# On the Euler characteristic of the commutative graph complex and the top
weight cohomology of $\mathcal{M}_{g}$

Michael Borinsky  Michael Borinsky
Institute for Theoretical Studies
ETH Zürich
8092 Zürich, Switzerland

###### Abstract.

We prove an asymptotic formula for the Euler characteristic of Kontsevich’s
commutative graph complex. This formula implies that the total amount of
commutative graph homology grows super-exponentially with the rank and, via a
theorem of Chan, Galatius, and Payne, that the dimension of the top weight
cohomology of the moduli space of curves, $\mathcal{M}_{g}$, grows super-
exponentially with the genus $g$.

##  1\. Introduction

A _graph_ $G$ is a one-dimensional CW complex. It has _rank_ $g$ if its
fundamental group is a free group of rank $g$, $\pi_{1}(G)\simeq F_{g}$, and
we call it _admissible_ if it has no vertices of degree $0$, $1$, and $2$. A
graph is _even-orientable_ if it has no automorphism inducing an odd
permutation on its edges.

Kontsevich defined various chain complexes that are generated by families of
graphs [16]. The arguably simplest of these _graph complexes_ ,
$\mathcal{GC}_{+}^{g}$, is rationally generated by all isomorphism classes of
connected, admissible, and even-orientable graphs of rank $g$. Each generator
is oriented by fixing an ordering of the graph’s edges. The $k$-chains in the
subspace $C_{k}(\mathcal{GC}_{+}^{g})\subset\mathcal{GC}_{+}^{g}$ are rational
linear combinations of such graphs with exactly $k$ edges and the boundary
maps,

$\displaystyle\partial_{k}:C_{k}(\mathcal{GC}_{+}^{g})\rightarrow
C_{k-1}(\mathcal{GC}_{+}^{g}),\quad G\mapsto\sum\nolimits_{e}\pm G/e,$

map a graph generator to the alternating sum of its edge contractions. The
sign in front of $G/e$ is the product of the parity of $e$ in the ordering of
$G$’s edges and the sign of the permutation that maps $G$’s edge ordering to
$G/e$’s. _Even commutative graph homology_ is the quotient

$H_{k}(\mathcal{GC}_{+}^{g})=\ker\partial_{k}/\operatorname{im}\partial_{k+1}.$

It has various applications, for instance, to deformation quantization [17,
11], group cohomology of automorphism groups of free groups [9], Drinfeld’s
theory of quasi-Hopf algebras [22], and to the cohomology of moduli spaces of
curves [8] (see §1.1).

The Euler characteristic of such a graph complex is the alternating sum of its
Betti numbers, $\chi(\mathcal{GC}_{+}^{g})=\sum_{k}(-1)^{k}\dim
H_{k}(\mathcal{GC}_{+}^{g})=\sum_{k}(-1)^{k}\dim C_{k}(\mathcal{GC}_{+}^{g})$.
As the dimension of the chain space $C_{k}(\mathcal{GC}_{+}^{g})$ is the
number of isomorphism classes of connected, admissible, and even-orientable
graphs of rank $g$ with $k$ edges, the computation of
$\chi(\mathcal{GC}_{+}^{g})$ is the combinatorial problem of counting such
graphs and taking the alternating sum of their numbers over $k$.

For two sequences $a_{1},a_{2},\ldots$ and $b_{1},b_{2},\ldots$ the asymptotic
notation ‘$a_{g}\sim b_{g}$ as $g\rightarrow\infty$’ means that
$\lim_{g\rightarrow\infty}a_{g}/b_{g}=1.$ The main result of this article is
an asymptotic expression for $\chi(\mathcal{GC}_{+}^{g})$:

######  Theorem 1.1.

The Euler characteristic, $\chi(\mathcal{GC}_{+}^{g})=\sum_{k}(-1)^{k}\dim
H_{k}(\mathcal{GC}_{+}^{g})$, behaves as

