# The immortal dyon index is positive

Martí Rosselló<EMAIL_ADDRESS>
###### Abstract.

The microstates of supersymmetric black holes in asymptotically flat four-
dimensional spacetime are expected to be bosonic due to the spherical symmetry
of their horizons. This implies that the index counting the difference between
bosonic and fermionic black hole microstates must be positive, as conjectured
by Sen in [1]. We show that the conjecture is satisfied by the index which
counts $\frac{1}{4}$-BPS states in $\mathcal{N}=4$ string theory forming
single centered black holes, called immortal dyons. The proof relies on the
relation between the indexed degeneracies and the coefficients of a family of
mock modular forms, thus showing that black holes predict the sign of the
Fourier coefficients of mock modular forms.

Institute of Mathematics, Academia Sinica, Taipei, Taiwan.

###### Contents

  1. 1 Introduction and statement of results
  2. 2 Black hole degeneracies in $\mathcal{N}=4$ string theory
    1. 2.1 Index definition
    2. 2.2 Single centered degeneracies
    3. 2.3 Exact expressions for the degeneracies
    4. 2.4 Polar coefficients
  3. 3 Proof of positivity of the index
    1. 3.1 Main idea
    2. 3.2 Notation
    3. 3.3 Preliminaries
    4. 3.4 Mock and higher $\gamma$ terms
    5. 3.5 Polar coefficients bounds
    6. 3.6 Leading term bounds
    7. 3.7 Final bounds
  4. A The first three cases $m=3,4,5$

##  1\. Introduction and statement of results

Black holes have been a major source of insight into quantum gravity. The
holographic principle and its concise realisation in the AdS/CFT
correspondence can ultimately be seen as a consequence of the Bekenstein-
Hawking area law for black hole entropy.

In this work, black holes are the source of a non-trivial mathematical
statement. Namely, we show that black holes predict the sign of the Fourier
coefficients of a meromorphic Siegel modular form, as conjectured by Sen in
[1], or equivalently, the sign of the Fourier coefficients of a family of mock
modular forms.

Such a precise statement originates from the study of supersymmetric black
holes in string theory, for which the microscopic origin of the Bekenstein-
Hawking entropy has long been understood as coming from the large degeneracy
of bound states of strings and branes [2, 3]. The quantum generalization of
the Bekenstein-Hawking-Wald entropy for extremal black holes, called the
quantum entropy function, can be defined as a path integral over string fields
by making use of the AdS2 factor in their near-horizon geometry [4]. This
definition of quantum entropy allows for reproducing the counting of
supersymmetric black hole microstates from the gravitational path integral by
using supersymmetric localization in supergravity [5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18]. The growing body of evidence points towards the exact
equivalence between the two complementary pictures of a black hole given by
string theory: a bound state of strings and branes, and a solution to the
supergravity equations of motion with an event horizon.

The enumeration of states (or multiplets) at weak string coupling is performed
through a supersymmetric index, which counts the difference between bosonic
and fermionic BPS states annihilated by a fraction of the supercharges. The
agreement between the number of indexed microstates in the limit of large
charges and the semiclassical answer for the entropy indicates that at strong
coupling, and in the large charge limit, the number of bosonic states must be
much larger than the number of fermionic states.

For supersymmetric black holes in 4d asymptotically flat spacetime, a stronger
argument can be made by exploiting the spherical symmetry of their
horizons,111The heuristic argument for extremal black holes in 4d
asymptotically flat spacetime is based on their solitonic behaviour. Being
solutions to the equations of motion, the curvature of the AdS2 factor in
their near-horizon geometry can only be cancelled by an $S^{2}$ factor. The
precise argument for the spherical symmetry of black holes preserving 4
supersymmetries follows by demanding the closure of the supersymmetry algebra
[19], which implies that the symmetry group contains an $SU(2)$ factor which
can be identified with a subgroup of spatial rotations. which implies that
they carry zero angular momentum. Furthermore, the quantization of the fermion
zero modes associated to the breaking of some supersymmetries by the black
hole does not affect the spherical symmetry of the horizon because these zero
modes live outside the horizon [20, 21]. Since extremal black holes possess an
AdS2 factor in their near horizon geometry, using AdS2/CFT1 arguments [4], Sen
postulated that supersymmetric black holes in four dimensions form an ensemble
of states all of which carry zero angular momentum [22], therefore providing a
sharp prediction for the sign of the index for all charges [1].222Naively, one
would expect the ensemble of black hole microstates to have zero average
angular momentum, but the AdS2 boundary conditions fix the ensemble to be the
microcanonical one where the charges are fixed.

The exact matching between the index counting states at weak string coupling
and the index defined by the quantum entropy function, computed by
supersymmetric localization of the path integral in supergravity, was obtained
in [15]. These results give additional support to the previous positivity
conjecture by arguing that the change of boundary conditions from the
gravitational index to the path integral computing the black hole microstate
degeneracies leaves the answer invariant.

The preceding discussion can be summarized in the following schematic form,

$\text{Tr}\,(-1)^{F}\;=\;\text{Tr}\,1\,,$ (1.1)

where the trace is taken over the black hole microstates with fixed charges
and $F$ is the fermion number, which can be related to the third component of
angular momentum $J$ through $(-1)^{F}=e^{2\pi iJ}$.

The previous arguments apply to the index counting horizon degrees of freedom
of a black hole. However, the counting of microstates is typically done
through a spacetime helicity supertrace [23, 24]. In the present work, we are
interested in the index counting $\frac{1}{4}$-BPS states in unorbifolded
$\mathcal{N}=4$ string theory compactified to four dimensions, meaning
heterotic string theory compactified on $T^{6}$, or its dual type II
compactified on $K3\times T^{2}$, for which a formula for the indexed
degeneracies was first conjectured in [25]. For these states, which break $12$
of the $16$ supersymmetries, the pertinent index is the sixth helicity
supertrace, which we introduce in the next section. The relevant point is that
the trace computes the index of asymptotic states, and not just horizon
states, therefore possibly receiving additional contributions external to the
horizon. The additional contributions to the helicity supertrace can be of
three types [1] – hair modes, multi-centered black holes and horizonless
smooth solutions. We expand on the treatment for the first two cases and refer
the reader to [1] for the reasons why no contributions from horizonless
solutions to the index are expected.

Hair modes are normalizable fluctuations which have support outside the
horizon of the black hole. The quantization of the fermionic hair modes which
are not the fermionic zero modes associated with broken supersymmetry can
produce states which preserve supersymmetry and contribute to the index,
therefore possibly spoiling the positivity argument. Since the hair modes
studied in [20] arise from momentum along the compact directions, in a frame
where all the charges of the black hole are pure D-brane charges there are no
expected extra hair modes (as in the case of $\frac{1}{8}$-BPS states in $4d$
$\mathcal{N}=8$ string theory). We assume that such frame exists for
$\frac{1}{4}$-BPS black holes. For these black holes, hair modes were studied
in the type IIB frame [5, 26] and shown to be bosonic thereby not spoiling the
positivity conjecture.333In fact, in this frame they provide a more stringent
conjecture than the one we prove in this work.

The previous arguments, which dealt with the horizon of a single black hole,
break down for multi-centered black holes since in this case the whole
configuration can have angular momentum. Since the helicity trace index at a
generic point in the moduli space and for generic charges can receive
contributions from multi-centered black holes, one must either subtract their
contribution to the index or work at a point in the moduli space where there
are no contributions from multi-centered black holes. For the index counting
$\frac{1}{4}$-BPS states in $\mathcal{N}=4$ string theory, the multi-centered
contributions come necessarily from bound states of two $\frac{1}{2}$-BPS
states [27]. These contributions are under control as there is a prescribed
way to extract the single centered contributions from the index [28]. Single
centered black holes, which carry both electric and magnetic charges, exist at
every point in the moduli space and are therefore called immortal dyons. We
call the index counting their degeneracies the immortal dyon index.

Postponing the precise expression of the index to the next section, it now
suffices to know that the charges carried by the $\frac{1}{4}$-BPS states can
be encoded in three integers $(m,n,\ell)$ and we denote immortal dyon index,
the indexed degeneracy for $\frac{1}{4}$-BPS states forming a single centered
black hole, by $d_{*}(m,n,\ell)$. Sen’s conjecture [1] then can be stated as

$d_{*}(m,n,\ell)\,>\,0\,,$ (1.2)

for the values $(m,n,\ell)$ for which there exists a single centered black
hole, which satisfy $m,n\geq 1$ and $4mn-\ell^{2}>0$. Evidence for the
conjecture (1.2) was given in the limit of large charges in [29, 19], for the
first few values of the charge bilinears in [1], and for $m=1,2$ in [30]. The
quantity $d_{*}(m,n,\ell)$ is defined in (2.23) as the Fourier coefficient of
a meromorphic Siegel modular form at a particular point in the Siegel upper-
half space which is charge dependent. For generic values of $(m,n,\ell)$, it
is not straightforwardly computed. However, as shown in Section 2, one can
define the following mock Jacobi form [31] which captures part of the spectrum
with fixed charge bilinear $m$,

$\psi_{m}^{F}(\sigma,v)\;=\;\sum_{\begin{subarray}{c}n,\ell\in\mathbb{Z}\\\
n\geq-1\end{subarray}}c_{m}^{F}(n,\ell)q^{n}y^{\ell}\,,$ (1.3)

where $q=e^{2\pi i\sigma}$ and $y=e^{2\pi iv}$. This provides an alternative
route for computing $d_{*}(m,n,\ell)$ for a restricted set of charge bilinears
$(m,n,\ell)$, since for the ranges $0\leq\ell<2m,2n$, the generating function
$\psi_{m}^{F}(\sigma,v)$ encodes the indexed degeneracies of the single
centered black hole microstates $d_{*}(m,n,\ell)$,

$d_{*}(m,n,\ell)\;=\;(-1)^{\ell+1}\,c^{F}_{m}(n,\ell)\,.$ (1.4)

Therefore, the conjecture (1.2) becomes a prediction for the sign of the
coefficients of a family of mock Jacobi forms. Our main result is the
following.

######  Theorem 1.

Let $m>0$, $n\geq m$ and $0\leq\ell\leq m$, then

$(-1)^{\ell+1}\,c^{F}_{m}(n,\ell)\,>\,0\,.$ (1.5)

The cases $m=1,2$ of this theorem were proved in [30], so in this work we
restrict ourselves to $m\geq 3$.

As we explain in Section 2, as a consequence of the $S$-duality invariance of
the single centered degeneracies, we have

######  Corollary 1.1.

Let $m,n,\ell\in\mathbb{Z}$ with $m,n\geq 1$ and $\Delta=4mn-\ell^{2}>0$.
Then,

$d_{*}(m,n,\ell)\,>\,0\,.$ (1.6)

Or, as stated in the title,

The index computing the single centered degeneracies of $\frac{1}{4}$-BPS
dyons
in type II string theory compactified on $K3\times T^{2}$ is positive.

The main idea behind the proof of the positivity conjecture for the single
centered $\frac{1}{4}$-BPS states comes from the following physical intuition.
The matching between the microscopic and macroscopic answers relies, so far,
on the modularity properties of the index counting the BPS degeneracies, which
allows for an analytic expression of the indexed degeneracies. This analytic
expression, known as the Rademacher expansion, takes the form of a convergent
series with a leading term and infinitely many suppressed corrections whose
sum yields an integer. This expansion is then matched to the localization
computation, where the path integral takes the form of a sum over saddles with
the leading term corresponding to the near-horizon geometry of the black hole
and the infinitely many exponentially suppressed contributions come from
orbifolds of the near-horizon geometry [32, 5, 33, 9, 16] having the same
boundary conditions as the unorbifolded one. The intuition is then that the
saddle corresponding to the near-horizon geometry gives the dominant, positive
contribution to the degeneracy which is not overshadowed by the remaining
ones.

The Rademacher expansion for the coefficients $(-1)^{\ell+1}c_{m}^{F}(n,\ell)$
[34, 35, 36, 37, 38] is given in (2.36). Following the previous discussion,
the proof of Theorem 1 simply consists in identifying the leading terms with a
positive contribution and bounding the remaining infinitely many contributions
against them.

We stress that the result in Corollary 1.1 does not imply that all microstates
are bosonic. It implies that, for each charge, the index receives more
contributions from bosonic states than fermionic ones. To show that all
microstates are bosonic, and furthermore that each of them carries zero
angular momentum, one has to follow a different route, possibly adapting the
results in [39, 40, 41] for $\frac{1}{8}$-BPS states in $\mathcal{N}=8$ string
theory to the case of $\frac{1}{4}$-BPS states in $\mathcal{N}=4$ string
theory.

The results of this work provide further evidence [1] for the description of
$\frac{1}{4}$-BPS states at strong coupling as black holes even for small
charges. A further check would be to find a proof of the conjecture in [1] for
CHL models [42], for which one could use the results about the sign of the
index and the hair modes [43, 44, 26, 45, 46].

The organization of the paper is as follows. In Section 2 we define the index
and give its expression in terms of the Fourier coefficients of a meromorphic
Siegel modular form. We review the connection to mock modular forms and
provide the exact expression for the indexed degeneracies with all the
relevant details for the proof, including some unknown aspects of the
formulae. In Section 3 we prove several small lemmas and combine them for the
proof of Theorem 1 for the cases $m\geq 6$. The proof of the cases $m=3,4,5$
is given in Appendix A.

##  2\. Black hole degeneracies in $\mathcal{N}=4$ string theory

The study of the BPS spectrum in string theory vacua with extended
supersymmetry has uncovered rich mathematical structures. These provide non-
trivial tests for dualities, as well as powerful tools to study black hole
entropy at zero temperature. For the case of type II string theory
compactified on $K3\times T^{2}$ (and supersymmetry preserving orbifolds,
known as CHL models [42]), the study of its BPS spectrum has led to new
findings in mathematics related to mock modularity [31] and is deeply related
to modern forms of moonshine (see [47] for a recent review). In the next
section we introduce the definition of the index counting $\frac{1}{4}$-BPS
states for type II string theory compactified on $K3\times T^{2}$ and refer
the reader to the reviews [29, 48, 49, 50] for more details.

###  2.1. Index definition

Heterotic string theory compactified on $T^{6}$ (or, equivalently, Type II
string theory compactified on $K3\times T^{2}$) has 28 gauge fields, 12 of
them arising from the components of the metric and 2-form field on $T^{6}$ and
16 from the Cartan generators of the heterotic gauge group. The electric and
magnetic charges of a generic state are described by the 28-dimensional
vectors $\vec{Q}$ and $\vec{P}$, respectively.

The vectors $\vec{Q},\vec{P}$ transform in the vector representation of the
$T$-duality group of the theory $O(6,22;\mathbb{Z})$ and therefore with an
$O(6,22;\mathbb{Z})$-invariant matrix one can define the charge bilinears
$\vec{Q}^{2},\vec{P}^{2},\vec{Q}\cdot\vec{P}$, which are invariant under
$T$-duality transformations. The following integer

$I\;=\;\gcd\\{Q_{i}P_{j}-Q_{j}P_{i}\,,\quad 1\leq i,j\leq 28\\}\,,$ (2.1)

is called the torsion of the state carrying charges $(\vec{Q},\vec{P})$. A
state with unit torsion, $I=1$, is completely specified by the charge
bilinears, $\vec{Q}^{2},\vec{P}^{2},\vec{Q}\cdot\vec{P}$ [51]. From now on we
focus on this class of states, since the degeneracies of dyons with torsion
greater than one are related to the degeneracies of unit torsion dyons [52,
53, 54].

The $U$-duality group also includes the $SL(2,\mathbb{Z})$ group which acts on
the doublet $(\vec{Q},\vec{P})$ by matrix multiplication. This group is the
electromagnetic duality group on the heterotic frame, and the modular group of
the $T^{2}$ factor in the type II frame, so that the full duality group
contains the following group

$SL(2,\mathbb{Z})\times O(6,22;\mathbb{Z})\,.$ (2.2)

To simplify the notation, it is standard to define

$m\;=\;\frac{\vec{P}^{2}}{2}\,,\qquad\qquad
n\;=\;\frac{\vec{Q}^{2}}{2}\,,\qquad\qquad\ell\;=\;\vec{Q}\cdot\vec{P}\,,$
(2.3)

which are integers since $\vec{Q}^{2},\vec{P}^{2}\in 2\mathbb{Z}$ with the
appropriate choice of inner product, and their associated discriminant is
given by

$\Delta\;=\;\vec{Q}^{2}\vec{P}^{2}-(\vec{Q}\cdot\vec{P})^{2}\;=\;4mn-\ell^{2}\,,$
(2.4)

which is invariant under the duality transformations (2.2).

The $\frac{1}{2}$-BPS states have charge vectors $\vec{Q},\vec{P}$ which are
parallel and therefore there is a duality frame in which one of them, say
$\vec{P}$, can be set to zero. The partition function for the degeneracies of
$\frac{1}{2}$-BPS states with charge bilinear $n$ is given by [55],

$\frac{1}{\eta^{24}(\sigma)}\;=\;q^{-1}\prod_{n=1}^{\infty}(1-q^{n})^{-24}\;=\;\sum_{n=-1}^{\infty}d(n)q^{n}\,,$
(2.5)

where $q=e^{2\pi i\sigma}$ with Im$(\sigma)>0$.

The indexed degeneracies of $\frac{1}{4}$-BPS states in heterotic string
theory compactified on $T^{6}$ are computed through the sixth helicity trace
index, defined as the following trace [23, 24, 22]

$B_{6}(\vec{Q},\vec{P})\;=\;\frac{1}{6!}\mathrm{Tr}\,(-1)^{F}(2h)^{6}\,,$
(2.6)

over states carrying charges $\vec{Q},\vec{P}$, where the helicity $h$ is the
third component of angular momentum in the rest frame. The $(2h)^{6}$ factor
is introduced to soak up the fermion zero modes associated to the 12 broken
supersymmetry generators. This factor guarantees that only states breaking 12
of the 16 supersymmetries contribute to the index (see [22, 19] for details).
The helicity trace index then takes the form

$B_{6}(\vec{Q},\vec{P})\;=\;\frac{1}{6!}\mathrm{Tr}\left((-1)^{F}(2h)^{6}\right)\;=\;-d(m,n,\ell)\,,$
(2.7)

where $d(m,n,\ell)$ is the Witten index over the remaining degrees of freedom
once the zero modes have been factored out, producing the overall minus sign.
Assuming that there are no extra hair modes, $d(m,n,\ell)$ captures the
indexed degeneracies of $\frac{1}{4}$-BPS states with charge bilinears
$(m,n,\ell)$ which form single centered or two-centered black holes [27].

