexp

2iπr²τ 4N

exp

2iπrb 2N

.

Now, note that exp ^2iπrb₈₄

doesn’t depend onr but only on its congruence class mod 84. Hence ga

42,₂^b(21τ) = 1 N

X

r≡a(N)

rq^r

2 4N exp

2iπrb 2N

= 1 N

X

r≡a(2N)

rq^r

2

4Ne^2iπab^2N + 1 N

X

r≡a+N(2N)

rq^r

2

4Ne^2iπ(a+N)b^2N .

Let us plug into this expression the different values of a, b that appear in (13). We obtain

√

42·ζ₈₄¹³·g13

42,−¹₂(21τ) = 1

√42



 X

r≡13(2N)

rq^r

2

4N − X

r≡55(2N)

rq^r

2 4N





√

42·ζ₈₄⁻¹·g⁴¹

42,−¹₂(21τ) = 1

√42



− X

r≡41(2N)

rq^r

2

4N + X

r≡83(2N)

rq^r

2 4N



.

(14)

For further computations, let us use the notation S_a(τ) := P

r∈Z r≡a(2N)

rq^r

2

4N, with N = 42 fixed, as above.

Then,

√

42·ζ₈₄⁻²⁹·g¹¹

42,−⁵

2(21τ) = 1

√42[S11−S53] (τ),

√

42·ζ₈₄⁴¹·g²⁵

42,−⁵

2(21τ) = 1

√

42[S25−S67](τ),

√

42·ζ₂₈⁻⁵·g²³

42−³₂(21τ) = 1

√

42[S23−S65](τ),

√

42·ζ₂₈⁹ ·g37

42,−³₂(21τ) = 1

√42[S₃₇−S₇₉](τ).

Putting everything into vector form, we get

ξ_1/2(H₇) = 1

√ 42







S13−S55−S41+S83

S11−S53+S25−S67

S₂₃−S₆₅+S₃₇−S₇₉





(τ). (16) Note that each of the Sa is exactly the coefficient of the a-th component of Θ³

2,42. Since ˜H7 has the same components as H₇, we can now write ξ¹

2( ˜H₇) as a sum of Atkin-Lehner involutions of Θ³

2,42. The defining equations for the Atkin-Lehner involution σ_c allow us to derive thatξ_1/2( ˜H₇) can be written as the sum of four unary theta series, namely that

√1 42

Θ³

2,42−Θ^σ3²

2,42+ Θ^σ3⁶

2,42−Θ^σ3³ 2,42

.

We will now use the result from Bruiner and Schwagenscheidt stated in Proposition 2.10. We evaluate Equation (7) at the four special points (1,42),(2,21),(3,14) and (6,7), and sum up our results with the appropriate signs to exhibit recurrence formulae for the coefficients of the holomorphic part of ˜H7, i.e. for the coefficients of F7. We obtain the equation

X

d|m

8 sgn m

d −42d

min m

d,42d

−8 sgn

2m d −21d

min

2m

d,21d

+ 8 sgn

6m d −7d

min

6m

d,7d

−8 sgn

3m d −14d

min

3m

d,14d

=X

n∈Z

am(−n)X

r∈Z

ra⁺_˜

H7

r, n− r² 4N

, (17)

where the notation is the same as in Proposition 2.10. Since a_m(−n) has been defined as the −n-th coefficient of 2jm+ 48σ1(m), it vanishes for 0 < n < m and m < n. Moreover, for n < 0, note that

(15)

n− _4N^r² ≤ −1. However, none of Ramanujan’s order 7 mock theta function admits a non-zero coefficient of order smaller than 0. Hence none of the components of F₇ admits a non-vanishing coefficient of order smaller or equal to −1. Therefore the right-hand side of Equation (17) can only be non-trivial for n= 0 or n=m.

Let us first simplify the right hand-side forn= 0. Notice that in this case the termn−₁₆₈^r² ≤0. However, F_i is holomorphic for i= 0,1,2 and the constant term ofF₁ vanishes. Therefore, onlyf1(τ) =q⁻¹⁶⁸¹ F₀(q) has a non-zero coefficient for a negative power of q. Notice that the term ⁻₁₆₈^r² < −₁₆₈¹ for |r| > 1 and that the coefficients of q⁻¹⁶⁸¹ F₀ vanish for powers of q strictly smaller than −₁₆₈¹ . Hence we only get contributions from the coefficient of order 0 of F₀ forr =±1.