$\displaystyle\chi(\mathcal{GC}_{+}^{g})\sim\begin{cases}(-1)^{\frac{g}{2}}\sqrt{\frac{8\pi}{g^{3}}}\left(\frac{g}{2\pi
e}\right)^{g}&\text{ if $g$ is even}\\\
\frac{\sqrt{2}}{g}\cos\left(\sqrt{\frac{\pi g}{4}}-\frac{\pi
g}{4}-\frac{\pi}{8}\right)e^{\sqrt{\frac{\pi g}{4}}}\left(\frac{g}{2\pi
e}\right)^{\frac{g}{2}}&\text{ if $g$ is odd}\end{cases}\qquad\text{as
}g\rightarrow\infty.$

This asymptotic behavior of the Euler characteristic implies lower bounds for
the total amount of even commutative graph homology,
$H(\mathcal{GC}_{+}^{g})=\bigoplus_{k}H_{k}(\mathcal{GC}_{+}^{g}):$

######  Theorem 1.2.

For every positive $C<(2\pi e)^{-1}=0.0585...,$ and all but finitely many
$g\geq 2$,

$\displaystyle\dim H(\mathcal{GC}_{+}^{g})>\begin{cases}(Cg)^{g}&\text{ if $g$
is even}\\\ (Cg)^{g/2}&\text{ if $g$ is odd}\end{cases}$

In the even $g$ case, the estimate above follows directly from Theorem 1.1.
The odd $g$ case is nontrivial and relies on a bound for the _irrationality
measure_ of $\pi$. By Theorem 1.2, the number of homology classes in
$\mathcal{GC}_{+}^{g}$ grows _super-exponentially_ with $g$. Because of the
wildly alternating signs in Theorem 1.1, many of these classes must have odd
degree. Both results refute the expectation formulated in [16, §7.B] that the
_orbifold_ and the classical Euler characteristic behave similarly for large
$g$, as the former vanishes for all odd $g$.

The homology $H_{k}(\mathcal{GC}_{+}^{g})$ vanishes outside the degree range
$2g\leq k\leq 3g-3$. The upper bound is due to the lack of admissible graphs
of rank $g$ with more than $3g-3$ edges. The lower bound was proven by
Willwacher, who also showed that the bottom degree homology
$H_{2g}(\mathcal{GC}_{+}^{g})$ is isomorphic to the degree $g$ subspace of the
_Grothendieck–Teichmüller Lie algebra_ , $\mathfrak{grt}_{1}$ [22]. A theorem
of Brown [5] implies that a free Lie algebra injects into
$\mathfrak{grt}_{1}$. Counting the number of generators of this free Lie
algebra shows that at least an exponentially growing amount of classes is
concentrated in degree $2g$. Khoroshkin, Willwacher, and Živković proved that
even graph homology, summed over all ranks and degrees larger than the bottom
degree $2g$, i.e., $\bigoplus_{g\geq 2,k>2g}H_{k}(\mathcal{GC}_{+}^{g})$, is
infinite-dimensional [15, Cor. 1]. Theorems 1.1 and 1.2 strengthen this
result. For instance, they directly imply that the dimension of
$\bigoplus_{k}H^{2k+1}(\mathcal{GC}_{+}^{4g+2})$ grows super-exponentially
with $g$. Also, Theorems 1.1 and 1.2 answer a related question posed after
[15, Cor. 6] in the negative. Moreover, Theorem 1.2 (negatively) answers
Brown’s question of whether a specific map into $H(\mathcal{GC}_{+}^{g})$ is
an isomorphism [6, Conj. 2.5 and §2.11], as the domain of this map is not
large enough to accommodate the super-exponentially growing dimension of the
image. Further, Theorem 1.2 implies the same bound on the dimension of the top
weight cohomology of the _handlebody group_ [13] (see, also [3, §1.4]).