This index was conjectured [25] and derived and rederived [56, 57, 58] to be
equal to

$d(m,n,\ell)\;=\;\int_{\mathcal{C}}d\rho d\sigma dv\,\frac{e^{-2\pi
i\left(m\rho+n\sigma+\ell\left(v+\frac{1}{2}\right)\right)}}{\Phi_{10}(\rho,\sigma,v)}\,,$
(2.8)

where the integration contour $\mathcal{C}$ is taken to be

$0\,\leq\,\rho_{1},\sigma_{1},v_{1}\,\leq
1\,,\qquad(\rho_{1},\sigma_{1},v_{1})\;=\;\mathrm{Re}(\rho,\sigma,v)\,,$ (2.9)

with the imaginary parts $\rho_{2},\sigma_{2}>0$ and $v_{2}$ kept fixed and
satisfying

$\rho_{2}\sigma_{2}-v_{2}^{2}\,\gg 0\,.$ (2.10)

The function $\Phi_{10}(\rho,\sigma,v)$ is the Igusa cusp form, which is a
Siegel modular form which transforms under $Sp(4,\mathbb{Z})$ transformations
with weight 10, i.e., satisfying

$\Phi_{10}\left((A\Omega+B)(C\Omega+D)^{-1}\right)\;=\;\det(C\Omega+D)^{10}\Phi_{10}\left(\Omega\right)\,,$
(2.11)

where

$\Omega\;=\;\begin{pmatrix}\rho&v\\\ v&\sigma\end{pmatrix}\,\quad\text{ and
}\quad\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\in Sp(4,\mathbb{Z})\,,$ (2.12)

(see Appendix A in [59] for a useful compendium on Siegel modular forms). In
order to define the Igusa cusp form, we first introduce the Jacobi form
$\varphi_{0,1}(\sigma,v)$ of weight $0$ and index 1 under
$SL(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$ transformations (see [60] for an
introduction to Jacobi forms)

$\varphi_{0,1}(\sigma,v)\;=\;4\left(\frac{\vartheta_{2}^{2}(\sigma,v)}{\vartheta_{2}^{2}(\sigma,0)}+\frac{\vartheta_{3}^{2}(\sigma,v)}{\vartheta_{3}^{2}(\sigma,0)}+\frac{\vartheta_{4}^{2}(\sigma,v)}{\vartheta_{4}^{2}(\sigma,0)}\right)\,.$
(2.13)

The functions ${\vartheta_{i}(\sigma,v)}$ for $i=1,2,3,4$ are the classical
Jacobi theta functions,

$\vartheta_{1}(\sigma,v)\;=\;\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{1}{2}\left(n-\frac{1}{2}\right)^{2}}y^{n-\frac{1}{2}}\,,\qquad\vartheta_{2}(\sigma,v)\;=\;\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}\left(n-\frac{1}{2}\right)^{2}}y^{n-\frac{1}{2}}\,,$
$\vartheta_{3}(\sigma,v)\;=\;\sum_{n\in\mathbb{Z}}q^{\frac{n^{2}}{2}}y^{n}\,,\qquad\vartheta_{4}(\sigma,v)\;=\;\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{n^{2}}{2}}y^{n}\,,$

where

$q=e^{2\pi i\sigma}\,,\quad y=e^{2\pi iv}\,.$ (2.14)

The Jacobi form $\varphi_{0,1}(\sigma,v)$ has the following Fourier expansion,

$\varphi_{0,1}(\sigma,v)\;=\;\sum_{\begin{subarray}{c}n,\ell\in\mathbb{Z}\\\
4n-\ell^{2}\geq-1\end{subarray}}C_{0}(4n-\ell^{2})q^{n}y^{\ell}\,,$ (2.15)

where the Fourier coefficients depend only on the combination $4n-\ell^{2}$
due to $\varphi_{0,1}(\sigma,v)$ being a Jacobi form of index 1. The Igusa
cusp form can then be defined as the multiplicative (or Borcherds) lift of
$2\varphi_{0,1}(\sigma,v)$ in the following way [61, 62],

$\Phi_{10}(\rho,\sigma,v)\;=\;pqy\prod_{\begin{subarray}{c}m,n,\ell\in\mathbb{Z}\\\
(m,n,\ell)>0\end{subarray}}(1-p^{m}q^{n}y^{\ell})^{2C_{0}(4mn-\ell^{2})}\,,$
(2.16)

where $(m,n,\ell)>0$ means that either $m>0$ or $m=0$ and $n>0$, or $m=n=0$
and $\ell<0$.

From the definition (2.13) one can see that $\varphi_{0,1}(\sigma,0)=12$ and
therefore

$\sum_{\ell}2C_{0}(4mn-\ell^{2})\;=\;\begin{cases}24\qquad\mathrm{if}\quad
m=0\quad\mathrm{or}\quad n=0\,,\\\ 0\qquad\mathrm{otherwise.}\end{cases}$
(2.17)

This, together with the value $C_{0}(-1)=1$ implies the following behaviour
for $\Phi_{10}(\rho,\sigma,v)$ near $v=0$,

${\Phi_{10}(\rho,\sigma,v)}\;=\;{(2\pi
iv)^{2}}{\eta^{24}(\rho)}{\eta^{24}(\sigma)}+\mathcal{O}(v^{4})\,,$ (2.18)

which means that near $v=0$ the function ${\Phi_{10}(\rho,\sigma,v)}^{-1}$
splits into a product of two $\frac{1}{2}$-BPS partition functions (2.5).

The Igusa cusp form vanishes at $v=0$ and all its images under the group of
$Sp(4,\mathbb{Z})$ transformations, implying that its reciprocal is a
meromorphic function with infinitely many poles. The locus of the poles is
parametrized by five integers $(m^{2},m^{1},j,n_{2},n_{1})$ [63] and is given
by an algebraic equation where the coefficients satisfy a constraint given by
the norm of the vector of integers,

$m^{2}-m^{1}\rho+n_{1}\sigma+n_{2}(\rho\sigma-v^{2})+jv\;=\;0\,,\qquad
j^{2}+4(m^{1}n_{1}+m^{2}n_{2})\;=\;1\,.$ (2.19)

The linear poles are the poles with $n_{2}=0$. They are the only ones
compatible with the defining conditions (2.9) and (2.10) of the contour
$\mathcal{C}$. The imaginary part of the locus of these poles is parametrized
by $SL(2,\mathbb{Z})$ matrices, and can be written as

$pq\sigma_{2}+rs\rho_{2}+(ps+qr)v_{2}\;=\;0\,,\quad\mathrm{where}\quad\begin{pmatrix}p&q\\\
r&s\end{pmatrix}\in SL(2,\mathbb{Z})\,.$ (2.20)

By changing the imaginary parts of the variables in (2.8) one can cross one of
these poles and therefore the value of the indexed degeneracies (2.8) depends
on the integration contour $\mathcal{C}$ through the value of the imaginary
parts of the integration variables $\rho,\sigma,v$. This phenomenon is an
instance of wall-crossing. The indexed degeneracies jump discontinuously as
one crosses a pole, with the jump being equal to the residue at the pole [64,
28, 65]. The poles (2.20) define planes in the space
$(\rho_{2},\sigma_{2},v_{2})$ which divide it in chambers where the indexed
degeneracy is constant. The residues for the linear poles can be computed from
the behaviour of $\Phi_{10}^{-1}$ near $v=0$, given in (2.18), together with
the $SL(2,\mathbb{Z})$ invariance of $\Phi_{10}(\rho,\sigma,v)$ under the
diagonal $SL(2,\mathbb{Z})$ subgroup of $Sp(4,\mathbb{Z})$. For the linear
pole (2.20), the residue of the integrand in (2.8) is

$\\\
(-1)^{\ell+1}(-2rs{n}\,-2pqm\,+(ps+qr){\ell})\,d(r^{2}{n}+\,p^{2}m-pr{\ell})d(s^{2}{n}+q^{2}m-qs{\ell})\,,$
(2.21)

which is consistent with the change in the indexed degeneracies being due to
the (dis)appearance of bound states of two $\frac{1}{2}$-BPS states [64, 28].

For each set of charges $m,n,\ell$ there is a special integration contour [28]
defined by

$\mathcal{C}_{*}=\\{\,0\,\leq\,\rho_{1},\sigma_{1},v_{1}\,\leq
1\,,\quad\rho_{2}\;=\;2n\Lambda,\,\sigma_{2}\;=\;2m\Lambda,\,v_{2}\;=\;-\ell\Lambda\,\\}\,,$
(2.22)

where $\Lambda\gg 0$, such that no linear pole contributes to the index. This
contour is called the attractor contour since it coincides with the one given
by the values that the moduli take at the black hole horizon through the
attractor mechanism [66]. With this choice of moduli, the degeneracy captured
by the index is the single centered one. The immortal dyon index is then
defined as

$d_{*}(m,n,\ell)\;=\;\int_{\mathcal{C}_{*}}d\rho d\sigma dv\,\frac{e^{-2\pi
i\left(m\rho+n\sigma+\ell\left(v+\frac{1}{2}\right)\right)}}{\Phi_{10}(\rho,\sigma,v)}\,,$
(2.23)

which, following the discussion in the introduction, is conjectured to
coincide with the degeneracy of states forming the black hole horizon and
therefore it must be positive as summarized in (1.2).

It follows from the definition of the contour $\mathcal{C}_{*}$ (2.22) that
the number of BPS indexed degeneracies associated to a single centered black
hole given by (2.23) is invariant under the duality group (2.2). The
discriminant $\Delta$ is also invariant under the duality transformations
(2.2), which leads to the following observations. The value of the single
centered degeneracies depends only on the $SL(2,\mathbb{Z})$ equivalence class
of the discriminant $\Delta$, while the inequivalent number of solutions is
given by the class number associated to the discriminant of $(m,n,\ell)$ [67].
Roughly, the class number grows like $\sqrt{\Delta}$, which is the same growth
for the area, and therefore the entropy, of the single centered black hole.

The coefficients of the Fourier expansion in $p\;=\;e^{2\pi i\rho}$ of the
Siegel modular form are two-variable functions which transform as Jacobi forms
due to the inherited symmetries from the symplectic transformations preserving
the Fourier expansion. That is, the coefficients in the Fourier expansion

$\frac{1}{\Phi_{10}(\rho,\sigma,v)}\;=\;\sum_{m=-1}^{\infty}\psi_{m}(\sigma,v)p^{m}\,,$
(2.24)

associated to $p^{m}$, which are the generating functions for the degeneracies
for fixed charge bilinear $m$, are Jacobi forms of weight $-10$ and index $m$.
This means that they satisfy the following transformations,

$\begin{split}\psi_{m}\left(\frac{a\sigma+b}{c\sigma+d},\frac{v}{c\sigma+d}\right)&\;=\;(c\sigma+d)^{-10}e^{2\pi
im\frac{c^{2}v}{c\sigma+d}}\,\psi_{m}(\sigma,v)\,,\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})\,,\\\
\psi_{m}\left(\sigma,v+\lambda\sigma+\mu\right)&\;=\;e^{-2\pi
im(\lambda^{2}\sigma+2\lambda
v)}\,\psi_{m}(\sigma,v)\,,\quad\lambda,\mu\in\mathbb{Z}\,.\end{split}$ (2.25)

The relation between $\Phi_{10}^{-1}$ and mock Jacobi forms arises by writing
the Jacobi forms in the Fourier expansion of $\Phi_{10}^{-1}$ as a sum of a
meromorphic function and a non-meromorphic one in the following canonical form
[31],

$\frac{1}{\Phi_{10}(\rho,\sigma,v)}\;=\;\sum_{m=-1}^{\infty}\left(\psi_{m}^{F}(\sigma,v)+\psi_{m}^{P}(\sigma,v)\right)p^{m}\,,$
(2.26)

where the function $\psi_{m}^{P}(\sigma,v)$, called the polar, is built out of
the poles of $\Phi_{10}(\rho,\sigma,v)^{-1}$ in the $(\sigma,v)$ plane,

$\psi_{m}^{P}(\sigma,v)\;=\;\frac{d(m)}{\eta^{24}(\sigma)}\mathcal{A}_{2,m}(\sigma,v)\;=\;\frac{d(m)}{\eta^{24}(\sigma)}\sum_{s\in\mathbb{Z}}\frac{q^{ms^{2}+s}\,y^{2ms+1}}{(1-q^{s}y)^{2}}\,.$
(2.27)

The Appell-Lerch sum $\mathcal{A}_{2,m}(\sigma,v)$ is a mock Jabobi form [68],
which means that it transforms properly under modular transformations only
after the addition of the non-holomorphic Eichler integral of another modular
object, which is called the shadow of the mock Jacobi form. More precisely,

$\widehat{\mathcal{A}}_{2,m}(\sigma,v)\;=\;\mathcal{A}_{2,m}(\sigma,v)+\sqrt{\frac{m}{4\pi}}\sum_{\ell\in\mathbb{Z}/2m\mathbb{Z}}\left(\vartheta_{m,\ell}^{0}\right)^{*}(\sigma)\vartheta_{m,\ell}(\sigma,v)\,,$
(2.28)

transforms as a Jacobi form of weight 2 and index $m$, where
$\vartheta_{m,\ell}(\sigma,v)$ are the index $m$ theta functions

$\vartheta_{m,\ell}(\sigma,v)\;=\;\sum_{\begin{subarray}{c}r\in\mathbb{Z}\\\
r=\ell\mod 2m\end{subarray}}q^{\frac{r^{2}}{4m}}y^{r}\,,$ (2.29)

and $\left(\vartheta_{m,\ell}^{0}\right)^{*}(\sigma)$ denotes the non-
holomorphic Eichler integral of the shadow
$\vartheta^{0}_{m,\ell}(\sigma)=\vartheta_{m,\ell}(\sigma,0)$,

$\left(\vartheta_{m,\ell}^{0}\right)^{*}(\sigma)\;=\;\left(\frac{i}{2\pi}\right)^{\frac{1}{2}}\int_{-\overline{\sigma}}^{\infty}(z+\sigma)^{-\frac{3}{2}}\,\overline{\vartheta_{m,\ell}^{0}(-\overline{z})}\,dz\,.$
(2.30)

Since the sum $\psi_{m}^{F}(\sigma,v)+\psi_{m}^{P}(\sigma,v)$ transforms as a
Jacobi form, the function $\psi_{m}^{F}(\sigma,v)$ is also a mock Jacobi444The
function $\psi_{m}^{P}(\sigma,v)$ is the product of a modular form and a mock
Jacobi form and therefore being precise the functions $\psi_{m}^{P}(\sigma,v)$
and $\psi_{m}^{F}(\sigma,v)$ are mixed mock Jacobi forms. form with shadow
having the opposite sign that for $\psi_{m}^{P}(\sigma,v)$. The mock Jacobi
form $\psi_{m}^{F}(\sigma,v)$ is called the finite part and since it is
holomorphic, it has the following well-defined Fourier expansion

$\psi_{m}^{F}(\sigma,v)\;=\;\sum_{\begin{subarray}{c}n,\ell\in\mathbb{Z}\\\
n\geq-1\end{subarray}}c_{m}^{F}(n,\ell)q^{n}y^{\ell}\,,$ (2.31)

having no wall-crossing in the $(\sigma,v)$ plane. Since
$\psi_{m}^{F}(\sigma,v)$ is a holomorphic function which satisfies the second
transformation in (2.25), it has the following theta expansion,

$\psi_{m}^{F}(\sigma,v)\;=\;\sum_{\ell\in\mathbb{Z}/2m\mathbb{Z}}f_{m,\ell}(\sigma)\vartheta_{m,\ell}(\sigma,v)\,,$
(2.32)

where $f_{m,\ell}(\sigma)$ are (mixed) mock modular forms. Thus, the
coefficients $c_{m}^{F}(n,\ell)$ can be seen as the Fourier coefficients of
the mixed mock modular forms $f_{m,\ell}(\sigma)$ [31].

###  2.2. Single centered degeneracies

The finite part $\psi_{m}^{F}$ as written in (2.31) is the generating function
with fixed magnetic bilinear $m$ for the single centered (or immortal) indexed
degeneracy of states carrying charge bilinears $m,n,\ell$ which satisfy
$0\leq\ell<2m,2n$ [31].

To see it, consider the chamber in the Siegel upper-half space bounded by the
walls $v_{2}=0$, $v_{2}=-\rho_{2}$ and $v_{2}=-\sigma_{2}$, known as the
$\mathcal{R}$ chamber [69]. This chamber coincides with the attractor chamber
for the charge bilinears which satisfy $0\leq\ell<2m,2n$ [1]. By performing
the Fourier expansion of $\Phi_{10}(\rho,\sigma,v)^{-1}$ in $p=e^{2\pi
i\rho}$, we have implicitly assumed that
$\rho_{2}\gg\sigma_{2},\mathinner{\\!\left\lvert v_{2}\right\rvert}$.
Expanding the Appell-Lerch sum appearing in (2.27),

$\mathcal{A}_{2,m}(\sigma,v)\;=\;\frac{1}{2}\sum_{s,\ell\in\mathbb{Z}}\ell\,\left[\mathrm{sign}(s+\frac{v_{2}}{\sigma_{2}})+\mathrm{sign}\,\ell\right]\,q^{ms^{2}+\ell
s}\,y^{2ms+\ell}\,,$ (2.33)

we see that for $0<-v_{2}/\sigma_{2}<1$, the Fourier coefficients of
$\mathcal{A}_{2,m}(\sigma,v)$ with $0<\ell<2m$ vanish, and the $\ell=0$
Fourier coefficient is zero. Thus, in the $\mathcal{R}$ chamber, where
$0<-v_{2}/\sigma_{2}<1$, the indexed degeneracies (2.8) with $0\leq\ell<2m$
are given by the Fourier coefficients $c_{m}^{F}(n,\ell)$ of the finite part
$\psi_{m}^{F}(\sigma,v)$. These coincide with the immortal degeneracies as
long as $\ell<2n$.

To further illustrate the necessity of these constraints, we provide an
example with $\ell\geq 2n$ where the contribution of multi-centered black
holes spoils the positivity of the degeneracies counted by
$\psi_{m}^{F}(\sigma,v)$ (this was emphasized recently in [70] for the case
$\ell\geq 2m$). Consider the set of charges $(m,n,\ell)=(10,1,6)$ with
$\Delta=4$ which satisfies $0\leq\ell<2m$ but does not satisfy the condition
$\ell<2n$. The corresponding Fourier coefficient of $\psi_{m}^{F}(\sigma,v)$
is

$c_{10}^{F}(1,6)\;=\;1393564656\,,$ (2.34)

and satisfies $(-1)^{6+1}c_{10}^{F}(1,6)<0$. So by not imposing the condition
$\ell<2n$, the positivity conjecture in this case is not satisfied due to the
contribution of multi-centered black holes to the index. The positivity is
recovered by subtracting the contributions from multi-centered configurations.
This can be done by subtracting the residues of $\Phi_{10}^{-1}$ at the poles
one crosses when going from the attractor chamber for the charges
$(m,n,\ell)=(10,1,6)$ to the $\mathcal{R}$ chamber. We obtain

$d_{*}(10,1,6)\;=\;(-1)^{7}\left(1393564656-4d(1)d(5)-2d(1)d(2)\right)\;=\;50064\,,$
(2.35)

which is equal to value for the degeneracy of the representative of the
equivalence class for which the attractor chamber is the $\mathcal{R}$
chamber, which is given by $(m,n,\ell)=(1,1,0)$ and satisfies
$(-1)c_{1}^{F}(1,0)=d_{*}(1,1,0)=d_{*}(10,1,6)$.