We compute that,

√

42·am(0)X

r∈Z

raF7

r,− r²

168

=am(0)

aF7

1,− 1

168

−aF7

−1,− 1 168

=√

42·am(0)

aq⁻¹⁶⁸¹ F₀

−1 168

+a

q⁻¹⁶⁸¹ F₀

− 1 168

= 96√

42·σ1(m).

We now siplify the right-hand side of Equation (7) specialized to ˜H7forn=m. Sincem >0, the expression m−₁₆₈^r² is positive for a finite number ofr ∈Z and we can expect contributions fromF₀,F₁ and F₂. We have

X

r∈Z

ra_F₇

r, m− r² 168

= X

r∈Z r²≡1 (168)

−84 r

raF0

m+1−r² 168

− X

r∈Z r²≡25 (168)

−84 r

raF1

m+ 25−r² 168

− X

r∈Z r²≡121 (168)

−84 r

raF2

m+−r²−47 168

,

where ⁻⁸⁴_r

denotes the Kroenecker symbol. Plugging this result into (17), we get Proposition 1.1.

(16)

4 Order 5 mock theta functions

We will apply the same methods as in the previous section to obtain a recurrence formula for the coefficients of the six order 5 mock-theta functions f0, f1, F0, F1, ψ0, ψ1, ϕ0 and ϕ1. We follow [GM12] and define them to be

f₀(q) =

∞

X

n=0

qⁿ² (−q;q)n

,

F0(q) =

∞

X

n=0

q²ⁿ² (q;q²)n

,

ψ₀(q) =

∞

X

n=0

q¹²^(n+1)(n+2)(−q;q)_n, ϕ₀(q) =

∞

X

n=0

qⁿ²(−q;q²)_n, f1(q) =

∞

X

n=0

qⁿ⁽ⁿ⁺¹⁾ (−q;q)n

,

F1(q) =

∞

X

n=0

q^2n(n+1) (q;q²)_n+1, ψ₁(q) =

∞

X

n=0

q¹²ⁿ⁽ⁿ⁺¹⁾(−q;q)_n, ϕ₁(q) =

∞

X

n=0

q⁽ⁿ⁺¹⁾²(−q;q²)_n, where (a;q)∞=Q∞

n=0(1−aqⁿ).

Our main results in this section, equating relations between the coefficients of Ramanujan’s order 5 mock theta functions to divisor sums, are stated in the following two propositions:

(17)

Proposition 4.1. For all m∈N, 96σ1(m) +X

d|m

4 sgn

3m d −20d

min

3m

d,20d

+ 4 sgn

4m d −15d

min

4m

d,15d

+ 4 sgn

5m d −12d

min

5m

d,12d

−4 sgn m

d −60d

min m

d,60d

= X

r∈Z r²≡1 (120) m−₂₄₀^r² +₂₄₀¹ 6=0

(−1)^br/30c2raF0

2m+ 1−r² 120

− X

r∈Z r²≡4(120)

(−1)^br/60cra_f₀

m+4−r² 240

+ X

r∈Z r²≡49 (120)

(−1)^br/30c2ra_F₁

2m−r²+ 71 120

− X

r∈Z r²≡76 (120)

(−1)^br/60cra_f₁

m− r²+ 44 240

(18)

Proposition 4.2. For all m∈N, 96σ₁(m) +X

d|m

4 sgn 3m

d −20d min

3m d,20d

+ 4 sgn 4m

d −15d min

4m d,15d + 4 sgn

5m

d −12d min

5m d,12d

−4 sgnm

d −60d

minm d,60d

=−2 X

r∈Z r²≡1(120)

(−1)^br/30c+^r

2−1 120 ra_ϕ₀

2

m−r²−1 240

+ 2 X

r∈Z r²≡4(120)

(−1)^br/60cra_ψ₀

m−r²−4 240

−2 X

r∈Z r²≡49(120)

(−1)^br/30c+^r

2−169 120 raϕ1

2

m−r²−49 240

+ 2 X

r∈Z r²≡76(120)

(−1)^br/60cra_ψ₁

m−44 +r² 240

(19) Similarly as before and following Zwegers, [Zwe02] pp. 74–83, we define the following vector-valued functions:

F_5,1(τ) :=







q⁻⁶⁰¹f0(q) q¹¹⁶⁰f1(q) q⁻²⁴⁰¹ (−1 +F0(q¹²))

q²⁴¹⁷¹F₁(q¹²) q⁻²⁴⁰¹ (−1 +F₀(−q¹²))

q²⁴¹⁷¹ F₁(−q¹²)