###  1.1. Implications on the cohomology of $\mathcal{M}_{g}$

Due to a recent result by Chan, Galatius, and Payne [8], Theorems 1.1 and 1.2
provide new information on the cohomology of the moduli space of curves
$\mathcal{M}_{g}.$ According to Deligne’s theory of mixed Hodge structures,
the cohomology of $\mathcal{M}_{g}$ admits a _weight grading_ in addition to
its grading by degree. The $j$-th weight-graded piece is the subspace
$\mathrm{Gr}^{W}_{j}H^{k}(\mathcal{M}_{g};{\mathbb{Q}})\subset
H^{k}(\mathcal{M}_{g};{\mathbb{Q}}).$ Chan, Galatius, and Payne’s theorem
establishes an isomorphism between the _top weight_ (i.e., the $(6g-6)$-th
weight-graded piece of the) cohomology of $\mathcal{M}_{g}$ and the homology
of $\mathcal{GC}_{+}^{g}.$ Translated to our degree conventions, they proved
$\mathrm{Gr}^{W}_{6g-6}H^{k}(\mathcal{M}_{g};{\mathbb{Q}})\simeq
H_{6g-6-k}(\mathcal{GC}^{g}_{+})$ [8, Thms. 1.2 and 1.3]. This result shed new
light on an older problem: In 1986, Harer and Zagier [14] proved an asymptotic
expression for the Euler characteristic of $\mathcal{M}_{g}$,
$\chi(\mathcal{M}_{g})=\sum_{k}(-1)^{k}\dim
H^{k}(\mathcal{M}_{g};{\mathbb{Q}}),$ implying an inequality of the shape
$\dim H(\mathcal{M}_{g};{\mathbb{Q}})>([\text{positive constant}]\cdot
g)^{2g},$ for all but finitely many $g\geq 2$. It is an open problem to
explain the nature of this super-exponentially growing amount of classes. The
only previously known family, the _tautological classes_ , grows less than
exponentially with $g$. Chan, Galatius, and Payne combined their insight on
the top weight cohomology of $\mathcal{M}_{g}$ with Brown and Willwacher’s
theorems on $\mathcal{GC}_{+}$ and $\mathfrak{grt}_{1}$ to establish the
existence of another _exponentially growing_ family of classes in the next-to-
top degree $H^{4g-6}(\mathcal{M}_{g};{\mathbb{Q}})$ (see [8, Thms. 1.1 and
2.7]).

Theorems 1.1 and 1.2 add a positive and a negative twist to this story:
Together with Chan, Galatius, and Payne’s result, Theorem 1.2 implies that the
dimension of the top weight cohomology
$\mathrm{Gr}^{W}_{6g-6}H^{\bullet}(\mathcal{M}_{g};{\mathbb{Q}})$ grows super-
exponentially with $g$:

######  Theorem 1.3.

The _top weight Euler characteristic_
$\sum_{k}(-1)^{k}\dim\mathrm{Gr}^{W}_{6g-6}H^{k}(\mathcal{M}_{g};{\mathbb{Q}})$
exhibits the same large $g$ asymptotic behavior as
$\chi(\mathcal{GC}_{+}^{g})$ in Theorem 1.1 and
$\dim\mathrm{Gr}^{W}_{6g-6}H(\mathcal{M}_{g};{\mathbb{Q}})$ respects the same
lower bounds as $\dim H(\mathcal{GC}_{+}^{g})$ in Theorem 1.2.

So, potentially Chan, Galatius, and Payne’s result can explain many more
classes than the exponentially growing family in degree $4g-6$. However, the
growth rate of the Euler characteristic of $\mathcal{GC}_{+}^{g}$, which is
dominated by a $g^{g}$ or $g^{\frac{g}{2}}$ term, is insufficient to explain
the even larger growth rate of the Euler characteristic of $\mathcal{M}_{g}$,
which is dominated by a $g^{2g}$ term. Further, as the Euler characteristic is
additive over direct sums, Theorem 1.1, in combination with the
Chan–Galatius–Payne and Harer–Zagier results, implies that the total dimension
of the rational, non-top weight cohomology of $\mathcal{M}_{g}$,
$\bigoplus_{j=0}^{6g-8}\mathrm{Gr}_{j}^{W}H(\mathcal{M}_{g};{\mathbb{Q}})$,
also grows super-exponentially with $g$.