###  2.3. Exact expressions for the degeneracies

The Fourier coefficients of the finite part $\psi_{m}^{F}(\sigma,v)$ possess
an exact expression in the form of a generalized Rademacher expansion [35, 38]
due to the mixed-mock nature of the function $\psi_{m}^{F}(\sigma,v)$,

$\begin{split}&(-1)^{\ell+1}c_{m}^{F}(n,\ell)\;=\;\\\
&(-1)^{\ell}\,i^{1/2}\,\sum_{\gamma=1}^{+\infty}\sum_{\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}}\left(2\pi\sum_{\begin{subarray}{c}\tilde{n}\geq-1,\\\
\tilde{\Delta}<0\end{subarray}}c^{F}_{m}(\tilde{n},\tilde{\ell})\frac{{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{\gamma
m}\sqrt{\Delta|\tilde{\Delta}|}\right)\right.\\\
&\qquad\qquad-\delta_{\tilde{\ell},0}\sqrt{2m}\,d(m)\frac{{\rm
Kl}(\frac{\Delta}{4m},-1;\gamma,\psi)_{\ell
0}}{\sqrt{\gamma}}\left(\frac{4m}{\Delta}\right)^{6}I_{12}\left(\frac{2\pi}{\gamma}\sqrt{\frac{\Delta}{m}}\right)\\\
&\qquad\qquad+\frac{1}{2\pi}d(m)\sum_{\begin{subarray}{c}g\in\mathbb{Z}/2m\gamma\mathbb{Z}\\\
g=\tilde{\ell}\text{ mod }2m\end{subarray}}\frac{{\rm
Kl}(\frac{\Delta}{4m},-1-\frac{g^{2}}{4m};\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma^{2}}\\\
&\qquad\qquad\left.\left(\frac{4m}{\Delta}\right)^{25/4}\int_{-1/\sqrt{m}}^{1/\sqrt{m}}dx^{\prime}f_{\gamma,g,m}(x^{\prime})(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right)\right),\end{split}$ (2.36)

where the discriminants $\Delta,\tilde{\Delta}$ are given by

$\Delta\;=\;4mn-\ell^{2}\,,\qquad\tilde{\Delta}\;=\;4m\tilde{n}-\tilde{\ell}^{2}\,,$
(2.37)

and $d(m)$ is the $m$-th coefficient of $\eta^{-24}(\sigma)$,

$\frac{1}{\eta^{24}(\sigma)}\;=\;\frac{1}{q\prod\limits_{n=1}^{\infty}(1-q^{n})^{24}}\
=\sum_{n=-1}^{+\infty}d(n)\,e^{2\pi i\sigma n},\;\;\;q\;=\;e^{2\pi
i\sigma}\,.$ (2.38)

The generalized Kloosterman sum has the following expression,

${\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}\;=\;\sum_{\begin{subarray}{c}0\leq-\delta<\gamma\\\
(\delta,\gamma)=1,\alpha\delta=1\text{ mod }\gamma\end{subarray}}e^{2\pi
i\left(\frac{\alpha}{\gamma}\frac{\tilde{\Delta}}{4m}+\frac{\delta}{\gamma}\frac{\Delta}{4m}\right)}\,{\psi}(\Gamma)_{\tilde{\ell}\ell}\,,$
(2.39)

where $\Gamma$ denotes the $SL(2,\mathbb{Z})$ matrix associated to $\gamma$,

$\Gamma\;=\;\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}\in
SL(2,\mathbb{Z})\,,$ (2.40)

and the multiplier system ${\psi}(\Gamma)_{{\tilde{\ell}}\ell}$ is the one of
the theta functions $\vartheta_{m,\ell}(\sigma,v)$ defined in (2.29), which
for $\gamma\geq 1$ it has the following form [71],

${\psi}(\Gamma)_{{\tilde{\ell}}\ell}\;=\;\frac{1}{\sqrt{2m\gamma
i}}\sum_{T\in\mathbb{Z}/\gamma\mathbb{Z}}e^{2\pi
i\left(\frac{\alpha}{\gamma}\frac{({\tilde{\ell}}-2mT)^{2}}{4m}-\frac{\ell({\tilde{\ell}}-2mT)}{2m\gamma}+\frac{\delta}{\gamma}\frac{\ell^{2}}{4m}\right)}\,.$
(2.41)

Finally, the function $f_{\gamma,g,m}(x^{\prime})$ inside the integral is
given by

$f_{\gamma,g,m}(x^{\prime})\,=\sum_{\begin{subarray}{c}p\in\mathbb{Z}\\\
2m\gamma p+g\neq 0\end{subarray}}\frac{\gamma^{2}}{\left(x^{\prime}-i\gamma
p-i\frac{g}{2m}\right)^{2}}=\left\\{\begin{aligned}
\frac{\pi^{2}}{\sinh^{2}\left(\frac{\pi x^{\prime}}{\gamma}-\frac{\pi
ig}{2m\gamma}\right)}&\;\;\text{ if}\;\;\;g\neq 0\text{ mod }2m\gamma\,,\\\
\frac{\pi^{2}}{\sinh^{2}\left(\frac{\pi
x^{\prime}}{\gamma}\right)}-\frac{\gamma^{2}}{x^{\prime 2}}&\;\;\text{
if}\;\;\;g=0\text{ mod }2m\gamma\,.\end{aligned}\right.$ (2.42)

Two different proofs of this formula can be found in [35] and in [38]. In the
derivation of the latter, the following explicit formula is found for the
coefficients $c^{F}_{m}(\tilde{n},\tilde{\ell})$ with $\tilde{\Delta}<0$,

$c^{F}_{m}(\tilde{n},\tilde{\ell})=\sum_{\begin{subarray}{c}r,s>0\\\
q\in\mathbb{Z}/s\mathbb{Z},\\\ ps-qr=1\\\
0\leq\frac{\tilde{\ell}}{2m}-\frac{q}{s}<\frac{1}{rs}\end{subarray}}(-2rs\tilde{n}\,-2pqm\,+(ps+qr)\tilde{\ell})\,d(r^{2}\tilde{n}+\,p^{2}m-pr\tilde{\ell})d(s^{2}\tilde{n}+q^{2}m-qs\tilde{\ell})\,.$
(2.43)

We expand on the meaning and implications of this formula in the following
subsection, which reviews known results and expands on some unknown aspects of
the formula which are used later for the proof of the positivity conjecture.

###  2.4. Polar coefficients

The coefficients $c^{F}_{m}(\tilde{n},\tilde{\ell})$ with
$\tilde{\Delta}=4m\tilde{n}-\tilde{\ell}^{2}<0$ appearing in (2.36) are called
polar coefficients. From the expression for the Rademacher expansion (2.36),
one can see that, together with the $m$-th coefficient of
$\eta^{-24}(\sigma)$, they completely determine the coefficients
$c^{F}_{m}({n},{\ell})$ with ${\Delta}=4m{n}-{\ell}^{2}>0$ of the mock Jacobi
form $\psi_{m}^{F}(\sigma,v)$.

A weakly holomorphic modular form $f(\sigma)$ for $SL(2,\mathbb{Z})$ (see,
e.g., [31]) has by definition a Fourier expansion of the form

$f(\sigma)\;=\;\sum_{n=n_{0}}^{\infty}a_{n}q^{n}\,,\qquad q=e^{2\pi i\sigma},$
(2.44)

where the finite number of coefficients $a_{n}$ with $n<0$ are called polar
coefficients, since they are responsible for the exponential growth of
$f(\sigma)$ as $\sigma\to i\infty$. That is, for the pole at the cusp
$i\infty$. The Rademacher expansion for forms of negative weight consists in
exploiting this behaviour to express the rest of the coefficients in terms of
the polar ones (see, e.g., [72, 73]). Thus, after finding the polar
coefficients by some means, the remaining infinitely-many coefficients are
determined through the Rademacher expansion. In general, given a set of weakly
holomorphic modular forms, one has to compute the polar coefficients on a
case-by-case basis. The case at hand is different in that the mock Jacobi
forms indexed by $m$ arise from a single Siegel modular form, and therefore
one might expect to find a general formula for the polar coefficients of this
family. This is indeed the case.

The polar coefficients $c^{F}_{m}(\tilde{n},\tilde{\ell})$ were systematically
studied in [36] following [69], where it was shown that they are
algorithmically computable and given by a finite sum involving
$SL(2,\mathbb{Z})$ matrices and the Fourier coefficients of
$\eta^{-24}(\sigma)$. The structure underlying the sum was elucidated in [37],
where it was found that the $SL(2,\mathbb{Z})$ matrices appearing in the sum
are the ones that are built out of the continued fraction expansion of
$\frac{\tilde{\ell}}{2m}$. The same continued fraction structure emerged in
[38] in the computation of the degeneracies as an infinite sum over residues
of $\Phi_{10}^{-1}$, yielding the expression (2.43). This equation can be
rewritten as

$c_{m}^{F}(\tilde{n},\tilde{\ell})_{\tilde{\Delta}<0}\;=\;\sum_{G\in
W(m,\tilde{n},\tilde{\ell})}\ell_{G}\,d(m_{G})d(n_{G})\,,$ (2.45)

where $(m_{G},n_{G},\ell_{G})$ are the images under $SL(2,\mathbb{Z})$ of the
triplet $(m,\tilde{n},\tilde{\ell})$,

$\begin{split}n_{G}&\;=\;s^{2}\tilde{n}+q^{2}m-qs\tilde{\ell}\,,\\\
m_{G}&\;=\;r^{2}\tilde{n}+\,p^{2}m-pr\tilde{\ell}\,,\\\
\ell_{G}&\;=\;-2rs\tilde{n}\,-2pqm\,+(ps+qr)\tilde{\ell}\,,\end{split}$ (2.46)

and $W(m,\tilde{n},\tilde{\ell})$ is the subset of $SL(2,\mathbb{Z})$
consisting of matrices which satisfy the continued fraction condition for
$\frac{\tilde{\ell}}{2m}$ with a non-zero contribution to
$c_{m}^{F}(\tilde{n},\tilde{\ell})$,

$\begin{split}&W(m,\tilde{n},\tilde{\ell})\;=\;\\\
&\left\\{G=\begin{pmatrix}p&q\\\ r&s\end{pmatrix}\in
SL(2,\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}r,s>0,\,0\leq\frac{\tilde{\ell}}{2m}-\frac{q}{s}<\frac{1}{rs},\;d(m_{G}),d(n_{G})\geq-1\right\\}\,,\end{split}$
(2.47)

where the continued fraction condition is given by the inequalities

$0\,\leq\,\frac{\tilde{\ell}}{2m}-\frac{q}{s}\,<\,\frac{1}{rs}\,.$ (2.48)

We call these inequalities the continued fraction condition since they are
satisfied by all the matrices built out of the successive convergents and
semiconvergents of the continued fraction of the rational number
$\frac{\tilde{\ell}}{2m}$. Given the continued fraction of
$\frac{\tilde{\ell}}{2m}$,

$\frac{\tilde{\ell}}{2m}\;=\;a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+a_{N}}}}\;=\;[a_{0};a_{1},\dots,a_{N}]\,,$
(2.49)

where $a_{I}\geq 1$ for $I\geq 1$, the convergents are the rational numbers
$[a_{0};a_{1},\dots,a_{K}]$ for $K\leq N$, while the semiconvergents are the
numbers of the form $[a_{0};a_{1},\dots,a_{K-1},J]$ with $J\leq a_{K}$.

The computation of the polar coefficients is reduced to the characterization
of the set $W(m,\tilde{n},\tilde{\ell})$. In the Rademacher formula (2.36) one
is instructed to sum $\tilde{\ell}$ over a set of representatives of
$\mathbb{Z}/2m\mathbb{Z}$. For simplicity we can take it to be
$-m<\tilde{\ell}\leq m$. Since the mock Jacobi form has even weight, it
satisfies the following identity

$\psi_{m}^{F}(\sigma,-v)\;=\;\psi_{m}^{F}(\sigma,v)\,,$ (2.50)

which implies that

$c_{m}^{F}(n,-\ell)\;=\;c_{m}^{F}(n,\ell)\,.$ (2.51)

Thus, we can reduce the range of $\tilde{\ell}$ in the polar coefficients
$c_{m}^{F}(\tilde{n},\tilde{\ell})$ to $0\leq\tilde{\ell}\leq m$. For
$\tilde{\ell}$ in this range, the set $W(m,\tilde{n},\tilde{\ell})$ contains
the matrix

$U\;=\;\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}\,,$ (2.52)

since the entries of the matrix $U$ satisfy the conditions (2.48) and the
contribution of this matrix to the polar coefficient
$c_{m}^{F}(\tilde{n},\tilde{\ell})$ is given by

$\ell_{U}\,d(m_{U})d(n_{U})\;=\;(\tilde{\ell}-2\tilde{n})d(m+\tilde{n}-\tilde{\ell})d(\tilde{n})\,,$
(2.53)

which is non-zero as long as $\tilde{n}\geq-1$ and $0\leq\tilde{\ell}\leq m$.
Therefore, the set $W(m,\tilde{n},\tilde{\ell})$ is non-empty, as it always
contains the $U$ matrix. We define the first contribution to the polar
coefficient as

$\hat{c}^{F}_{m}(\tilde{n},\tilde{\ell})\;=\;(\tilde{\ell}-2\tilde{n})d(m+\tilde{n}-\tilde{\ell})d(\tilde{n})\,.$
(2.54)

This corresponds to the contribution from the leading pole to the degeneracies
[74, 56, 32, 38], or, equivalently, to the contribution from the
AdS${}_{2}\times S^{2}$ saddle to the gravitational path integral [16], as
first shown in [12]. The other saddles, which are subleading, correspond to
orbifolds of the near-horizon geometry [32, 33, 16].

The contribution from the other terms in $W(m,\tilde{n},\tilde{\ell})$ follows
from the continued fraction description. Since $0\leq\tilde{\ell}\leq m$, the
first coefficient of the continued fraction of $\frac{\tilde{\ell}}{2m}$
vanishes, $a_{0}=0$. Ignoring the $a_{0}$ coefficient,555The matrix associated
to $a_{0}$ is $T^{a_{0}}$, which does not satisfy the conditions in (2.47).
from the remaining coefficients $a_{I}$ one can build the following ordered
set of $SL(2,\mathbb{Z})$ matrices

$\begin{pmatrix}1&0\\\ 1&1\end{pmatrix},\,\dots,\,\begin{pmatrix}1&0\\\
1&1\end{pmatrix}^{a_{1}},\,\begin{pmatrix}1&0\\\
1&1\end{pmatrix}^{a_{1}}\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\dots,\,\begin{pmatrix}1&0\\\
1&1\end{pmatrix}^{a_{1}}\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}^{a_{2}},\dots$
(2.55)

Note that the first matrix is the $U$ matrix (2.52). In general, a matrix in
the set (2.55) has the form

$\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}^{a_{1}}\begin{pmatrix}1&1\\\
0&1\end{pmatrix}^{a_{2}}\cdots\begin{pmatrix}1&1\\\
0&1\end{pmatrix}^{a_{K}}\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}^{J}$ (2.56)

with $K$ even, $K\leq N$ and $J\leq a_{K+1}$ (with $a_{N+1}=0$). There are
$\sum_{i=1}^{N}a_{i}\leq 2m$ matrices of this form, and all of them satisfy
the continued fraction condition (2.48), furnishing all possible elements of
the set $W(m,\tilde{n},\tilde{\ell})$ (see [37] for a detailed description and
possible subtleties when $\tilde{\ell}=m$, which do not affect the later
analysis).

From this point of view, the reason why the saddles associated to the matrices
which are not the $U$ matrix are subleading is that the other possible
elements in the set $W(m,\tilde{n},\tilde{\ell})$ have smaller arguments in
the coefficients of $\eta^{-24}(\sigma)$. Therefore, their contribution to the
polar coefficient is exponentially suppressed with respect to the first one
(2.54). To see that the arguments of the coefficients of $\eta^{-24}(\sigma)$
decrease, first notice that we can write

$\frac{n_{G}}{s^{2}}\;=\;\frac{\tilde{\Delta}}{4m}+m\left(\frac{\tilde{\ell}}{2m}-\frac{q}{s}\right)^{2}\,,\qquad\frac{m_{G}}{r^{2}}\;=\;\frac{\tilde{\Delta}}{4m}+m\left(\frac{p}{r}-\frac{\tilde{\ell}}{2m}\right)^{2}\,.$
(2.57)

At each step in building the matrices of the form (2.56), either $r$ or $s$
increases, making the difference $\frac{p}{r}-\frac{\tilde{\ell}}{2m}$ or
$\frac{\tilde{\ell}}{2m}-\frac{q}{s}$ smaller by the properties of the
continued fraction. Since $\tilde{\Delta}<0$, when $r$ increases $m_{G}$
becomes smaller and when $s$ increases $n_{G}$ becomes smaller. In other
words, writing the pairs $(m_{G},n_{G})$ by $(m_{i},n_{i})$ with
$i=1,\dots,i_{\mathrm{max}}$, where they are ordered by their appearance from
the continued fraction, either $m_{i+1}$ or $n_{i+1}$ is strictly smaller than
$m_{i}$ or $n_{i}$, respectively. Furthermore, the finiteness of the sum in
(2.43) follows from the fact that the Fourier coefficients $d(r)$ vanish for
$r<-1$.