 ,

(18)

and

G5,1(τ) := 1 2







2ζ₁₂R¹

30,⁵₂ + 2ζ₁₂⁻¹R¹¹

30,⁵₂

2ζ12R¹³

30,⁵₂ + 2ζ₁₂⁻¹R²³

30,⁵₂

−R19

60,0−R²⁹

60,0+R⁴⁹

60,0+R⁵⁹

60,0

−R13

60,0−R23

60,0+R43

60,0+R53 60,0

ζ₂₄⁻⁵R¹⁹

60,⁵₂ +ζ₂₄⁵ R²⁹

60,⁵₂ +ζ24R⁴⁹

60,⁵₂ +ζ₂₄⁻¹R⁵⁹

60,⁵₂

ζ₂₄R¹³

60,⁵₂ +ζ₂₄⁻¹R²³

60,⁵₂ +ζ₂₄⁻⁵R⁴³

60,⁵₂ +ζ₂₄⁵ R⁵³

60,⁵₂





 (30τ).

Then, it holds that:

Proposition 4.3 ([Zwe02], Proposition 4.10). The function H5,1:=F5,1−G5,1

is a vector-valued real-analytic modular form of weight 1/2and is annihilated by the weight-1/2 Laplacian

∆¹

2. More precisely, we have

H5,1(τ + 1) =







ζ₆₀⁻¹ 0 0 0 0 0

0 ζ₆₀¹¹ 0 0 0 0

0 0 0 0 ζ₂₄₀⁻¹ 0

0 0 0 0 0 ζ₂₄₀⁷¹

0 0 ζ₂₄₀⁻¹ 0 0 0

0 0 0 ζ₂₄₀⁷¹ 0 0







H5,1(τ),

H5,1

−1 τ

=√

−iτ 2

√

5M5H5,1(τ), where

M5 =







0 0 √

2 sin^π₅ √

2 sin^2π₅ 0 0

0 0 √

2 sin^2π₅ −√

2 sin^π₅ 0 0

√1

2sin^π₅ ^√¹

2sin^2π₅ 0 0 0 0

√1

2sin^2π₅ −^√¹

2sin^π₅ 0 0 0 0

0 0 0 0 sin^2π₅ sin^π₅

0 0 0 0 sin^π₅ −sin^2π₅





 .

Moreover G5,1 is bounded whenever τ ↓ξ, forξ ∈Q. In other words,H5,1 is a harmonic weak Maass form of weight 1/2.

(19)

Similarly, we define

F_5,2(τ) :=







2q⁻⁶⁰¹ ψ0(q) 2q¹¹⁶⁰ψ1(q) q⁻²⁴⁰¹ ϕ₀(−q¹²)

−q⁻²⁴⁰⁴⁹ϕ₁(−q¹²) q⁻²⁴⁰¹ ϕ₀(q¹²) q⁻²⁴⁰⁴⁹ϕ1(q¹²)





 ,

and

G5,2(τ) :=−G_5,1(τ).

Zwegers proved that

Proposition 4.4 ([Zwe02], Proposition 4.13). The function H_5,2:=F_5,2−G_5,2

is a vector-valued real-analytic modular form of weight 1/2and is annihilated by the weight-1/2 Laplacian

∆1

2. More precisely, we have

H_5,2(τ + 1) =







ζ₆₀⁻¹ 0 0 0 0 0

0 ζ₆₀¹¹ 0 0 0 0

0 0 0 0 ζ₂₄₀⁻¹ 0

0 0 0 0 0 ζ₂₄₀⁷¹

0 0 ζ₂₄₀⁻¹ 0 0 0

0 0 0 ζ₂₄₀⁷¹ 0 0







H_5,2(τ),

H5,2

−1 τ

=√

−iτ 2

√

5M5H5,2(τ),

where M5 is defined as is Proposition 4.1. Moreover, the function G5,2 is bounded whenever τ ↓ ξ, for ξ ∈Q. In other words, H_5,2 is a harmonic weak Maass form of weight 1/2.

Using Equation (10), we now computeξ_1/2(H5,1(τ)) =ξ_1/2(−G_5,1(τ)).We have

∂

∂τ h

2ζ₁₂R¹

30,⁵₂ + 2ζ₁₂⁻¹R¹¹

30,⁵₂

i

(30τ) =−i r60

v h

ζ₁₂g¹

30,−⁵₂ +ζ₁₂⁻¹g¹¹

30,−⁵₂

i(−30τ)

Hence, we obtain