###  1.2. The odd commutative graph complex

Kontsevich defined his graph complexes in two different flavors: The _odd_ and
the _even_ one.111 In the original article [16], the even/odd nomenclature was
used in precisely the reversed way. We agree with most recent publications
that use the present convention.  An _odd orientation_ of a graph $G$ is given
by an ordering of its edges together with bases for its zeroth and first
homology $H_{0}(G,{\mathbb{Z}})$ and $H_{1}(G,{\mathbb{Z}})$. A given
automorphism $\alpha$ of a graph induces a permutation $\alpha_{E}$ on $G$’s
set of edges and automorphisms
$\alpha_{H_{0}}:H_{0}(G,{\mathbb{Z}})\rightarrow H_{0}(G,{\mathbb{Z}})$,
$\alpha_{H_{1}}:H_{1}(G,{\mathbb{Z}})\rightarrow H_{1}(G,{\mathbb{Z}})$. A
graph is _odd-orientable_ if it has no automorphism $\alpha$ for which
${\operatorname{sign}}(\alpha_{E})\cdot\det(\alpha_{H_{0}})\cdot\det(\alpha_{H_{1}})=-1$.
The _odd commutative graph complex_ , $\mathcal{GC}_{-}^{g}$, is defined
analogously to the even graph complex described above. Here, generators are
graded by the number of vertices instead of the edge number. So, connected,
odd-oriented graphs of genus $g$ with $k$ vertices span the space of
$k$-chains $C_{k}(\mathcal{GC}_{-}^{g})$. The boundary maps,
$\partial_{k}:C_{k}(\mathcal{GC}_{-}^{g})\rightarrow
C_{k-1}(\mathcal{GC}_{-}^{g})$ are again defined on generators by
$G\mapsto\sum\nolimits_{e}\pm G/e$, where the sign in front of $G/e$ is the
parity of $e$ in the edge ordering times the sign of the permutation that
translates from $G$’s edge ordering to $G/e$’s and the determinants of the
canonical isomorphisms that map the bases of $H_{0}(G,{\mathbb{Z}})$ and
$H_{1}(G,{\mathbb{Z}})$ to the chosen bases of $H_{0}(G/e,{\mathbb{Z}})$ and
$H_{1}(G/e,{\mathbb{Z}})$. Odd commutative graph homology
$H_{k}(\mathcal{GC}_{-}^{g})$ has support in the degree range $1\leq k\leq
2g-2$, as admissible graphs at rank $g$ have at most $2g-2$ vertices.

We will prove that the Euler characteristic of the odd commutative complex,
$\chi(\mathcal{GC}_{-}^{g})=\sum_{k}(-1)^{k}\dim H_{k}(\mathcal{GC}_{-}^{g})$,
behaves similarly to the even one up to a sign:

######  Theorem 1.4.

$\chi(\mathcal{GC}_{-}^{g})\sim-\chi(\mathcal{GC}_{+}^{g})$ as
$g\rightarrow\infty$. Moreover, $\dim H(\mathcal{GC}^{g}_{-})$ respects the
same lower bounds as $\dim H(\mathcal{GC}^{g}_{+})$ in Theorem 1.2.

The odd commutative graph complex was studied extensively due to its role in
knot theory and finite type invariants (see, e.g., [1]). Many classes are
known in the top-degree $2g-2$ due to a relation to Chern–Simons theory. It is
conjectured that odd commutative graph cohomology in this degree is generated
by a specific commutative algebra [21].

Aside from showing that the total amount of odd graph homology grows rapidly
with $g$, Theorem 1.4 explains an observation in [23] on the similarity of the
Euler characteristics of $\mathcal{GC}_{+}^{g}$ and $\mathcal{GC}_{-}^{g}$:
Asymptotically, they are equal up to a sign.

In many articles, $\mathcal{GC}_{+}$ is denoted as $\mathcal{GC}_{0}$, or it
appeared as the, up to degree-shifts equivalent, $\mathcal{GC}_{2}$.
Similarly, $\mathcal{GC}_{-}$ corresponds to $\mathcal{GC}_{1}$ in the
conventional notation. Here, we use the $\pm$-notation to avoid repetition, as
our statements on $\mathcal{GC}_{+}$ and $\mathcal{GC}_{-}$ often only differ
by signs.