There is a more geometric way to see the decrease of the quantities
$m_{G},n_{G}$ at each step in the continued fraction algorithm. One can check
that treating $n_{G}$ ($m_{G}$) as a function of $q,s$ (resp. $p,r$),

$\left(p\frac{\partial}{\partial q}+r\frac{\partial}{\partial
s}\right)n_{G}\;=\;-\ell_{G}\,,\qquad\left(q\frac{\partial}{\partial
p}+s\frac{\partial}{\partial r}\right)m_{G}\;=\;-\ell_{G}\,.$ (2.58)

The first equation can be written as $(p,r)\cdot\nabla n_{G}(q,s)=-\ell_{G}$,
so that the derivative of $n_{G}$ in the $(q,s)$ plane in the direction
$(p,r)$ is given by $-\ell_{G}$. For the matrices satisfying the continued
fraction condition, one has

$0\,\leq\,\frac{\tilde{\ell}}{2m}-\frac{q}{s}\,<\,\frac{1}{rs}\,\Rightarrow\,\ell_{G}>0\,.$
(2.59)

Thus, for the pairs $(p,r)$ and $(q,s)$ forming an $SL(2,\mathbb{Z})$ matrix
and satisfying the continued fraction condition, $(p,r)\cdot\nabla n_{G}(q,s)$
and $(q,s)\cdot\nabla m_{G}(p,r)$ are always negative. Given the
semiconvergents $\frac{p}{r}$ and $\frac{q}{s}$ of $\frac{\tilde{\ell}}{2m}$,
the next semiconvergent in the continued fraction algorithm, which substitutes
either $\frac{p}{r}$ or $\frac{q}{s}$ in the successive $SL(2,\mathbb{Z})$
matrix, is given by Farey addition

$\frac{p}{r}\oplus\frac{q}{s}\;=\;\frac{p+q}{r+s}\,.$ (2.60)

Therefore, $(p,q)$ changes in the $(q,s)$ direction and $(p,r)$ changes in the
$(q,s)$ direction, along which $m_{G}$ and $n_{G}$ are decreasing by equations
(2.58) and (2.59).666There is an even more visual way to see the decrease of
$m_{G}$ and $n_{G}$ by using the topograph of a quadratic form, a visual
representation of a quadratic form introduced by Conway in [75]. One can view
the triplet $(m,\tilde{n},\tilde{\ell})$ with
$\tilde{\Delta}=4m\tilde{n}-\tilde{\ell}^{2}<0$ as defining the indefinite
binary quadratic form $Q(x,y)=mx^{2}-\tilde{\ell}xy+\tilde{n}y^{2}$, since
$(m,\tilde{n},\tilde{\ell})$ and $Q(x,y)$ transform in the same way under
$SL(2,\mathbb{Z})$. It is shown in [75] that there is a river in the topograph
connecting the zeros of $Q(x,y)$, given by
$\frac{x}{y}=\frac{\tilde{\ell}\pm\sqrt{|\tilde{\Delta}|}}{2m}$, along which
$Q(x,y)$ is periodic. Furthermore, there is a result, called the Climbing
Lemma, which states that as one moves away from the river, $Q(x,y)$ is
strictly increasing or decreasing. The continued fraction of
$\frac{\tilde{\ell}}{2m}$ defines a path in the topograph transverse to the
river, from the positive to the negative side. Therefore, by the Climbing
Lemma the values that the quadratic form takes along the path, which
correspond to $m_{G}$ or $n_{G}$, are decreasing. In other words, the path
along the topograph defined by the continued fraction of
$\frac{\tilde{\ell}}{2m}$ goes downhill.

We conclude this section by characterizing the possible values of the
discriminant $\tilde{\Delta}<0$ and the coefficients with the most negative
one. The Fourier coefficients $c_{m}^{F}(\tilde{n},\tilde{\ell})$ of the mock
Jacobi form $\psi_{m}^{F}$, are only non-zero for $\tilde{n}\geq-1$. Since
they also satisfy the elliptic symmetry

$c_{m}^{F}(\tilde{n},\tilde{\ell})\;=\;c_{m}^{F}(\tilde{n}+mT^{2}+T,\tilde{\ell}+2mT)\,,$
(2.61)

for any $T\in\mathbb{Z}$, this implies that they are non-zero only for

$\tilde{\Delta}\;=\;4m\tilde{n}-\tilde{\ell}^{2}\,\geq\,-4m-m^{2}\,.$ (2.62)

The polar coefficients $c_{m}^{F}(\tilde{n},\tilde{\ell})$ with
$\tilde{\Delta}<0$ saturating the previous bound are the extreme polar
coefficients. Their contribution to the Rademacher expansion is the one with
the largest argument for the modified Bessel function of the first kind with
index $23/2$. In the Rademacher formula (2.36), one is instructed to sum
$\tilde{\ell}$ over the equivalence class $\mathbb{Z}/2m\mathbb{Z}$. As
earlier, for simplicity we can take it to be $-m<\tilde{\ell}\leq m$, and in
this case the extreme polar coefficient is

$c_{m}^{F}(-1,m)\;=\;\hat{c}_{m}^{F}(-1,m)\;=\;m+2\,,$ (2.63)

which indeed has discriminant $\tilde{\Delta}=-4m-m^{2}$. The value (2.63) is
easily computed by noting that $W(m,-1,m)$ has only one element, $U$, since
any further contribution must contain $d(r)$ with $r<-1$.

##  3\. Proof of positivity of the index

###  3.1. Main idea

Since the single centered degeneracies are $S$-duality invariant, it is enough
to prove the positivity for the representatives of one $S$-duality equivalence
class. That is, it is enough proving it for states satisfying $n\geq m$,
$0\leq\ell<2m$, and then it is automatically proved for all the
$SL(2,\mathbb{Z})$ images. Furthermore, since the invariance group can be
enlarged to $GL(2,\mathbb{Z})$ (see, e.g. [59], or notice that the Igusa cusp
form has even weight), it is enough to prove it for charge configurations
satisfying

$n\geq m,\hskip 22.76219pt0\leq\ell\leq m,$ (3.1)

which for $m\geq 1$ automatically satisfy $\Delta>0$.

Thus, to prove that the indexed degeneracies of single centered $\frac{1}{4}$
BPS states in $\mathcal{N}=4$ string theory (given by (2.23)) are positive, we
use the Rademacher expansion (2.36) for the set of representatives defined by
(3.1). The Rademacher expansion takes the form of a leading term plus
exponentially suppressed corrections given by the sum over $\gamma$, which
converge to an integer. Therefore, the crux of the proof consists in showing
that the leading term is positive.

Figure 1. Walls of marginal stability in the upper half plane parametrized by
$\tau=-\frac{v_{2}}{\sigma_{2}}+i\frac{\sqrt{\rho_{2}\sigma_{2}-v_{2}^{2}}}{\sigma_{2}}$,
with $\rho_{2}\sigma_{2}-v_{2}^{2}\gg 1$, which for the attractor values
becomes $\tau=\frac{\ell}{2m}+i\frac{\sqrt{\Delta}}{2m}$. The $\mathcal{R}$
chamber is the one delimited by the walls connecting $0$, $1$ and $i\infty$.
The delimited area in grey is a fundamental domain for $GL(2,\mathbb{Z})$,
which lies inside the $\mathcal{R}$ chamber and contains the attractor values
allowed by our choice of representatives (3.1).

Our proof is similar to the proof of Bringmann-Murthy [30] for the case $m=2$.
They show that the term given by the most polar state in the non-mock part of
the Rademacher expansion (the first line in the right hand side of (2.36), as
opposed to the second and third lines which are the mock part since they
originate from the anomalous modular transformation) gives the leading
contribution to the degeneracy, being always positive and much larger than the
rest of the terms combined. Instead, we identify several contributions which
are positive, and their sum is greater than the rest combined. First,
inspecting the non-mock part

$(-1)^{\ell}\,i^{1/2}\,2\pi\sum_{\gamma=1}^{+\infty}\sum_{\begin{subarray}{c}\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}\\\
\tilde{n}\geq-1,\\\
\tilde{\Delta}<0\end{subarray}}c^{F}_{m}(\tilde{n},\tilde{\ell})\frac{{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{\gamma
m}\sqrt{\Delta|\tilde{\Delta}|}\right)\,,$

the leading term of the first sum is given by the $\gamma=1$ term, since all
the other contributions are exponentially suppressed with respect to it. When
$\gamma=1$, the associated $SL(2,\mathbb{Z})$ matrix is

$\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ (3.2)

and the generalized Kloosterman sum gets simplified to

${\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},1,\psi)_{\ell\tilde{\ell}}\;=\;\frac{e^{2\pi
i\frac{\ell\tilde{\ell}}{2m}}}{\sqrt{2mi}}\,.$ (3.3)

When $\tilde{\ell}=m$, we have

${\rm Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},1,\psi)_{\ell
m}\;=\;\frac{(-1)^{\ell}}{\sqrt{2mi}}\,.$ (3.4)

Therefore, the $\gamma=1$ contribution to the Rademacher expansion for each of
the terms with $\tilde{\ell}=m$ is positive. We write it as

$C_{\mathrm{pos.}}(\Delta,m,\tilde{n})\;=\;\frac{2\pi}{\sqrt{2m}}c^{F}_{m}(\tilde{n},m)\left(\frac{|\tilde{\Delta}_{m}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{m}|}\right)\,>\,0\,,$
(3.5)

where we introduced the notation for the negative discriminant with
$\tilde{\ell}=m$,

$|\tilde{\Delta}_{m}|\;=\;4m\tilde{n}-m^{2}\,,$ (3.6)

leaving the dependence on $\tilde{n}$ implicit. From (2.63), the expression
(3.5) for the extreme polar coefficient is

$C_{\mathrm{pos.}}(\Delta,m,-1)\;=\;2\pi\,\frac{m+2}{\sqrt{2m}}\left(\frac{4m+m^{2}}{\Delta}\right)^{23/4}I_{23/2}\left(2{\pi}\sqrt{\frac{\Delta}{m}\left(1+\frac{m}{4}\right)}\right)\,.$
(3.7)

The argument of the Bessel function is larger than the arguments in any other
Bessel function appearing in (2.36). This term is responsible for the leading
growth of $d(m,n,\ell)$ for $\Delta\gg 1$, correctly reproducing the
Bekenstein-Hawking entropy $d(m,n,\ell)\sim e^{\pi\sqrt{\Delta}}$ since the
area of the black holes with charge discriminant $\Delta$ is given by
$4\pi\sqrt{\Delta}$ [76, 77]. In particular, over the course of the proof this
term is shown to dominate over the mock-part of (2.36) given by terms with
modified Bessel functions of the first kind of indices 12 and 25/2.

An advantage of the choice of representatives given by (3.1) is the following
inequality for the discriminant,

$\Delta\;=\;4mn-\ell^{2}\,\geq\,4m^{2}-\ell^{2}\geq 3m^{2},$ (3.8)

That is, the conditions $n\geq m$ and $0\leq\ell\leq m$ imply that $\Delta$
grows at least quadratically with $m$. This fact turns out to be crucial and
will be used several times along the proof.

After bounding the mock part versus the non-mock part, the main part of the
proof consists in showing that the estimates $e_{23/2}(\Delta,m,\tilde{n})$
defined by the following ratios

$e_{23/2}(\Delta,m,\tilde{n})\;=\;\frac{1}{C_{\mathrm{pos.}}(\Delta,m,\tilde{n})}\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{4\pi}{\sqrt{2m}}\,{c}_{m}^{F}(\tilde{n},\tilde{\ell})\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)\,,$
(3.9)

or, more explicitly, by,

$e_{23/2}(\Delta,m,\tilde{n})\;=\;\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,,$
(3.10)

are smaller than one.

###  3.2. Notation

To simplify the exposition we introduce the following notation

$\begin{split}&C_{23/2}(m,n,\ell,\gamma)\;=\;\\\
&(-1)^{\ell}\,i^{1/2}2\pi\sum_{\begin{subarray}{c}\tilde{n}\geq-1,\\\
\tilde{\Delta}<0\\\
\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}\end{subarray}}c^{F}_{m}(\tilde{n},\tilde{\ell})\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}\frac{{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma}I_{23/2}\left(\frac{\pi}{\gamma
m}\sqrt{\Delta|\tilde{\Delta}|}\right)\,,\end{split}$
$C_{12}(m,n,\ell,\gamma)\;=\;(-1)^{\ell+1}\,i^{1/2}\sqrt{2m}\,d(m)\frac{{\rm
Kl}(\frac{\Delta}{4m},-1;\gamma,\psi)_{\ell
0}}{\sqrt{\gamma}}\left(\frac{4m}{\Delta}\right)^{6}I_{12}\left(\frac{2\pi}{\gamma}\sqrt{\frac{\Delta}{m}}\right)\,,$
$\begin{split}C_{25/2}(m,n,\ell,\gamma)\;=\;&(-1)^{\ell}\,i^{1/2}\frac{1}{2\pi}d(m)\sum_{\begin{subarray}{c}g\in\mathbb{Z}/2m\gamma\mathbb{Z}\\\
g=\tilde{\ell}\text{ mod }2m\\\
\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}\end{subarray}}\frac{{\rm
Kl}(\frac{\Delta}{4m},-1-\frac{g^{2}}{4m};\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma^{2}}\left(\frac{4m}{\Delta}\right)^{25/4}\\\
&\int_{-1/\sqrt{m}}^{1/\sqrt{m}}dx^{\prime}f_{\gamma,g,m}(x^{\prime})(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right)\,,\end{split}$

so that

$c_{m}^{F}(n,\ell)\;=\;\sum_{\gamma=1}^{+\infty}\left(C_{23/2}(m,n,\ell,\gamma)+C_{12}(m,n,\ell,\gamma)+C_{25/2}(m,n,\ell,\gamma)\right)\,.$
(3.11)

Furthermore, in what follows we will use the following definitions

$\Delta\;=\;4mn-\ell^{2}\,,\qquad\tilde{\Delta}\;=\;4m\tilde{n}-\tilde{\ell}^{2}\,,\qquad\tilde{\Delta}_{{m}}\;=\;4m\tilde{n}-m^{2}\,.$
(3.12)

###  3.3. Preliminaries

In this subsection we state results that are used several times in the
following. The first one is a bound for ratios of modified Bessel functions of
the first kind.

######  Proposition 1.

For $0<x<y$ and $\nu\geq\frac{1}{2}$, we have

$\frac{I_{\nu}(x)}{I_{\nu}(y)}\,<\,e^{x-y}\left(\frac{x}{y}\right)^{1/2}\,.$
(3.13)

The proof can be found in [78]. The following proposition is used to estimate
the growth of the polar coefficients.

The second result is a rough bound for the Fourier coefficients of
$\eta^{-24}(\tau)$ which is used for studying the growth of the polar
coefficients of $\psi_{m}^{F}(\sigma,v)$.

######  Proposition 2.

Let $n\in\mathbb{N}_{0}$, the Fourier coefficients $d(n)$ of
$\eta^{-24}(\tau)$ satisfy

$\left(1-\frac{1}{12}\right)2\pi
n^{-13/2}I_{13}(4\pi\sqrt{n})\,<\,d(n)\,<\,\left(1+\frac{1}{12}\right)2\pi
n^{-13/2}I_{13}(4\pi\sqrt{n})\,.$ (3.14)

###### Proof.

The coefficients $d(n)$ with $n\geq 1$ have the following Rademacher
expansion,

$d(n)\;=\;2\pi\sum_{c=1}^{\infty}\frac{\mathrm{Kl}(n,-1,c)}{c}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)\,,$
(3.15)

where

$\mathrm{Kl}(a,b,c)\;=\;\sum_{\begin{subarray}{c}0\leq x\leq c-1\\\
\gcd(c,x)=1\end{subarray}}e^{\frac{2\pi i}{c}\left(ax+bx^{*}\right)}\,,$
(3.16)

is the classical Kloosterman sum, with $x^{*}$ being the inverse of $x$ modulo
$c$. The expression (3.15) is also defined for $n=0$ by taking the limit $n\to
0$ from above. The value of the Kloosterman sum for $c=1$ is simply $1$, while
for $c\geq 1$ it satisfies the following bound

$\left|\frac{\mathrm{Kl}(n,-1,c)}{c}\right|\,\leq\,1\,.$ (3.17)

Thus, we can split

$d(n)\;=\;2\pi
n^{-13/2}I_{13}\left({4\pi}\sqrt{n}\right)+2\pi\sum_{c=2}^{\infty}\frac{\mathrm{Kl}(n,-1,c)}{c}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)\,,$
(3.18)

and bound the absolute value of the series by

$\left|2\pi\sum_{c=2}^{\infty}\frac{\mathrm{Kl}(n,-1,c)}{c}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)\right|\,\leq\,2\pi\sum_{c=2}^{\infty}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)\,.$
(3.19)

Since ${I_{\nu}(x)}$ and $\frac{I_{\nu}(x)}{x^{\nu}}$ are monotonically
increasing functions of $x$ for $x,\nu>0$, we can bound

$\begin{split}&2\pi\sum_{c=2}^{\infty}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)\,\leq\,2\pi\int_{1}^{\infty}n^{-13/2}I_{13}\left(\frac{4\pi}{c}\sqrt{n}\right)dc\\\
&\,\leq\,2\pi
n^{-13/2}I_{13}\left({4\pi}\sqrt{n}\right)\int_{1}^{\infty}c^{-13}dc\;=\;\frac{1}{12}2\pi
n^{-13/2}I_{13}\left({4\pi}\sqrt{n}\right).\end{split}$ (3.20)

∎

###  3.4. Mock and higher $\gamma$ terms

We start by bounding the higher $\gamma$ terms in the Rademacher expansion
with respect to the $\gamma=1$ counterparts and finding bounds for the terms
coming from the anomalous transformation due to the mock modular properties of
the Jacobi forms, which as stated earlier are the last two lines in the right
hand side of (2.36) and we call them the mock terms.

######  Lemma 1.

The following quantities appearing in the Rademacher expansion (2.36) satisfy
the bounds

$\left|\sum_{\gamma=2}^{\infty}\frac{{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma}I_{23/2}\left(\frac{\pi}{\gamma
m}\sqrt{\Delta|\tilde{\Delta}|}\right)\right|\,<\,\frac{2}{21}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)\,,$
(3.21) $\left|\sum_{\gamma=2}^{+\infty}\frac{{\rm
Kl}(\frac{\Delta}{4m},-1;\gamma,\psi)_{\ell
0}}{\sqrt{\gamma}}I_{12}\left(\frac{2\pi}{\gamma}\sqrt{\frac{\Delta}{m}}\right)\right|\,<\,\frac{2}{21}I_{12}\left({2\pi}{}\sqrt{\frac{\Delta}{m}}\right)\,,$
(3.22)

and

$\begin{split}&\left|\frac{1}{2\pi}d(m)\sum_{\gamma=1}^{+\infty}\sum_{\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}}\sum_{\begin{subarray}{c}g\in\mathbb{Z}/2m\gamma\mathbb{Z}\\\
g=\tilde{\ell}\text{ mod }2m\end{subarray}}\frac{{\rm
Kl}(\frac{\Delta}{4m},-1-\frac{g^{2}}{4m};\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma^{2}}\left(\frac{4m}{\Delta}\right)^{25/4}\right.\\\
&\left.\int_{\frac{-1}{\sqrt{m}}}^{\frac{1}{\sqrt{m}}}dx^{\prime}f_{\gamma,g,m}(x^{\prime})(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right)\right|\\\
&\leq\,\left(1+\frac{2}{21}\right){d(m)}\left(\frac{4m}{\Delta}\right)^{25/4}m^{3/2}\pi\left(\frac{\pi^{2}}{3}+\frac{1}{3}\right)I_{25/2}\left(\frac{2\pi}{\sqrt{m}}\sqrt{\Delta}\right)\,.\end{split}$
(3.23)

###### Proof.

Since ${\psi}(\Gamma)_{{\tilde{\ell}}\ell}$ is a unitary matrix, it satisfies
$\left|{\psi}(\Gamma)_{{\tilde{\ell}}\ell}\right|\leq 1$, from which

$\left|{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},\gamma,\psi)_{\ell\tilde{\ell}}\right|\,\leq\,\gamma\,,$
(3.24)

follows. The proof of the first two bounds (3.21) and (3.22) is completed by
following the same steps as in Proposition 2.

For the last bound (3.23), we begin by finding a bound for the integral.
Adapting the bounds in Lemma 3.2 of [34] to our function
$f_{\gamma,g,m}(x^{\prime})$, we have

$\left|\sinh\left(\frac{\pi x^{\prime}}{\gamma}-\frac{\pi
ig}{2m\gamma}\right)\right|\;=\;\left|\cosh\left(\frac{\pi
x^{\prime}}{\gamma}-\pi
i\left(\frac{g}{2m\gamma}+\frac{1}{2}\right)\right)\right|\,\geq\,\left|\sin\left(\frac{\pi
g}{2m\gamma}\right)\right|\,,$ (3.25)

so for $-m\gamma<g\leq m\gamma$ and $g\neq 0$,

$\left|\frac{1}{\sin\left(\frac{\pi
g}{2m\gamma}\right)}\right|\,\leq\,\frac{\pi^{2}}{4\left(\frac{\pi
g}{2m\gamma}\right)^{2}}\;=\;\frac{\gamma^{2}m^{2}}{g^{2}}\,.$ (3.26)

For $g=0$ we can simply use the bound

$\left|\frac{1}{\sinh^{2}(x)}-\frac{1}{x^{2}}\right|\,\leq\,\frac{1}{3}\,,$
(3.27)

to obtain

$\left|f_{\gamma,0,m}(x^{\prime})\right|\,\leq\,\frac{\pi^{2}}{3}\,.$ (3.28)

Defining

$h_{\gamma,g,m}\;=\;\left\\{\begin{aligned}
\frac{\gamma^{2}m^{2}}{g^{2}}&\;\;\text{ if}\;\;\;g\neq 0\text{ mod
}2m\gamma\\\ \frac{1}{3}&\;\;\text{ if}\;\;\;g=0\text{ mod
}2m\gamma\,,\end{aligned}\right.$ (3.29)

we have shown

$f_{\gamma,g,m}(x^{\prime})\,\leq\,\pi^{2}h_{\gamma,g,m}\,.$ (3.30)

The function $(1-mx^{\prime 2})$ is non-negative and bounded by 1 in the
interval $[\frac{-1}{\sqrt{m}},\frac{1}{\sqrt{m}}]$, therefore

$\begin{split}&\left|\left.\int_{-1/\sqrt{m}}^{1/\sqrt{m}}dx^{\prime}f_{\gamma,g,m}(x^{\prime})(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right)\right)\right|\,\leq\,\\\
&\frac{2}{\sqrt{m}}\pi^{2}h_{\gamma,g,m}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta}\right)\,.\end{split}$
(3.31)

Using the bound (3.24), we can combine the sums over $\tilde{\ell}$ and $g$ in
(3.23) into a sum over $g\in\mathbb{Z}/2m\gamma\mathbb{Z}$. Moreover, from

$\sum_{g=1}^{m\gamma}h_{\gamma,g,m}\;=\;\gamma^{2}m^{2}\sum_{g=1}^{m\gamma}\frac{1}{g^{2}}\,<\,\frac{\pi^{2}}{6}\gamma^{2}m^{2}\,,$
(3.32)

the sum over the remaining $g$ dependent part satisfies the bound

$\sum_{g=-m\gamma+1}^{m\gamma}h_{\gamma,g,m}\,<\,\left(\frac{\pi^{2}}{3}+\frac{1}{3}\right)\gamma^{2}m^{2}\,.$
(3.33)

The last part of the proof consists in bounding the sum over $\gamma\geq 2$,
which can be done as in the previous cases. ∎

The following Lemma shows that the mock terms are suppressed with respect to
the largest positive non-mock contribution—the one with $\tilde{n}=-1$.

######  Lemma 2.

Let $m\geq 6$, $n\geq m$ and $0\leq\ell\leq m$, $\Delta=4mn-\ell^{2}$, then

$e_{12}(\Delta,m)\;=\;\frac{\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|}{C_{\mathrm{pos.}}(\Delta,m,-1)}\,<\,5\cdot
10^{-4}\,,$ (3.34)

and

$e_{25/2}(\Delta,m)\;=\;\frac{\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|}{C_{\mathrm{pos.}}(\Delta,m,-1)}\,<\,0.015\,.$
(3.35)

###### Proof.

From (3.22), we have

${\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|}\,<\,{\frac{23}{21}\sqrt{2m}\,d(m)\left(\frac{4m}{\Delta}\right)^{6}I_{12}\left(2\pi\sqrt{\frac{\Delta}{m}}\right)}\,.$
(3.36)

Since $\Delta\geq 3m^{2}$ and $m\geq 6$, we have

$\begin{split}&\frac{{\frac{23}{21}\sqrt{2m}\,d(m)\left(\frac{4m}{\Delta}\right)^{6}I_{12}\left(2\pi\sqrt{\frac{\Delta}{m}}\right)}}{C_{\mathrm{pos.}}(\Delta,m,-1)}\,\leq\,\frac{\frac{23}{21}\sqrt{2m}\,d(m)\left(\frac{4m}{3m^{2}}\right)^{6}I_{12}\left(2\sqrt{3}\pi\sqrt{m}\right)}{C_{\mathrm{pos.}}(3m^{2},m,-1)}\\\
&\;=\;\,\frac{\frac{23}{21}\sqrt{2m}\,d(m)\left(\frac{4m}{3m^{2}}\right)^{6}I_{12}\left(2\sqrt{3}\pi\sqrt{m}\right)}{2\pi\,(m+2)\frac{1}{\sqrt{2m}}\left(\frac{4m+m^{2}}{3m^{2}}\right)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{4m+m^{2}}\right)}\,.\end{split}$
(3.37)

The last expression is a decreasing function of $m$, being bounded by its
value at $m=6$, which is smaller than $5\cdot 10^{-4}$.

The other bound is proved in the same way. From (3.23) and $\Delta\geq
3m^{2}$, we have

$\frac{\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|}{C_{\mathrm{pos.}}(3m^{2},m,-1)}\,\leq\,\frac{\left(1+\frac{2}{21}\right){d(m)}\left(\frac{4m}{3m^{2}}\right)^{25/4}m^{3/2}\pi\left(\frac{\pi^{2}}{3}+\frac{1}{3}\right)I_{25/2}\left({2\sqrt{3}\pi}{\sqrt{m}}\right)}{2\pi\,(m+2)\frac{1}{\sqrt{2m}}\left(\frac{4m+m^{2}}{3m^{2}}\right)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{4m+m^{2}}\right)}\,.$

The last expression is a decreasing function of $m$, being bounded by its
value at $m=6$ which is smaller than $0.015$. ∎

###  3.5. Polar coefficients bounds

In this subsection we bound the polar coefficients
$c_{m}^{F}(\tilde{n},\tilde{\ell})$ with
$\tilde{\Delta}=4m\tilde{n}-\tilde{\ell}^{2}<0$ with respect to the first
contribution defined in (2.54) coming from the $U$ matrix. We start with the
case $\tilde{n}=0$ for which there is an exact expression for the polar
coefficients. This expression can be derived by noticing that the matrices of
the form $U^{r}$ with $r\geq 1$ are enough to find a path from the
$\mathcal{R}$ chamber to a zero degeneracy chamber. Alternatively, one can
also compute the coefficients with $\tilde{n}=0$ as the
$p^{m}y^{\tilde{\ell}}$ coefficient of the function
$\frac{d(0)}{\eta^{24}(\rho)}\mathcal{A}_{2,0}(\rho,v)$. The explicit form is,

$c^{F}_{m}(0,\tilde{\ell})\;=\;\tilde{\ell}\,d(0)\sum_{r=1}^{\lfloor\frac{m+1}{\tilde{\ell}}\rfloor}d(m-r\tilde{\ell})\,,$
(3.38)

which is valid for $0<\tilde{\ell}\leq m$. From this expression we can prove
the following.

######  Lemma 3.

Let $m\geq 6$ and $1\leq k<m$. Then

${c}^{F}_{m}(0,m-k)\,\leq\,\left(1+\frac{25}{d(k)}+\kappa_{1}k^{1/2}\right)\hat{c}^{F}_{m}(0,m-k)\,,$
(3.39)

where $\kappa_{1}=\frac{91}{88\pi}$.

###### Proof.

The ratio vs. the first contribution reads

$\frac{c^{F}_{m}(0,\tilde{\ell})}{\hat{c}^{F}_{m}(0,\tilde{\ell})}\;=\;\frac{\sum\limits_{r=1}^{\lfloor\frac{m+1}{\tilde{\ell}}\rfloor}d(m-r\tilde{\ell})}{d(m-\tilde{\ell})}\;=\;\frac{1}{d(k)}\sum_{r=1}^{\lfloor\frac{m+1}{m-k}\rfloor}d(m-rm+rk)\,,$
(3.40)

where $k=m-\tilde{\ell}$. There will be extra contributions starting at $r=2$
only if

$m-2m+2k\,\geq\,-1\Longrightarrow k\,\geq\,\frac{m-1}{2}\,>\,2\,,$ (3.41)

since we are only considering $m\geq 6$. Thus, the following equality is
satisfied

${c}^{F}_{m}(0,m-k)\;=\;\hat{c}^{F}_{m}(0,m-k)\qquad\text{ for }k=1,2,3\,.$
(3.42)

The following analysis is restricted to $k\geq 3$. For these cases, we have

$\frac{1}{d(k)}\sum_{r=1}^{\lfloor\frac{m+1}{m-k}\rfloor}d(m-rm+rk)\leq\frac{1}{d(k)}\left(d(-1)+d(0)+\sum_{r=1}^{\lfloor\frac{m-1}{m-k}\rfloor}d(m-rm+rk)\right)\,.$
(3.43)

Using the second inequality in (3.14), the discrete sum over $r$ can be
bounded by an integral and we obtain

$\begin{split}&\sum_{r=1}^{\lfloor\frac{m-1}{m-k}\rfloor}d(m-rm+rk)\,\leq\,\\\
&d(k)+\int_{1}^{\frac{m-1}{m-k}}\left(1+\frac{1}{12}\right)2\pi(m-rm+rk)^{-13/2}I_{13}(4\pi\sqrt{m-rm+rk})dr\,,\end{split}$
(3.44)

where we have used that the sum over $r$ is monotonically decreasing. The
integral can be evaluated to yield the following bound

$\frac{1}{d(k)}\sum_{r=1}^{\lfloor\frac{m-1}{m-k}\rfloor}d(m-rm+rk)\leq
1+\frac{1}{d(k)}\left(1+\frac{1}{12}\right)\frac{2\pi}{m-k}\left(\frac{I_{12}(4\pi\sqrt{k})}{2k^{6}\pi}-\frac{I_{12}(4\pi)}{2\pi}\right)\,.$
(3.45)

Applying the first inequality in (3.14) to $d(k)$ in the denominator, together
with the bound

$\frac{I_{12}(4\pi\sqrt{n})}{I_{13}(4\pi\sqrt{n})}\leq\frac{7}{4}\qquad\text{
for }n\geq 3\,,$ (3.46)

we obtain

$\sum_{r=1}^{\lfloor\frac{m+1}{m-k}\rfloor}d(m-rm+rk)\,\leq\,d(-1)+d(0)+d(k)+d(k)\frac{1+\frac{1}{12}}{1-\frac{1}{12}}k^{1/2}\frac{7}{8\pi}\,,$
(3.47)

where we have used the rough bound $(m-k)^{-1}\leq 1$. We can write this bound
as

$\frac{{c}^{F}_{m}(0,m-k)}{\hat{c}^{F}_{m}(0,m-k)}\leq\frac{1}{d(k)}\left(d(-1)+d(0)+d(k)+d(k)\frac{1+\frac{1}{12}}{1-\frac{1}{12}}k^{1/2}\frac{7}{8\pi}\right)\,,$
(3.48)

which is equivalent to (3.39). ∎

We now bound the polar coefficients for $\tilde{n}\geq 1$. For this we use the
general form of the polar coefficients (2.45).

######  Lemma 4.

Let $m\geq 6$, $\tilde{n}\geq 1$, $1\leq k\leq m$ and
$\tilde{\Delta}=4m\tilde{n}-(m-k)^{2}<0$. Then

$c_{m}^{F}(\tilde{n},m-k)\,\leq\,\left(\kappa_{2}+\kappa_{3}k\right)\,\hat{c}_{m}^{F}(\tilde{n},m-k)\,,$
(3.49)

where $\kappa_{2}=181/100$ and $\kappa_{3}=243/400$.

###### Proof.

Consider first the second contribution to $W(m,\tilde{n},\tilde{\ell})$, where
$\tilde{\ell}=m-k$. For all charge bilinears satisfying $0<\tilde{\ell}<m$, we
have $0<\frac{\tilde{\ell}}{2m}<\frac{1}{2}$ and therefore the second matrix
coming from the continued fraction expansion of $\frac{\tilde{\ell}}{2m}$ is
always

$U^{2}=\begin{pmatrix}1&0\\\ 2&1\end{pmatrix},\;\;\;\;\text{ with
}\;\;\;{\ell}_{U^{2}}d(m_{U^{2}})d({n}_{U^{2}})\;=\;(\tilde{\ell}-4\tilde{n})d(4\tilde{n}+m-2\tilde{\ell})d(\tilde{n})\,.$
(3.50)

In terms of $k=m-\tilde{\ell}$ the contribution becomes

${\ell}_{U^{2}}d(m_{U^{2}})d(\tilde{n}_{U^{2}})\;=\;(m-k-4\tilde{n})d(4\tilde{n}-m+2k)d(\tilde{n})\,.$
(3.51)

The ratio of the second contribution and the first contribution satisfies the
following bound,

$\frac{(m-k-4\tilde{n})d(4\tilde{n}-m+2k)d(\tilde{n})}{(m-k-2\tilde{n})d(\tilde{n}+k)d(\tilde{n})}\,\leq\,\frac{d(4\tilde{n}-m+2k)}{d(\tilde{n}+k)}\,,$
(3.52)

since $\tilde{n}\geq 1$. We use this inequality to bound the rest of the
contributions. As explained in Section 2.4, $m_{G}$ and ${n}_{G}$ are
decreasing in terms of the sequence defined by the continued fraction.
Moreover, since ${n}_{U}=\tilde{n}\geq 1$, $m_{U}=\tilde{n}+k\geq 1$ and the
discriminant $\tilde{\Delta}$ is invariant under $SL(2,\mathbb{Z})$
transformations,

$\tilde{\Delta}\;=\;4m_{U}{n}_{U}-{\ell}_{U}^{2}\;=\;4m_{G}{n}_{G}-{\ell}_{G}^{2}\,,$
(3.53)

the value of ${\ell}_{G}$ cannot be greater than ${\ell}_{U}$,

${\ell}_{U}\,\geq\,{\ell}_{G}\,.$ (3.54)

By the description of the set $W(m,\tilde{n},m-k)$ in terms of the continued
fraction of $\frac{\tilde{\ell}}{2m}$ given in section 2.4, there are less
than $2m$ contributions to the polar coefficients, so we can bound the sum of
all contributions by

$\sum_{G\in
W(m,\tilde{n},m-k)\setminus\\{U\\}}\frac{{\ell}_{G}\,d(m_{G})\,d({n}_{G})}{{\ell}_{U}\,d(m_{U})\,d({n}_{U})}\,<\,2m\frac{d(4\tilde{n}-m+2k)}{d(\tilde{n}+k)}\,.$
(3.55)

The right hand side is decreasing with $m$. Thus, we can bound it by the
smallest possible value of $m$. From the conditions

$\tilde{\Delta}=4m\tilde{n}-(m-k)^{2}<0,\;\;\;1\leq k<m\,,$ (3.56)

one finds

$m>k+2\tilde{n}+\sqrt{(k+2\tilde{n})^{2}-k^{2}}\,,$ (3.57)

and therefore

$2m\frac{d(4\tilde{n}-m+2k)}{d(\tilde{n}+k)}\leq
2\left(k+2\tilde{n}+\sqrt{(k+2\tilde{n})^{2}-k^{2}}\right)\frac{d\left(\left\lfloor
2\tilde{n}+k-\sqrt{(k+2\tilde{n})^{2}-k^{2}}\right\rfloor\right)}{d(\tilde{n}+k)}\,.$
(3.58)

This quantity is maximized by the value $\tilde{n}=1$,

$2m\frac{d(4\tilde{n}-m+2k)}{d(\tilde{n}+k)}\,<\,(6k+8)\frac{d(\lfloor
2+k-2\sqrt{k}\rfloor)}{d(1+k)}\,,$ (3.59)

and the ratio of the $\eta^{-24}(\tau)$ coefficients is maximized by its value
at $k=1$,

$\frac{d(\lfloor
2+k-2\sqrt{k}\rfloor)}{d(1+k)}\,\leq\,\frac{d(1)}{d(2)}\;=\;\frac{324}{3200}\,,$
(3.60)

yielding the desired bound

$\frac{1}{\hat{c}_{m}^{F}(m,\tilde{n},m-k)}\sum_{G\in
W(m,\tilde{n},m-k)\setminus\\{U\\}}\tilde{\ell}_{G}\,d(m_{G})d(\tilde{n}_{G})\,\leq\,\frac{324}{3200}(6k+8)\,.$
(3.61)

∎

The previous result is a rough bound, mostly due to the use of the inequality
(3.55). Numerical experiments show that there is no linear growth in $k$ and
presumably using the precise form of the polar coefficients (2.45) one could
prove a tighter bound. For our purposes, however, the bound is enough.

The remaining bounds concern the polar coefficients with $\tilde{n}=-1$. The
coefficients $c_{m}^{F}(-1,\tilde{\ell})$ with $0\leq\tilde{\ell}\leq m$ have
the following exact expression

$c_{m}^{F}(-1,\tilde{\ell})\;=\;\sum_{r\geq
1}(2r+\tilde{\ell})d(m-r^{2}-r\tilde{\ell})\,,$ (3.62)

where the sum over $r$ is bounded by

$r_{\max}\;=\;{\lfloor\frac{-\tilde{\ell}+\sqrt{\tilde{\ell}^{2}+4m+4}}{2}\rfloor}\,,$
(3.63)

due to the vanishing of $d(n)$ for $n<-1$. The expression (3.62) can be
obtained directly from the product representation of
$\Phi_{10}(\rho,\sigma,v)$ (2.16), since for $\Phi_{10}(\rho,\sigma,v)^{-1}$
the $p^{-1}$ coefficient is given by the quantities $C_{0}(-1)=1$ and
$C_{0}(0)=10$, yielding the following negative index Jacobi form

$\begin{split}\psi_{-1}(\sigma,v)\;=\;&\frac{q^{-1}y}{(1-y)^{2}}\prod_{n=1}^{\infty}(1-yq^{n})^{-2}(1-y^{-1}q^{n})^{-2}(1-q^{n})^{-20}\\\
\;=\;&\frac{1}{\eta^{24}(\sigma)}\frac{1}{\varphi_{-2,1}(\sigma,v)}\,.\end{split}$
(3.64)

The function $\varphi_{-2,1}(\sigma,v)^{-1}$ can be expressed as a derivative
of an Appell-Lerch sum [79, 59], giving the coefficients (3.62) [37].