###  1.3. Generating functions for commutative graph complex Euler
characteristics

We will prove Theorems 1.1 and 1.4 after establishing _generating functions_
for $\chi(\mathcal{GC}_{+}^{g})$ and $\chi(\mathcal{GC}_{-}^{g})$.

Let $\mu(n)$ denote the _Möbius function_ , defined by $\mu(1)=1$ and
$\mu(n)=-\sum_{d|n,d\neq n}\mu(d)$ for all $n\geq 2$. For any integer $k\geq
1$, we write $M_{k}(z)=\frac{1}{k}\sum_{d|k}\mu(d)z^{k/d}$,
$L_{k}(z)=\log(kM_{k}(z)/z^{k})$ and $\delta_{2|n}=(1+(-1)^{n})/2$. The
_Bernoulli numbers_ , $B_{m}$, are defined by
$\frac{x}{e^{x}-1}=\sum_{m=0}^{\infty}{B_{m}}\frac{x^{m}}{m!}$.

Each member of the following family of power series is meant to be expanded in
terms of $1/z$,

(1)
$\displaystyle\Psi_{k}^{\pm}(z)=\pm\left((1-L_{k}(z))M_{k}(z)-\frac{z^{k}}{k}+\frac{\delta_{2|k}}{2k}\right)-\frac{1}{2}L_{k}(z)\mp\sum_{m=1}^{\infty}\frac{B_{m+1}}{m(m+1)}\frac{1}{M_{k}(z)^{m}}.$

We write $\lceil x\rceil$ for the smallest integer $\geq x$.

######  Theorem 1.5.

For all $k\geq 1$, we have $\Psi_{k}^{\pm}(z)\in z^{-\lceil
k/6\rceil}{\mathbb{Q}}[[\frac{1}{z}]]$ and the identity in
${\mathbb{Q}}[[\frac{1}{z}]]$ holds,

$\sum_{g\geq 2}\chi(\mathcal{GC}_{\pm}^{g})z^{1-g}=\sum_{k,\ell\geq
1}\frac{\mu(\ell)}{\ell}\Psi^{\pm}_{k}(z^{\ell}).$

Generating functions for the Euler characteristic of
$\chi(\mathcal{GC}_{\pm}^{g})$ were previously obtained by Willwacher and
Živković [23]. Their formulas do not seem amendable to an asymptotic analysis.
Tsopméné and Turchin extended this formula to hold for graph complexes where
the graphs are allowed to have _leaves (also called legs or hairs)_ [19].
Chan, Faber, Galatius, and Payne [7] proved a formula for the Euler
characteristic of the complex of even-orientable graphs with $n$ leaves or
equivalently the (equivariant) top weight Euler characteristic of the moduli
space of curves with $n$ marked points, $\mathcal{M}_{g,n}$. This formula,
which also does not seem amendable to an asymptotic analysis, was conjectured
in an unpublished note by Zagier based on computations by Faber [24]. On the
same note, Zagier derived a simplified expression from this formula that for
$n=0$ reduces to the expression in Theorem 1.5 in the even,
$\mathcal{GC}_{+}$, case.

###  1.4. Kontsevich’s other graph complexes

In addition to the commutative graph complex, Kontsevich defined the
_associative_ and the _Lie_ graph complex [16]. The former is generated by
oriented _ribbon_ graphs, i.e., graphs with a cyclic ordering attached to each
vertex. Getzler and Kapranov obtained a generating function for the Euler
characteristic of this graph complex [12, Theorem 9.18] that is remarkably
similar to the formula in Theorem 1.5. Let $\chi(\mathcal{AGC}^{g})$ denote
the Euler characteristic of the associative graph complex. Applying the
methods developed in the present article to Getzler and Kapranov’s result (see
Section 9.1) gives

######  Theorem 1.6.

$\chi(\mathcal{AGC}^{g})\sim-\chi(\mathcal{GC}_{+}^{g})$ as
$g\rightarrow\infty$ and associative graph homology $\dim
H(\mathcal{AGC}^{g})$ respects the same lower bounds as $\dim
H(\mathcal{GC}^{g}_{+})$ in Theorem 1.2.