In (3.62) the second contribution coming from $r=2$ is only non-vanishing if

$m-4-2\tilde{\ell}=m-4-2(m-k)\geq-1\Rightarrow k\geq\frac{3+m}{2}\geq 5\,,$
(3.65)

since $m\geq 6$. Thus, in the following we only consider the case $k\geq 5$.
We have

######  Lemma 5.

Let $m\geq 6$ and $5\leq k\leq m$, then

$c_{m}^{F}(-1,m-k)\,\leq\,(\kappa_{4}+\kappa_{5}k^{1/2})\,\hat{c}_{m}^{F}(-1,m-k)\,,$
(3.66)

where the constants $\kappa_{4},\kappa_{5}$ are $\kappa_{4}=3+2\cdot 10^{-5}$
and $\kappa_{5}=\frac{1+\frac{1}{12}}{1-\frac{1}{12}}\frac{1}{4\pi}$.

###### Proof.

The proof is similar to the one for the $\tilde{n}=0$ case. We have

$\frac{c_{m}^{F}(-1,m-k)}{\hat{c}_{m}^{F}(-1,m-k)}\,\leq\,1+\sum_{r\geq
2}\frac{(2r+m-k)d(m-r^{2}-r(m-k))}{(2+m-k)d(k-1)}\,.$ (3.67)

We isolate the possible contribution containing the coefficient $d(-1)$ in
order to use the bounds (3.14). Denoting by $r_{*}$ the positive value of $r$
satisfying $m-r^{2}-r(m-k)=-1$, if $r_{*}$ is an integer, the corresponding
summand in (3.67) can be bounded by

$\frac{(2r_{*}+m-k)d(m-r_{*}^{2}-r_{*}(m-k))}{(2+m-k)d(k-1)}\;=\;\frac{\sqrt{(m-k)^{2}+4m+4}}{(2+m-k)d(k-1)}\,\leq\,\frac{\sqrt{1+k}}{d(k-1)}<2\cdot
10^{-5}\,,$

where the last bound follows from the ratio being decreasing with $k$, with
the minimum value given by $k=5$.

It follows that the right hand side of (3.67) is less than

$1+2\cdot 10^{-5}+\sum_{r\geq
2}^{r_{\max}^{\prime}}\frac{(2r+m-k)d(m-r^{2}-r(m-k))}{(2+m-k)d(k-1)}\,,$
(3.68)

where

$r_{\max}^{\prime}\;=\;\left\lfloor\frac{-(m-k)+\sqrt{(m-k)^{2}+4m}}{2}\right\rfloor\,,$
(3.69)

with the convention that the sum is zero if $r_{\max}^{\prime}<2$. From the
second inequality in (3.14), the decreasing sum satisfies the following bound

$\begin{split}&\sum_{r=2}^{r_{\max}^{\prime}}{(2r+m-k)d(m-r^{2}-r(m-k))}\,\leq\,{(4+m-k)d(m-4-2(m-k))}\\\
&+\int_{2}^{r_{max}^{\prime}}dr\,(2r+m-k)\left(1+\frac{1}{12}\right)2\pi\frac{I_{13}(4\pi\sqrt{m-r^{2}-r(m-k)})}{(m-r^{2}-r(m-k))^{13/2}}\,.\end{split}$
(3.70)

The integral can be evaluated using the following equality

$\int_{2}^{r_{max}^{\prime}}dr\,(2r+m-k)\frac{I_{13}(4\pi\sqrt{m-r^{2}-r(m-k)})}{(m-r^{2}-r(m-k))^{13/2}}\;=\;\frac{I_{12}(4\pi\sqrt{2k-m-4})}{2\pi(2k-m-4)^{6}}-\frac{(2\pi)^{11}}{12!}\,,$

which yields,

$\begin{split}&\sum_{r\geq
2}^{r_{\max}^{\prime}}\frac{(2r+m-k)d(m-r^{2}-r(m-k))}{(2+m-k)d(k-1)}\\\
<\,&\,\frac{(4+m-k)d(2k-m-4)}{(2+m-k)d(k-1)}+\frac{\left(1+\frac{1}{12}\right)\frac{I_{12}(4\pi\sqrt{2k-m-4})}{(2k-m-4)^{6}}}{(2+m-k)d(k-1)}\,.\end{split}$
(3.71)

The first summand is decreasing with $m$. Since for fixed $k$ the least
possible value of $m$ is $k$, this term is bounded by $2$,

$\frac{(4+m-k)d(2k-m-4)}{(2+m-k)d(k-1)}\,<\,2\,.$ (3.72)

Now we examine the second term, which is again decreasing with $m$ and bounded
by the value $m=k$,

$\frac{\left(1+\frac{1}{12}\right)\frac{I_{12}(4\pi\sqrt{2k-m-4})}{(2k-m-4)^{6}}}{(2+m-k)d(k-1)}\,\leq\,\frac{\left(1+\frac{1}{12}\right)\frac{I_{12}(4\pi\sqrt{k-4})}{(k-4)^{6}}}{2d(k-1)}\,.$
(3.73)

We use the first inequality in (3.14) to write

$\frac{\left(1+\frac{1}{12}\right)\frac{I_{12}(4\pi\sqrt{k-4})}{(k-4)^{6}}}{2d(k-1)}\,\leq\,\frac{1+\frac{1}{12}}{1-\frac{1}{12}}\frac{(k-1)^{13/2}}{4\pi{(k-4)^{6}}}\frac{I_{12}(4\pi\sqrt{k-4})}{I_{13}(4\pi\sqrt{k-1})}\,.$
(3.74)

This last expression is an unbounded function of $k$ which is smaller than the
function capturing its asymptotic growth,

$\frac{1+\frac{1}{12}}{1-\frac{1}{12}}\frac{(k-1)^{13/2}}{4\pi{(k-4)^{6}}}\frac{I_{12}(4\pi\sqrt{k-4})}{I_{13}(4\pi\sqrt{k-1})}\,\leq\,\frac{1+\frac{1}{12}}{1-\frac{1}{12}}\frac{k^{1/2}}{4\pi}\,.$
(3.75)

Putting the previous bounds together, we have

$\frac{c_{m}^{F}(-1,m-k)}{\hat{c}_{m}^{F}(-1,m-k)}\,<\,3+2\cdot
10^{-5}+\frac{1+\frac{1}{12}}{1-\frac{1}{12}}\frac{k^{1/2}}{4\pi}\,.$ (3.76)

∎

###  3.6. Leading term bounds

The remaining bounds concern the the leading term in the Rademacher expansion,
by which we mean the first line in the right hand side of (2.36). We show that
the positive contribution (3.5) for each $\tilde{n}$ is larger than the sum of
the remaining contributions with $\tilde{\ell}<m$. We first derive a bound for
the cases $\tilde{n}\geq 0$ by considering only the first contribution to the
polar coefficients (2.54) and then we correct it with the bounds for the polar
coefficients derived in the previous subsection separately for $\tilde{n}=0$
and $\tilde{n}\geq 1$. The bounds for the more involved case $\tilde{n}=-1$
are derived at the end.

######  Lemma 6.

Let $m\geq 6$, $n\geq m$, $0\leq\ell\leq m$, and $\tilde{n}\geq 0$. Then

$\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,\hat{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,<\,0.123\,.$
(3.77)

###### Proof.

Using the inequality (3.8), $\Delta\geq 3m^{2}$, we can bound each summand in
(3.77) by

$\frac{2\,\hat{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\leq\frac{2\,\hat{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{|\tilde{\Delta}_{{m}}|}\right)}.$

The right hand side of this equation has the following explicit expression

$\frac{2\,(m-k-2\tilde{n})d(k+\tilde{n})d(\tilde{n})((m-k)^{2}-4m\tilde{n})^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m-k)^{2}-4m\tilde{n}}\right)}{(m-2\tilde{n})d(\tilde{n})d(\tilde{n})(m^{2}-4m\tilde{n})^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{m^{2}-4m\tilde{n}}\right)}$

where $k\;=\;m-\tilde{\ell}$. From the inequality (3.13) we obtain

$\begin{split}&\frac{2\,(m-k-2\tilde{n})d(k+\tilde{n})d(\tilde{n})((m-k)^{2}-4m\tilde{n})^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m-k)^{2}-4m\tilde{n}}\right)}{(m-2\tilde{n})d(\tilde{n})d(\tilde{n})(m^{2}-4m\tilde{n})^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{m^{2}-4m\tilde{n}}\right)}\\\
&<\frac{2\,(m-k-2\tilde{n})d(k+\tilde{n})d(\tilde{n})\,}{(m-2\tilde{n})d(\tilde{n})d(\tilde{n})}\left(\frac{(m-k)^{2}-4m\tilde{n}}{m^{2}-4m\tilde{n}}\right)^{6}e^{\sqrt{3}{\pi}\left(\sqrt{(m-k)^{2}-4m\tilde{n}}-\sqrt{m^{2}-4m\tilde{n}}\right)}.\end{split}$
(3.78)

Since $\tilde{n}\geq 0$, the right hand side of the previous equation is an
increasing function of $m$. Therefore, it is smaller or equal than its
$m\to\infty$ limit, which is given by

$\frac{2\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}.$ (3.79)

The sum over $\tilde{\ell}$ is restricted to the values which satisfy
$\tilde{\Delta}=4m\tilde{n}-\tilde{\ell}^{2}<0$, so the sum over $k$ is from
$k=1$ to $k=\lfloor m-2\sqrt{m\tilde{n}}\rfloor$. Therefore, by taking the
limit $m\to\infty$, we have shown

$\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,\hat{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,<\,\sum_{k=1}^{\infty}\frac{2\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}\,.$
(3.80)

The last expression is a decreasing function of $\tilde{n}$. We can write it
as

$\sum_{k=1}^{\infty}\frac{2\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}\;=\;\frac{2e^{\sqrt{3}\pi\tilde{n}}}{d(\tilde{n})}\left(\eta^{-24}\left(\frac{i\sqrt{3}}{2}\right)-\sum_{r=-1}^{\tilde{n}}d(r)e^{-\sqrt{3}\pi
r}\right)\,,$ (3.81)

which we can compute for any value of $\tilde{n}$. E.g., for $\tilde{n}=1$ the
previous quantity is smaller than $0.09$, and for $\tilde{n}=0$ it gives the
desired bound

$\frac{2}{24}\left(\eta^{-24}\left(\frac{i\sqrt{3}}{2}\right)-e^{\sqrt{3}\pi}-24\right)\;=\;0.122...<0.123\,.$
(3.82)

∎

The next step consists in using the bounds for the polar coefficients found in
the previous subsection to find a bound for the complete contribution.

######  Lemma 7.

Let $m\geq 6,n\geq m,0\leq\ell\leq m$, and $\tilde{n}\geq 1$. Then

$e_{23/2}(\Delta,m,\tilde{n})\;=\;\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,<\,0.23\,.$
(3.83)

###### Proof.

Substituting ${c}_{m}^{F}(\tilde{n},m-k)$ by its bound from Lemma 4,
$(\kappa_{2}+\kappa_{3}k)\hat{c}_{m}^{F}(\tilde{n},m-k)$, following the steps
in the proof of Lemma 6, one arrives at

$\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,(\kappa_{2}+\kappa_{3}k)\hat{c}_{m}^{F}(\tilde{n},\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(\tilde{n},m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,<\,\sum_{k=1}^{\infty}\frac{2(\kappa_{2}+\kappa_{3}k)\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}\,.$
(3.84)

We split the right hand side in two terms. The first one is proportional to
the expression obtained in Lemma 6,

$\sum_{k=1}^{\infty}\frac{2\kappa_{2}\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}\;=\;\frac{2\kappa_{2}e^{\sqrt{3}\pi\tilde{n}}}{d(\tilde{n})}\left(\eta^{-24}\left(\frac{i\sqrt{3}}{2}\right)-\sum_{r=-1}^{\tilde{n}}d(r)e^{-\sqrt{3}\pi
r}\right)\,,$ (3.85)

which for $\tilde{n}=1$ has a value which is smaller than 0.17. The second
term in (3.84) is also decreasing with $\tilde{n}$ and can be written as

$\sum_{k=1}^{\infty}\frac{2\kappa_{3}k\,d(k+\tilde{n})}{d(\tilde{n})}e^{-\sqrt{3}{\pi}k}\;=\;\frac{2\kappa_{3}e^{\sqrt{3}\pi\tilde{n}}}{d(\tilde{n})}\left(-\frac{E_{2}\left(\frac{i\sqrt{3}}{2}\right)+\tilde{n}}{\eta^{24}\left(\frac{i\sqrt{3}}{2}\right)}-\sum_{r=-1}^{\tilde{n}}(r-\tilde{n})d(r)e^{-\sqrt{3}\pi
r}\right)\,,$ (3.86)

in terms of the function

$\frac{E_{2}(\tau)}{\eta^{24}(\tau)}\;=\;-\sum_{n=-1}^{\infty}n\,d(n)e^{2\pi
i\tau n}\,,$ (3.87)

where

$E_{2}(\tau)\;=\;\frac{1}{2\pi
i}\frac{d}{d\tau}\log\eta^{24}(\tau)\;=\;1-24\sum_{n=1}^{\infty}\sigma_{1}(n)e^{2\pi
i\tau n}\,,$ (3.88)

is the Eisenstein series of weight 2 and $\sigma_{1}(n)$ is the sum-of-
divisors function. The quantity (3.86) is a decreasing function of $\tilde{n}$
with a value at $\tilde{n}=1$ which is smaller than $0.06$. The sum of the
bounds for the two contributions in the right hand side of (3.84) yields the
inequality (3.83). ∎

######  Lemma 8.

Let $m\geq 6,n\geq m,0\leq\ell\leq m$, and $\tilde{n}=0$. Then

$e_{23/2}(\Delta,m,0)\;=\;\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2\,{c}_{m}^{F}(0,\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(0,m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}<0.125\,.$
(3.89)

###### Proof.

The proof is essentially the same as for the previous Lemma, so we skip some
details. Following the reasoning for Lemma 6 for the extra contribution due to
the subleading terms in ${c}^{F}_{m}(0,m-k)$ given in Lemma 3, which is only
taken into account for $k\geq 3$, one can bound the extra contribution to
$e_{23/2}(\Delta,m,0)$ by

$\begin{split}&\frac{2}{d(0)}\sum_{k=3}^{\infty}\left(25+\kappa_{1}d(k)k^{1/2}\right)e^{-\sqrt{3}\pi
k}\,<\,\\\ &\frac{50}{d(0)}\sum_{k=3}^{\infty}e^{-\sqrt{3}\pi
k}+\frac{2\kappa_{1}}{d(0)}\left(-\frac{E_{2}\left(\frac{i\sqrt{3}}{2}\right)}{\eta^{24}\left(\frac{i\sqrt{3}}{2}\right)}-\sum_{r=-1}^{2}rd(r)e^{-\sqrt{3}\pi
r}\right)\,<\,0.02\,.\end{split}$ (3.90)

∎

Now we turn to the final case $\tilde{n}=-1$. Equipped with the bound for the
polar coefficients from the previous subsection, we prove the following
result, which is the last preliminary result for the proof of the main
theorem.

######  Lemma 9.

Let $m\geq 6,n\geq m,0\leq\ell\leq m$ and $\tilde{n}=-1$. Then

$e_{23/2}(\Delta,m,-1)\;=\;\sum_{0\leq\tilde{\ell}<m}\frac{2\,{c}_{m}^{F}(-1,\tilde{\ell})|\tilde{\Delta}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}{{c}_{m}^{F}(-1,m)|\tilde{\Delta}_{{m}}|^{23/4}\,{I}_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}_{{m}}|}\right)}\,<\,0.3\,.$
(3.91)

###### Proof.

In (3.62) the second contribution to the polar coefficient, given by $r=2$, is
only non-vanishing if

$m-4-2\tilde{\ell}\;=\;m-4-2(m-k)\,\geq\,-1\quad\Longrightarrow\quad
k\,\geq\,\frac{3+m}{2}\,\geq\,5\,,$ (3.92)

since $m\geq 6$. Thus, for the first four cases $k=1,2,3,4$, which are the
ones with the largest contribution to (3.91), we can substitute
${c}_{m}^{F}(-1,\tilde{\ell})$ for $\hat{c}_{m}^{F}(-1,\tilde{\ell})$. For
$k=m-\tilde{\ell}=1,2,3,4$, the summands are given by

$\begin{split}&\frac{(m+2-k)\,d(k-1)\,2\,((m-k)^{2}+4m)^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m-k)^{2}+4m}\right)}{(m+2)\,(m^{2}+4m)^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{m^{2}+4m}\right)}\\\
\leq\,&\,\frac{d(k-1)\,2\,((m-k)^{2}+4m)^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m-k)^{2}+4m}\right)}{(m^{2}+4m)^{23/4}\,{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{m^{2}+4m}\right)}\,,\end{split}$
(3.93)

where we have also used that, as a function of $\Delta$, the summands in
(3.91) are less or equal than their value for $\Delta=3m^{2}$. Unlike the
previous cases with $\tilde{n}>-1$, the ratio

$\frac{{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m-k)^{2}+4m}\right)}{{I}_{23/2}\left(\sqrt{3}{\pi}\sqrt{m^{2}+4m}\right)}\,,$
(3.94)

is a decreasing function of $m$, and therefore we cannot bound (3.93) by its
value at $m\to\infty$, which is $2d(k-1)e^{-\sqrt{3}\pi k}$. As a function of
$m$, the right hand side of (3.93) is decreasing at $m=6$ and increasing for
large enough $m$. Since for $m\geq 6$ it only has one local extremum, which is
a minimum, (3.93) is bounded by its values at $m=6$ and by the limiting value
$m\to\infty$. Computing the value of (3.93) for $m=6$ and $m\to\infty$, for
all four cases $k=1,2,3,4$ we find that the maximum value is given by the
value at $m=6$. These numerical values are smaller than
$0.231,0.022,0.003,6\cdot 10^{-4}$ for $k=1,2,3,4$, respectively. In other
words, we have

$\sum_{k=1}^{4}\frac{2\,c_{m}^{F}(-1,m-k)((m-k))^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta((m-k))^{2}+4m)}\right)}{c_{m}^{F}(-1,m)(m^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta(m^{2}+4m)}\right)}\,<\,0.255$
(3.95)

for all $m\geq 6$.

We analyse the cases $k=5,6$ separately. For this we first compute the Fourier
coefficients $c_{m}^{F}(-1,m-k)$ for $m=6$ and $k=5,6$,

$c^{F}_{6}(-1,1)\;=\;3d(4)+5d(0)\;=\;52888,\;\;\;c^{F}_{6}(-1,0)\;=\;2d(5)+4d(2)\;=\;2160240\,.$

With these we can bound $c^{F}_{m}(-1,m-5)<1.0003\,\hat{c}^{F}_{m}(-1,m-5)$
and $c^{F}_{m}(-1,m-6)<1.006\,\hat{c}^{F}_{m}(-1,m-6)$ for $m\geq 6$. To bound
the quantities

$\frac{2\,c_{m}^{F}(-1,m-k)((m-k))^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta((m-k))^{2}+4m)}\right)}{c_{m}^{F}(-1,m)(m^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta(m^{2}+4m)}\right)}\,,$
(3.96)

for $k=5,6$, we follow the same reasoning as before. Again, the ratio

$\frac{((m-k))^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{((m-k))^{2}+4m)}\right)}{(m^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m^{2}+4m)}\right)}\,,$
(3.97)

is initially decreasing with $m$ and asymptotes to $e^{-\sqrt{3}\pi k}$ for
$m\to\infty$, having only one minimum in this range. Thus, we can bound (3.96)
by the values at the extrema which turn out to be given by the values at
$m=6$, which are smaller than $0.0012$ and $0.003$ for $k=5,6$ respectively.
Thus, we have

$\sum_{k=1}^{6}\frac{2\,c_{m}^{F}(-1,m-k)((m-k)^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta((m-k)^{2}+4m)}\right)}{c_{m}^{F}(-1,m)(m^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta(m^{2}+4m)}\right)}\,<\,0.26$
(3.98)

for all $m\geq 6$.

Finally, we bound the remaining the contributions

$\sum_{k=7}^{m}\frac{2\,c_{m}^{F}(-1,m-k)((m-k)^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{((m-k)^{2}+4m)}\right)}{c_{m}^{F}(-1,m)(m^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m^{2}+4m)}\right)}\,,$
(3.99)

for $m\geq 7$.

From Lemma 5, we can use the inequality (3.66) to show that each summand in
(3.99) is smaller than

$\frac{2(\kappa_{4}+\kappa_{5}\,{k^{1/2}})d(k-1)((m-k)^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{((m-k)^{2}+4m)}\right)}{(m^{2}+4m)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(m^{2}+4m)}\right)}\,.$
(3.100)

In the limit $m\to\infty$ the quantity in (3.100) becomes

$2(\kappa_{4}+\kappa_{5}\,{k^{1/2}})d(k-1)e^{-\sqrt{3}\pi k}\,.$ (3.101)

To bound the summands in (3.99) for all possible values of $m$ one can proceed
as in the previous cases. In the range $m\geq k$ the quantity (3.100) has only
one minimum, being bounded by its values at the extrema of the domain. The
limit $m\to\infty$ is given by (3.101) and the first possible value $m$ can
take for each $k$ is $m=k$, i.e.,

$\frac{2(\kappa_{4}+\kappa_{5}\,{k^{1/2}})d(k-1)(4k)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{4k)}\right)}{(k^{2}+4k)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(k^{2}+4k)}\right)}\,.$
(3.102)

Using the bound (3.13) and the second inequality in (3.14), we can write

$\begin{split}&\frac{2(\kappa_{4}+\kappa_{5}\,{k^{1/2}})d(k-1)(4k)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{4k)}\right)}{(k^{2}+4k)^{23/4}I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(k^{2}+4k)}\right)}<\\\
&\left(1+\frac{1}{12}\right)\frac{4\pi(\kappa_{4}+\kappa_{5}\,{k^{1/2}})(4k)^{11/2}}{(k-1)^{13/2}(k^{2}+4k)^{11/2}}\,I_{13}\left(4\pi\sqrt{k-1}\right)\,e^{-\sqrt{3}\pi(\sqrt{k^{2}+4k}-2\sqrt{k})}=e_{f}(k)\,.\end{split}$
(3.103)

This final estimate $e_{f}(k)$ is larger than (3.101) for all $k\geq 7$, and
decreasing with $k$. Therefore we can bound (3.99) by

$\sum_{k=7}^{\infty}e_{f}(k)\,<\,\int_{6}^{\infty}dk\,e_{f}(k)\,<\,0.005\,,$
(3.104)

so that

$\sum_{k=1}^{m}\frac{2\,c_{m}^{F}(-1,m-k)((m-k))^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta((m-k))^{2}+4m)}\right)}{c_{m}^{F}(-1,m)(m^{2}+4m)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta(m^{2}+4m)}\right)}<0.3\,.$
(3.105)

∎

###  3.7. Final bounds

The proof of Theorem 1 for the cases $m\geq 6$ follows from combining the
results of the previous lemmas, as follows

###### Proof of Theorem 1 for $m\geq 6$.

From the Rademacher expansion for the coefficients $c_{m}^{F}({n},{\ell})$
(2.36) with $\Delta>0$, and $n\geq m$, $0\leq\ell\leq m$, we can bound
$(-1)^{\ell+1}c_{m}^{F}({n},{\ell})$ from below by the positive contribution
minus the absolute value of the other contributions,

$\begin{split}(-1)^{\ell+1}c_{m}^{F}({n},{\ell})\,\geq\,&\sum_{\begin{subarray}{c}\tilde{n}\geq-1\\\
\tilde{\Delta}<0\end{subarray}}C_{\mathrm{pos.}}(\Delta,m,\tilde{n})\left(1-\frac{2}{21}\right)\\\
&-2\pi\left(1+\frac{2}{21}\right)\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{n}\geq-1,\\\
\tilde{\Delta}<0\end{subarray}}\frac{2}{\sqrt{2m}}\,c^{F}_{m}(\tilde{n},\tilde{\ell})\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)\\\
&-\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|-\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|\,,\end{split}$
(3.106)

where we have used Lemma 1 for writing the bounds on the absolute values in
the first and second line. The positive contribution with $\tilde{n}=-1$ is
the dominant term which was used in Lemma 2 to bound the last line. We have

$\begin{split}&C_{\mathrm{pos.}}(\Delta,m,-1)\left(1-\frac{2}{21}\right)-\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|-\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|\\\
\;=\;\,&C_{\mathrm{pos.}}(\Delta,m,-1)\left(1-e_{12}(\Delta,m)-e_{25/2}(\Delta,m)-\frac{2}{21}\right)\,.\end{split}$
(3.107)

Finally, from Lemmas 2, 7, 8 and 9 we have

$\begin{split}(-1)^{\ell+1}c_{m}^{F}({n},{\ell})\,\geq\,&C_{\mathrm{pos.}}(\Delta,m,-1)\left(1-e_{23/2}(\Delta,m,-1)\left(1+\frac{2}{21}\right)\right.\\\
&-e_{12}(\Delta,m)-e_{25/2}(\Delta,m)-\frac{2}{21}\Bigg{)}\\\
&+C_{\mathrm{pos.}}(\Delta,m,0)\left(1-e_{23/2}(\Delta,m,0)\left(1+\frac{2}{21}\right)-\frac{2}{21}\right)\\\
&+\sum_{\begin{subarray}{c}\tilde{n}\geq 1\\\
\tilde{\Delta}<0\end{subarray}}C_{\mathrm{pos.}}(\Delta,m,\tilde{n})\left(1-e_{23/2}(\Delta,m,\tilde{n})\left(1+\frac{2}{21}\right)-\frac{2}{21}\right)\,>0\,,\end{split}$
(3.108)

since for $m\geq 6$ and $\Delta\geq 3m^{2}$,

$e_{23/2}(\Delta,m,-1)\left(1+\frac{2}{21}\right)+e_{12}(\Delta,m)+e_{25/2}(\Delta,m)+\frac{2}{21}\,<\,0.45\,<\,1\,,$
(3.109)
$e_{23/2}(\Delta,m,0)\left(1+\frac{2}{21}\right)+\frac{2}{21}\,<\,0.25\,<\,1\,,$
(3.110)

and

$e_{23/2}(\Delta,m,\tilde{n})\left(1+\frac{2}{21}\right)+\frac{2}{21}\,<\,0.35\,<\,1\,.$
(3.111)

∎

The cases $m=1,2$ were proven in [30] and the proof for the remaining cases
$m=3,4,5$ is given in Appendix A.

## Acknowledgements

I would like to thank the participants of the GASP workshop at IPMU Tokyo,
Justin David, and Sameer Murthy for useful and enjoyable discussions. I would
like to thank IPMU Tokyo and the University of Amsterdam for hospitality, and
the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support
and hospitality during the programme Black holes: bridges between number
theory and holographic quantum information, where work on this paper was
undertaken. This work was supported by EPSRC grant EP/R014604/1.

##  Appendix A The first three cases $m=3,4,5$

In this Appendix we prove the statement of the main Theorem 1 for the cases
$m=3,4,5$. The proof is illustrative of the power of the Rademacher expansion
to easily prove the positivity on a case by case basis when the polar
coefficients are known.

###### Proof of Theorem 1 for $m=3,4,5$.

Starting with $m=3$, we first compute the polar coefficients
$c^{F}_{3}(\tilde{n},\tilde{\ell})$ with $\tilde{\Delta}<0$ that enter the
formula (2.36). As earlier, since we are instructed to sum over
$\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}$, we can choose the set of
representatives $-m<\tilde{\ell}\leq m$, which we can further reduce to
$0\leq\tilde{\ell}\leq m$ due to the symmetry
$c^{F}_{m}(\tilde{n},\tilde{\ell})=c^{F}_{m}(\tilde{n},-\tilde{\ell})$. We
have

$c^{F}_{3}(\tilde{n},\tilde{\ell})$ | $\tilde{\ell}=3$ | $\tilde{\ell}=2$ | $\tilde{\ell}=1$ | $\tilde{\ell}=0$
---|---|---|---|---
$\tilde{n}=-1$ | 5 | 96 | 972 | 6404
$\tilde{n}=0$ | 1728 | 15600 | 85176 |

where these coefficients are computed from the formula (2.45). More
explicitly,

$\begin{split}c_{3}^{F}(-1,3)_{\tilde{\Delta}=-21}&\;=\;5d(-1)d(-1)\;=\;5,\\\
c_{3}^{F}(-1,2)_{\tilde{\Delta}=-16}&\;=\;4d(-1)d(0)\;=\;96,\\\
c_{3}^{F}(-1,1)_{\tilde{\Delta}=-13}&\;=\;3d(-1)d(1)\;=\;972,\\\
c_{3}^{F}(-1,0)_{\tilde{\Delta}=-12}&\;=\;2d(-1)d(2)+4d(-1)d(-1)\;=\;6404,\end{split}$
(A.1)

and

$\begin{split}c_{3}^{F}(0,3)_{\tilde{\Delta}=-9}&\;=\;3d(0)d(0)\;=\;1728,\\\
c_{3}^{F}(0,2)_{\tilde{\Delta}=-4}&\;=\;2d(0)d(1)+2d(0)d(-1)\;=\;15600,\\\
c_{3}^{F}(0,1)_{\tilde{\Delta}=-1}&\;=\;d(0)d(2)+d(0)d(1)+d(0)d(0)+d(0)d(-1)\;=\;85176.\end{split}$
(A.2)

From these values one can directly see that, roughly, for smaller
$|\tilde{\Delta}|$ the polar coefficients are larger and therefore could
compete with the Bessel function growth. As in the main text, we want to show
now that, for each $\tilde{n}$, the contribution with $\tilde{\ell}=3$ to the
Rademacher expansion is larger than the contributions with $\tilde{\ell}<3$,
being enough to prove it for $\gamma=1$. That is, for a given $m$, we want to
show that for each $\tilde{n}$, the term

$2\pi\,c^{F}_{m}(\tilde{n},m)\frac{1}{\sqrt{2m}}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)$
(A.3)

majorates

$\left|(-1)^{\ell+1}2\pi\sum_{\begin{subarray}{c}-m<\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}c^{F}_{m}(\tilde{n},\tilde{\ell}){{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},1,\psi)_{\ell\tilde{\ell}}}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)\right|\,.$
(A.4)

For $\gamma=1$,

$\left|{{\rm
Kl}(\frac{\Delta}{4m},\frac{\tilde{\Delta}}{4m},1,\psi)_{\ell\tilde{\ell}}}\right|\,\leq\,\frac{1}{\sqrt{2m}}\,,$
(A.5)

so that the ratio to bound is

$\sum_{\begin{subarray}{c}-m<\tilde{\ell}<m\\\
\tilde{\Delta}<0\end{subarray}}\frac{2c^{F}_{m}(\tilde{n},\tilde{\ell}){|4m\tilde{n}-\tilde{\ell}^{2}|}^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|4m\tilde{n}-\tilde{\ell}^{2}|}\right)}{c^{F}_{m}(\tilde{n},m){|4m\tilde{n}-m^{2}|}^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|4m\tilde{n}-m^{2}|}\right)}\,.$
(A.6)

For the case $m=3$ we can just evaluate each numerical value. Using the fact
that the function

$\frac{I_{\nu}(ax)}{I_{\nu}(bx)}$ (A.7)

is a decreasing function of $x$ for $0<a<b$ and $\nu>0$, we can use the
inequality $\Delta=4mn-\ell^{2}\geq 3m^{2}=27$ to bound

$\begin{split}e_{23/2}(\Delta,3,-1)\;=\;&\sum_{\tilde{\ell}=0}^{2}\frac{2c_{3}^{F}(-1,\tilde{\ell})}{c_{3}^{F}(-1,3)}\frac{|12+\tilde{\ell}^{2}|^{23/4}}{|12+9^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(12+\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(12+9^{2})}\right)}\,\leq\,\\\
&\sum_{\tilde{\ell}=0}^{2}\frac{2c_{3}^{F}(-1,\tilde{\ell})}{c_{3}^{F}(-1,3)}\frac{|12+\tilde{\ell}^{2}|^{23/4}}{|12+9^{2}|^{23/4}}\frac{I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(12+\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\sqrt{3}{\pi}\sqrt{(12+9^{2})}\right)}\,<\,0.44\,.\end{split}$
(A.8)

For $\tilde{n}=0$ applying the same reasoning we get

$e_{23/2}(\Delta,3,0)\;=\;\sum_{\tilde{\ell}=1}^{2}\frac{2c_{3}^{F}(0,\tilde{\ell})}{c_{3}^{F}(0,3)}\frac{|\tilde{\ell}^{2}|^{23/4}}{|9^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(9^{2})}\right)}<0.0002\,.$
(A.9)

For the case $m=4$ we follow exactly the same reasoning as previously, being
therefore less explicit. The non-zero polar coefficients
$c^{F}_{4}(\tilde{n},\tilde{\ell})$ are

$c^{F}_{4}(\tilde{n},\tilde{\ell})$ | $\tilde{\ell}=4$ | $\tilde{\ell}=3$ | $\tilde{\ell}=2$ | $\tilde{\ell}=1$ | $\tilde{\ell}=0$
---|---|---|---|---|---
$\tilde{n}=-1$ | 6 | 120 | 1296 | 9600 | 51396
$\tilde{n}=0$ | 2304 | 23328 | 154752 | 700776 |

With this information we can compute

$e_{23/2}(\Delta,4,-1)\;=\;\sum_{\tilde{\ell}=0}^{3}\frac{2c_{4}^{F}(-1,\tilde{\ell})}{c_{4}^{F}(-1,4)}\frac{|16+\tilde{\ell}^{2}|^{23/4}}{|16+16^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(16+\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(16+16^{2})}\right)}\,<\,0.28\,,$
(A.10)

and

$e_{23/2}(\Delta,4,0)\;=\;\sum_{\tilde{\ell}=1}^{3}\frac{2c_{4}^{F}(0,\tilde{\ell})}{c_{4}^{F}(0,4)}\frac{|\tilde{\ell}^{2}|^{23/4}}{|16^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(16^{2})}\right)}<0.002\,,$
(A.11)

where we have used again $\Delta\geq 3m^{2}$.

Finally, the polar coefficients for $m=5$ are

$c^{F}_{5}(\tilde{n},\tilde{\ell})$ | $\tilde{\ell}=5$ | $\tilde{\ell}=4$ | $\tilde{\ell}=3$ | $\tilde{\ell}=2$ | $\tilde{\ell}=1$ | $\tilde{\ell}=0$
---|---|---|---|---|---|---
$\tilde{n}=-1$ | 7 | 144 | 1620 | 12800 | 76955 | 353808
$\tilde{n}=0$ | 2880 | 31104 | 230472 | 1246800 | 4930920 |
$\tilde{n}=1$ | 315255 |  |  |  |  |

which yield

$e_{23/2}(\Delta,5,-1)\;=\;\sum_{\tilde{\ell}=0}^{4}\frac{2c_{5}^{F}(-1,\tilde{\ell})}{c_{5}^{F}(-1,5)}\frac{|20+\tilde{\ell}^{2}|^{23/4}}{|16+16^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(20+\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(20+25^{2})}\right)}\,<\,0.24\,,$
(A.12)

and

$e_{23/2}(\Delta,5,0)\;=\;\sum_{\tilde{\ell}=1}^{4}\frac{2c_{5}^{F}(0,\tilde{\ell})}{c_{5}^{F}(0,5)}\frac{|\tilde{\ell}^{2}|^{23/4}}{|25^{2}|^{23/4}}\frac{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(\tilde{\ell}^{2})}\right)}{I_{23/2}\left(\frac{\pi}{3}\sqrt{\Delta(25^{2})}\right)}\,<\,0.005\,.$
(A.13)

To complete the proof of positivity of these Fourier coefficients we need to
also bound the contribution coming from the mock terms with Bessel functions
of different index. As in the cases $m\geq 6$, we bound them with respect to
the largest contribution coming from the $\tilde{n}=-1$ term. For the $I_{12}$
terms, from (3.22), and following the same reasoning as in Lemma 2, the
quantity

$e_{12}(\Delta,m)\;=\;\frac{\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|}{2\pi\,c^{F}_{m}(-1,m)\frac{1}{\sqrt{2m}}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}$
(A.14)

can be shown to decrease with $m$, with the values for $m=3,4,5$ satisfying
$e_{12}(\Delta,3)<0.15$, $e_{12}(\Delta,4)<0.03$, and
$e_{12}(\Delta,5)<0.004$.