The Lie graph complex computes the rational group cohomology of
${\operatorname{Out}}(F_{g})$, the outer automorphism group of the free group,
and the cohomology of the moduli space of graphs. Vogtmann and the author
proved that the associated Euler characteristic behaves like
$\chi({\operatorname{Out}}(F_{g}))\sim-e^{-1/4}(g/e)^{g}/(g\log g)^{2}$ as
$g\rightarrow\infty$ [4], which is strikingly different from the oscillatory
behavior exhibited by the other graph complexes. The results of the present
paper hence complete the analysis of the Euler characteristics of Kontsevich’s
original graph complexes.

###  1.5. Outline

Section 2 starts with a brief digression into the diophantine properties of
the asymptotic expression from Theorem 1.1, which implies Theorem 1.2.

Sections 3–8 are devoted to the proof of Theorem 1.5. In Sections 3–5, we
develop the necessary enumerative combinatorial tools to capture the value of
the Euler characteristics. The methods of these sections are closely related
to techniques used by Vogtmann and the author in [4].

Section 6 will introduce quantum field theory-inspired _partition functions_
to define fully analytic generating functions for the graph complex Euler
characteristics $\chi(\mathcal{GC}_{\pm}^{g})$. Families of specific integrals
will define these partition functions. The technology is inspired by Getzler
and Kapranov’s work on modular operads [12]. Here, however, we avoid using
operads and construct explicitly _convergent_ families of integrals suitable
for manipulation via analytic tools.

In Section 7, we use this additional flexibility to manipulate the partition
functions and simplify them via a variable transformation. Eventually, in
Section 8, we can evaluate these simplified integrals due to a relation to the
asymptotic expansion of the $\Gamma$ function and prove Theorem 1.5.

In Section 9, we prove Theorems 1.1, 1.4, and 1.6 via an asymptotic analysis
of the generating function from Theorem 1.5. The key technical step (Lemma 9.6
and Lemma 9.5) is to express a highly alternating sum as an integral and
evaluate this integral in the desired limit.

### Acknowledgements

I am grateful to Francis Brown, Thomas Willwacher, and Karen Vogtmann for
illuminating discussions and to Don Zagier for sharing his unpublished notes.
This research was supported by Dr. Max Rössler, the Walter Haefner Foundation,
and the ETH Zürich Foundation.

##  2\. The oscillating asymptotic behavior of $\chi(\mathcal{GC}_{\pm}^{g})$
and the irrationality of $\pi$

The magnitude of the Euler characteristic,
$\chi(\mathcal{GC}_{+}^{g})=\sum_{k}(-1)^{k}\dim H_{k}(\mathcal{GC}_{+}^{g})$,
gives a lower bound for the total dimension of the homology $\sum_{k}\dim
H_{k}(\mathcal{GC}_{+}^{g})$ (and for $\mathcal{GC}_{-}^{g}$ and
$\mathcal{AGC}^{g}$ analogously). So, if there are sequences $a_{g},b_{g}$
such that $\lim_{g\rightarrow\infty}{\chi(\mathcal{GC}_{+}^{g})/a_{g}}=1$ and
$|a_{g}|\geq|b_{g}|$ for all but finitely many $g$, then $\sum_{k}\dim
H_{k}(\mathcal{GC}_{+}^{g})>|b_{g}|$ for all but finitely many $g\geq 2$.

In the even $g$ case, Theorem 1.1 directly implies the even $g$ case of
Theorem 1.2: If $C<1/(2\pi e)$, then the inequality $g^{-3/2}(g/(2\pi
e))^{g}\geq(Cg)^{g}$ holds for all but finitely many $g\geq 2.$ The odd $g$
case is more complicated, as the cosine in the statement of Theorem 1.1 can
become arbitrarily small. However, known bounds on the measure of the
irrationality of $\pi$ imply that the cosine cannot become so small that it
cancels the super-exponential growth rate from the $g^{g/2}$ term. The
following paragraphs will explain this mechanism.