The contribution from the terms containing a Bessel function of index $25/2$
is treated differently that for the cases $m\geq 6$, since a tighter bound is
needed. We first focus on the $\gamma=1$ contribution. Its absolute value,
using (A.5), can be written as

$\begin{split}&\left|C_{25/2}(m,n,\ell,1)\right|\;=\;\frac{1}{2\pi}d(m)\sum_{\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}}\frac{1}{\sqrt{2m}}\left(\frac{4m}{\Delta}\right)^{25/4}\\\
&\int_{\frac{-1}{\sqrt{m}}}^{\frac{1}{\sqrt{m}}}dx^{\prime}\left|f_{1,\tilde{\ell},m}(x^{\prime})\right|(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right).\\\ \end{split}$ (A.15)

The ratio of this quantity against the positive contribution (3.5) with
$\tilde{n}=-1$ can be majorated by

$\begin{split}&\frac{1}{C_{\mathrm{pos}}(3m^{2},m,-1)}\frac{1}{2\pi}d(m)\sum_{\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}}\frac{1}{\sqrt{2m}}\left(\frac{4m}{3m^{2}}\right)^{25/4}\\\
&\int_{\frac{-1}{\sqrt{m}}}^{\frac{1}{\sqrt{m}}}dx^{\prime}\left|f_{1,\tilde{\ell},m}(x^{\prime})\right|(1-mx^{\prime
2})^{25/4}I_{25/2}\left({2\sqrt{3}\pi}\sqrt{m(1-mx^{\prime 2})}\right).\\\
\end{split}$ (A.16)

We can compute this quantity for $m=3,4,5$, yielding values which are smaller
than $0.06,0.01$ and $0.001$, respectively. For the higher gamma contributions
we use the previous bounds (3.24) and (3.30) together with the bounds for the
sum over $\gamma$ as in Proposition 2 to obtain the inequality

$\begin{split}&\left|\frac{1}{2\pi}d(m)\sum_{\gamma=2}^{+\infty}\sum_{\tilde{\ell}\in\mathbb{Z}/2m\mathbb{Z}}\sum_{\begin{subarray}{c}g\in\mathbb{Z}/2m\gamma\mathbb{Z}\\\
g=\tilde{\ell}\text{ mod }2m\end{subarray}}\frac{{\rm
Kl}(\frac{\Delta}{4m},-1-\frac{g^{2}}{4m};\gamma,\psi)_{\ell\tilde{\ell}}}{\gamma^{2}}\left(\frac{4m}{\Delta}\right)^{25/4}\right.\\\
&\left.\int_{\frac{-1}{\sqrt{m}}}^{\frac{1}{\sqrt{m}}}dx^{\prime}f_{\gamma,g,m}(x^{\prime})(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\gamma\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right)\right|\\\
&\,\leq\,\frac{1}{2\pi}d(m)\pi^{2}\left(m^{2}\frac{2}{21}\frac{\pi^{2}}{3}+\frac{2}{25}\frac{1}{3}\right)\left(\frac{4m}{\Delta}\right)^{25/4}\\\
&\int_{\frac{-1}{\sqrt{m}}}^{\frac{1}{\sqrt{m}}}dx^{\prime}(1-mx^{\prime
2})^{25/4}I_{25/2}\left(\frac{2\pi}{\sqrt{m}}\sqrt{\Delta(1-mx^{\prime
2})}\right).\end{split}$ (A.17)

The ratio of this quantity over the positive contribution (3.5) can be bounded
using $\Delta\geq 3m^{2}$ value, and the ratios for $\Delta=3m^{2}$ have
values smaller than $0.04,0.008,0.002$ for $m=3,4,5$, respectively, so that we
have shown that

$e_{25/2}(\Delta,m)\;=\;\frac{\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|}{2\pi\,c^{F}_{m}(-1,m)\frac{1}{\sqrt{2m}}\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)}\,,$
(A.18)

for $m=3,4,5$, satisfies $e_{25/2}(\Delta,3)<0.1$, $e_{25/2}(\Delta,4)<0.02$
and $e_{25/2}(\Delta,5)<0.003$.

We conclude by following the same steps as in the proof of Theorem 1 for
$m\geq 6$. We have

$\begin{split}(-1)^{\ell+1}c_{m}^{F}({n},{\ell})\,\geq\,&\sum_{\begin{subarray}{c}\tilde{n}\geq-1\\\
\tilde{\Delta}<0\end{subarray}}C_{\mathrm{pos.}}(\Delta,m,\tilde{n})\left(1-\frac{2}{21}\right)\\\
&-2\pi\left(1+\frac{2}{21}\right)\sum_{\begin{subarray}{c}0\leq\tilde{\ell}<m\\\
\tilde{n}\geq-1,\\\
\tilde{\Delta}<0\end{subarray}}2c^{F}_{m}(\tilde{n},\tilde{\ell})\left(\frac{|\tilde{\Delta}|}{\Delta}\right)^{23/4}I_{23/2}\left(\frac{\pi}{m}\sqrt{\Delta|\tilde{\Delta}|}\right)\\\
&-\left|\sum\limits_{\gamma=1}^{+\infty}C_{12}(m,n,\ell,\gamma)\right|-\left|\sum\limits_{\gamma=1}^{+\infty}C_{25/2}(m,n,\ell,\gamma)\right|\,,\end{split}$
(A.19)

which we can write as follows for the three cases at hand

$\begin{split}(-1)^{\ell+1}c_{m}^{F}({n},{\ell})\,\geq\,&C_{\mathrm{pos.}}(\Delta,m,-1)\left(1-e_{23/2}(\Delta,m,-1)\left(1+\frac{2}{21}\right)\right.\\\
&-e_{12}(\Delta,m)-e_{25/2}(\Delta,m)-\frac{2}{21}\Bigg{)}\\\
&+C_{\mathrm{pos.}}(\Delta,m,0)\left(1-e_{23/2}(\Delta,m,0)\left(1+\frac{2}{21}\right)-\frac{2}{21}\right)\,>\,0\,,\end{split}$
(A.20)

where the last inequality follows from the previous numerical values for the
bounds.

∎

## References

  * [1] A. Sen, “How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?,” Gen. Rel. Grav. 43 (2011) 2171–2183, 1008.4209.
  * [2] A. Sen, “Extremal black holes and elementary string states,” Mod. Phys. Lett. A 10 (1995) 2081–2094, hep-th/9504147.
  * [3] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B 379 (1996) 99–104, hep-th/9601029.
  * [4] A. Sen, “Quantum Entropy Function from AdS(2)/CFT(1) Correspondence,” Int. J. Mod. Phys. A 24 (2009) 4225–4244, 0809.3304.
  * [5] N. Banerjee, S. Banerjee, R. K. Gupta, I. Mandal, and A. Sen, “Supersymmetry, Localization and Quantum Entropy Function,” JHEP 02 (2010) 091, 0905.2686.
  * [6] A. Dabholkar, J. Gomes, and S. Murthy, “Quantum black holes, localization and the topological string,” JHEP 06 (2011) 019, 1012.0265.
  * [7] A. Dabholkar, J. Gomes, and S. Murthy, “Localization & Exact Holography,” JHEP 04 (2013) 062, 1111.1161.
  * [8] R. K. Gupta and S. Murthy, “All solutions of the localization equations for N=2 quantum black hole entropy,” JHEP 02 (2013) 141, 1208.6221.
  * [9] A. Dabholkar, J. Gomes, and S. Murthy, “Nonperturbative black hole entropy and Kloosterman sums,” JHEP 03 (2015) 074, 1404.0033.
  * [10] R. K. Gupta, Y. Ito, and I. Jeon, “Supersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets,” JHEP 11 (2015) 197, 1504.01700.
  * [11] S. Murthy and V. Reys, “Functional determinants, index theorems, and exact quantum black hole entropy,” JHEP 12 (2015) 028, 1504.01400.
  * [12] S. Murthy and V. Reys, “Single-centered black hole microstate degeneracies from instantons in supergravity,” JHEP 04 (2016) 052, 1512.01553.
  * [13] I. Jeon and S. Murthy, “Twisting and localization in supergravity: equivariant cohomology of BPS black holes,” JHEP 03 (2019) 140, 1806.04479.
  * [14] B. de Wit, S. Murthy, and V. Reys, “BRST quantization and equivariant cohomology: localization with asymptotic boundaries,” JHEP 09 (2018) 084, 1806.03690.
  * [15] L. V. Iliesiu, S. Murthy, and G. J. Turiaci, “Black hole microstate counting from the gravitational path integral,” 2209.13602.
  * [16] G. Lopes Cardoso, A. Kidambi, S. Nampuri, V. Reys, and M. Rosselló, “The Gravitational Path Integral for $N=4$ BPS Black Holes from Black Hole Microstate Counting,” Annales Henri Poincare 24 (2023), no. 10, 3305–3346, 2211.06873.
  * [17] A. Sen, “Revisiting localization for BPS black hole entropy,” 2302.13490.
  * [18] A. González Lezcano, I. Jeon, and A. Ray, “Supersymmetry and complexified spectrum on Euclidean AdS2,” Phys. Rev. D 108 (2023), no. 4, 045018, 2305.12925.
  * [19] A. Dabholkar, J. Gomes, S. Murthy, and A. Sen, “Supersymmetric Index from Black Hole Entropy,” JHEP 04 (2011) 034, 1009.3226.
  * [20] N. Banerjee, I. Mandal, and A. Sen, “Black Hole Hair Removal,” JHEP 07 (2009) 091, 0901.0359.
  * [21] D. P. Jatkar, A. Sen, and Y. K. Srivastava, “Black Hole Hair Removal: Non-linear Analysis,” JHEP 02 (2010) 038, 0907.0593.
  * [22] A. Sen, “Arithmetic of Quantum Entropy Function,” JHEP 08 (2009) 068, 0903.1477.
  * [23] C. Bachas and E. Kiritsis, “F(4) terms in N=4 string vacua,” Nucl. Phys. B Proc. Suppl. 55 (1997) 194–199, hep-th/9611205.
  * [24] A. Gregori, E. Kiritsis, C. Kounnas, N. A. Obers, P. M. Petropoulos, and B. Pioline, “R**2 corrections and nonperturbative dualities of N=4 string ground states,” Nucl. Phys. B 510 (1998) 423–476, hep-th/9708062.
  * [25] R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, “Counting dyons in N=4 string theory,” Nucl. Phys. B 484 (1997) 543–561, hep-th/9607026.
  * [26] A. Chattopadhyaya and J. R. David, “Horizon states and the sign of their index in ${\mathcal{N}}=4$ dyons,” JHEP 03 (2021) 106, 2010.08967.
  * [27] A. Dabholkar, M. Guica, S. Murthy, and S. Nampuri, “No entropy enigmas for N=4 dyons,” JHEP 06 (2010) 007, 0903.2481.
  * [28] M. C. N. Cheng and E. Verlinde, “Dying Dyons Don’t Count,” JHEP 09 (2007) 070, 0706.2363.
  * [29] A. Sen, “Black Hole Entropy Function, Attractors and Precision Counting of Microstates,” Gen. Rel. Grav. 40 (2008) 2249–2431, 0708.1270.
  * [30] K. Bringmann and S. Murthy, “On the positivity of black hole degeneracies in string theory,” Commun. Num. Theor Phys. 07 (2013) 15–56, 1208.3476.
  * [31] A. Dabholkar, S. Murthy, and D. Zagier, “Quantum Black Holes, Wall Crossing, and Mock Modular Forms,” 1208.4074.
  * [32] N. Banerjee, D. P. Jatkar, and A. Sen, “Asymptotic Expansion of the N=4 Dyon Degeneracy,” JHEP 05 (2009) 121, 0810.3472.
  * [33] S. Murthy and B. Pioline, “A Farey tale for N=4 dyons,” JHEP 09 (2009) 022, 0904.4253.
  * [34] K. Bringmann and J. Manschot, “From sheaves on $P^{2}$ to a generalization of the Rademacher expansion,” Am. J. Math. 135 (2013), no. 4, 1039–1065, 1006.0915.
  * [35] F. Ferrari and V. Reys, “Mixed Rademacher and BPS Black Holes,” JHEP 07 (2017) 094, 1702.02755.
  * [36] A. Chowdhury, A. Kidambi, S. Murthy, V. Reys, and T. Wrase, “Dyonic black hole degeneracies in $\mathcal{N}=4$ string theory from Dabholkar-Harvey degeneracies,” JHEP 10 (2020) 184, 1912.06562.
  * [37] G. L. Cardoso, S. Nampuri, and M. Rosselló, “Arithmetic of decay walls through continued fractions: a new exact dyon counting solution in $\mathcal{N}$ = 4 CHL models,” JHEP 03 (2021) 154, 2007.10302.
  * [38] G. L. Cardoso, S. Nampuri, and M. Rosselló, “Rademacher expansion of a Siegel modular form for $\mathcal{N}=4$ counting,” 2112.10023.
  * [39] A. Chowdhury, R. S. Garavuso, S. Mondal, and A. Sen, “BPS State Counting in N=8 Supersymmetric String Theory for Pure D-brane Configurations,” JHEP 10 (2014) 186, 1405.0412.
  * [40] A. Chowdhury, R. S. Garavuso, S. Mondal, and A. Sen, “Do All BPS Black Hole Microstates Carry Zero Angular Momentum?,” JHEP 04 (2016) 082, 1511.06978.
  * [41] S. Maji and A. Chowdhury, “Counting $\mathcal{N}=8$ Black Holes as Algebraic Varieties,” 2311.04786.
  * [42] S. Chaudhuri, G. Hockney, and J. D. Lykken, “Maximally supersymmetric string theories in D $<$ 10,” Phys. Rev. Lett. 75 (1995) 2264–2267, hep-th/9505054.
  * [43] A. Chattopadhyaya and J. R. David, “Dyon degeneracies from Mathieu moonshine symmetry,” Phys. Rev. D 96 (2017), no. 8, 086020, 1704.00434.
  * [44] A. Chattopadhyaya and J. R. David, “Properties of dyons in $\mathcal{N}$ = 4 theories at small charges,” JHEP 05 (2019) 005, 1810.12060.
  * [45] S. Govindarajan, S. Samanta, P. Shanmugapriya, and A. Virmani, “Positivity of discrete information for CHL black holes,” Nucl. Phys. B 987 (2023) 116095, 2205.08726.
  * [46] A. Chattopadhyaya, “Sign of BPS index for ${\mathcal{N}}=4$ dyons,” 2303.03913.
  * [47] S. M. Harrison, J. A. Harvey, and N. M. Paquette, “Snowmass White Paper: Moonshine,” 2201.13321.
  * [48] M. C. N. Cheng, The Spectra of Supersymmetric States in String Theory.  PhD thesis, Amsterdam U., 2008.  0807.3099.
  * [49] I. Mandal and A. Sen, “Black Hole Microstate Counting and its Macroscopic Counterpart,” Class. Quant. Grav. 27 (2010) 214003, 1008.3801.
  * [50] A. Dabholkar and S. Nampuri, “Quantum black holes,” Lect. Notes Phys. 851 (2012) 165–232, 1208.4814.
  * [51] S. Banerjee and A. Sen, “Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on T**6,” JHEP 03 (2008) 022, 0712.0043.
  * [52] S. Banerjee and A. Sen, “S-duality Action on Discrete T-duality Invariants,” JHEP 04 (2008) 012, 0801.0149.
  * [53] S. Banerjee, A. Sen, and Y. K. Srivastava, “Partition Functions of Torsion $>$ 1 Dyons in Heterotic String Theory on T**6,” JHEP 05 (2008) 098, 0802.1556.
  * [54] A. Dabholkar, J. Gomes, and S. Murthy, “Counting all dyons in N =4 string theory,” JHEP 05 (2011) 059, 0803.2692.
  * [55] A. Dabholkar and J. A. Harvey, “Nonrenormalization of the Superstring Tension,” Phys. Rev. Lett. 63 (1989) 478.
  * [56] J. R. David and A. Sen, “CHL Dyons and Statistical Entropy Function from D1-D5 System,” JHEP 11 (2006) 072, hep-th/0605210.
  * [57] D. Shih, A. Strominger, and X. Yin, “Recounting Dyons in N=4 string theory,” JHEP 10 (2006) 087, hep-th/0505094.
  * [58] D. Gaiotto, “Re-recounting dyons in N=4 string theory,” hep-th/0506249.
  * [59] G. Bossard, C. Cosnier-Horeau, and B. Pioline, “Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds,” SciPost Phys. 7 (2019), no. 3, 028, 1806.03330.
  * [60] M. Eichler and D. Zagier, The Theory of Jacobi Forms.  Progress in Mathematics. Birkhäuser Boston, 2013.
  * [61] V. A. Gritsenko and V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras. Part 1.,”.
  * [62] V. A. Gritsenko and V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras. Part 2,” alg-geom/9611028.
  * [63] R. E. Borcherds, “Automorphic forms on os + 2,2(r) and infinte products.,” Inventiones mathematicae 120 (1995), no. 1, 161–214.
  * [64] A. Sen, “Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories,” JHEP 05 (2007) 039, hep-th/0702141.
  * [65] S. Banerjee, A. Sen, and Y. K. Srivastava, “Genus Two Surface and Quarter BPS Dyons: The Contour Prescription,” JHEP 03 (2009) 151, 0808.1746.
  * [66] S. Ferrara, R. Kallosh, and A. Strominger, “N=2 extremal black holes,” Phys. Rev. D 52 (1995) R5412–R5416, hep-th/9508072.
  * [67] G. W. Moore, “Arithmetic and attractors,” hep-th/9807087.
  * [68] S. Zwegers, “Mock theta functions,” 2008.
  * [69] A. Sen, “Negative discriminant states in N=4 supersymmetric string theories,” JHEP 10 (2011) 073, 1104.1498.
  * [70] A. Bhand, A. Sen, and R. K. Singh, “Mock Modularity In CHL Models,” 2311.16252.
  * [71] D. Zagier and N.-P. Skoruppa, “A trace formula for jacobi forms.,” Journal für die reine und angewandte Mathematik 393 (1989) 168–198.
  * [72] H. Rademacher, “On the partition function p(n),” Proceedings of the London Mathematical Society s2-43 (1938), no. 1, 241–254, https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s2-43.4.241.
  * [73] T. Apostol, Modular Functions and Dirichlet Series in Number Theory.  Graduate Texts in Mathematics. Springer New York, 2012.
  * [74] G. L. Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, “Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy,” JHEP 12 (2004) 075, hep-th/0412287.
  * [75] J. H. Conway and F. Y. C. Fung, The Sensual Quadratic Form, vol. 26.  Mathematical Association of America, 1 ed., 1997.
  * [76] M. Cvetic and A. A. Tseytlin, “Solitonic strings and BPS saturated dyonic black holes,” Phys. Rev. D 53 (1996) 5619–5633, hep-th/9512031. [Erratum: Phys.Rev.D 55, 3907 (1997)].
  * [77] M. Cvetic and D. Youm, “Dyonic BPS saturated black holes of heterotic string on a six torus,” Phys. Rev. D 53 (1996) 584–588, hep-th/9507090.
  * [78] A. Baricz, “Bounds for modified bessel functions of the first and second kinds,” Proceedings of the Edinburgh Mathematical Society 53 (2010), no. 3, 575–599.
  * [79] K. Bringmann, L. Rolen, and S. Zwegers, “On the Fourier coefficients of negative index meromorphic Jacobi forms,” 1501.04476.