By a theorem of Mahler [18], there is a number $\mu\leq 42$, the
_irrationality measure of $\pi$_, such that for all but finitely many pairs of
integers $p,q$ the inequality $\left|\pi-{p}/{q}\right|>{q^{-\mu}}$ holds. The
most recent result shows that $\mu\leq 7.103...$ [25]. We can use this bound
to prove the following lower bound on the cosine in the statement of Theorem
1.1:

######  Proposition 2.1.

Let $\mu^{*}$ be strictly larger than the irrationality measure $\mu$ of
$\pi$, then

$\left|\cos\left(\sqrt{\frac{\pi g}{4}}-\frac{\pi
g}{4}-\frac{\pi}{8}\right)\right|\geq g^{\frac{1}{2}-\mu^{*}}\text{ for all
but finitely many integers $g\geq 2$}.$

###### Proof.

For any given $g$, let $k_{g}$ be the (unique) integer that minimizes the
magnitude of $\mathcal{R}_{g}$ in

$\sqrt{\frac{\pi g}{4}}=\frac{\pi k_{g}}{8}+\mathcal{R}_{g}.$

If $m=k_{g}-2g-1$ is not an integer of the form $8n+4$ for some
$n\in{\mathbb{Z}}$, then $|\cos\left(\sqrt{\frac{\pi g}{4}}-\frac{\pi
g}{4}-\frac{\pi}{8}\right)|=|\cos(\frac{\pi
m}{8}+\mathcal{R}_{g})|=|\sin(\pi\frac{m-4}{8}+\mathcal{R}_{g})|\geq\sin(\frac{\pi}{16})$,
because $|\mathcal{R}_{g}|\leq\frac{\pi}{16}$. So, the statement is only
nontrivial if $m=8n+4$ for some $n\in{\mathbb{Z}}$, which we will assume.
Using that together with the inequality $|\sin(x)|\geq|x|(1-\frac{|x|}{\pi})$,
valid for all $|x|\leq\pi$, gives $|\cos\left(\sqrt{\frac{\pi g}{4}}-\frac{\pi
g}{4}-\frac{\pi}{8}\right)|=|\sin\left(\mathcal{R}_{g}\right)|\geq|\mathcal{R}_{g}|(1-\frac{|\mathcal{R}_{g}|}{\pi})\geq\frac{15}{16}|\mathcal{R}_{g}|$.
It remains to prove a lower bound on $|\mathcal{R}_{g}|$. The bounds on the
irrationality measure of $\pi$ discussed above also imply
$|\pi-16{g}/{k_{g}^{2}}|>{k_{g}^{-2\mu}}$ for all but finitely many pairs
$g,k_{g}$. By definition of $\mathcal{R}_{g}$,
$|\pi-16{g}/{k_{g}^{2}}|=|16{\mathcal{R}_{g}}/({\pi
k_{g}})+8^{2}{\mathcal{R}_{g}^{2}}/({\pi^{2}k_{g}^{2}})|\leq
20{|\mathcal{R}_{g}|}/{(\pi k_{g})}.$ Hence,
$|\mathcal{R}_{g}|>\frac{\pi}{20}{k_{g}^{1-2\mu}}$ for all but finitely many
$g$. Using $k_{g}\geq\frac{8}{\pi}\left(\sqrt{\frac{\pi
g}{4}}-\frac{1}{2}\right)$ and $\mu^{*}>\mu$ gives the statement. ∎

##  3\. Connected and disconnected graphs

All graphs in this article are admissible. So, we will drop this adjective
from now on. Let $\mathbf{G}_{+}$ ($\mathbf{G}_{-}$) be the set of isomorphism
classes of even(-odd)-orientable graphs and $\mathbf{cG}_{+}$
($\mathbf{cG}_{-}$) be the subsets of such graphs that are connected. The
elements of $\mathbf{cG}_{\pm}$ generate the complexes
$\mathcal{GC}_{\pm}^{g}$, and these complexes’ Euler characteristics are
signed sums over these elements:

######  Proposition 3.1.

With $|E_{G}|$ and $|V_{G}|$ the number of edges and vertices of $G$,

$\displaystyle\chi(\mathcal{GC}_{+}^{g})$
$\displaystyle=\sum_{\begin{subarray}{c}G\in\mathbf{cG}_{+}\\\
\operatorname{{rank}}(G)=g\end{subarray}}(-1)^{|E_{G}|}$
$\displaystyle\chi(\mathcal{GC}_{-}^{g})$
$\displaystyle=\sum_{\begin{subarray}{c}G\in\mathbf{cG}_{-}\\\
\operatorname{{rank}}(G)=g\end{subarray}}(-1)^{|V_{G}|},$

where we sum over all connected graphs with fixed rank.

Only connected graphs generate the graph complexes $\mathcal{GC}_{+}^{g}$ and
$\mathcal{GC}_{-}^{g}$, but it is easier to evaluate expressions like the two
above if we sum over all, possibly disconnected, graphs. Further, it is
convenient to index such graphs by their Euler characteristic and define
numbers for all $n\geq 0$,

(2) $\displaystyle\chi^{+}_{n}$
$\displaystyle=\sum_{\begin{subarray}{c}G\in\mathbf{G}_{+}\\\
\chi(G)=-n\end{subarray}}(-1)^{|E_{G}|}$ $\displaystyle\chi^{-}_{n}$
$\displaystyle=\sum_{\begin{subarray}{c}G\in\mathbf{G}_{-}\\\
\chi(G)=-n\end{subarray}}(-1)^{|V_{G}|},$

where we sum over all graphs with fixed Euler characteristic
$\chi(G)=|E_{G}|-|V_{G}|$. Admissibility of a graph implies that $2|E_{G}|\geq
3|V_{G}|$. Hence, $\chi(G)<0$ if $G$ is non-empty, and there are only finitely
many graphs $G$ with specific $\chi(G)=-n$. So, the sequences above are well-
defined, and $\chi^{\pm}_{0}=1$ due to the empty graph for which $\chi(G)=0$.

The following statement translates between the connected and disconnected
case.

######  Proposition 3.2.

In the ring of power series ${\mathbb{Q}}[[\frac{1}{z}]]$,

$\displaystyle\sum_{n\geq 0}\chi^{\pm}_{n}~{}z^{-n}=\prod_{g\geq
2}\left(\frac{1}{1-z^{1-g}}\right)^{\chi(\mathcal{GC}_{\pm}^{g})}.$

The proof is a variant of the one for Theorem 2.5 in [4].

###### Proof.

Each disconnected even/odd-orientable graph can be written as a unique
disjoint union of connected even/odd-orientable graphs. However, not every
such disjoint union is necessarily an even/odd-orientable graph. If a
disconnected graph contains two copies of the same connected graph, it has an
automorphism that interchanges these components. This automorphism may flip
the graph’s orientation, making it non-orientable. For even orientability,
this happens if a disconnected graph, $G^{\prime}$, contains two copies of the
same graph, $G$, with an odd number of edges. The automorphism that switches
the two components induces an odd permutation on the edge set of the whole
graph, which is thereby not even-orientable. For the odd-orientability, the
same situation arises if a disconnected graph, $G^{\prime}$, contains two
copies of a graph $G$ with the sum $|E_{G}|+\dim H_{0}(G,{\mathbb{Z}})+\dim
H_{1}(G,{\mathbb{Z}})$ being an odd number, as then, with $\alpha$ the
automorphism of $G^{\prime}$ that switches the two copies of $G$, we have
${\operatorname{sign}}(\alpha_{E})\det(\alpha_{H_{0}})\det(\alpha_{H_{1}})=-1$.

Let $\deg^{+}(G)=|E_{G}|$ and $\deg^{-}(G)=|V_{G}|$ and note that
$\deg^{-}(G)=|E_{G}|+\chi(G)=|E_{G}|+\dim H_{0}(G,{\mathbb{Z}})-\dim
H_{1}(G,{\mathbb{Z}})$ has the same parity as $|E_{G}|+\dim
H_{0}(G,{\mathbb{Z}})+\dim H_{1}(G,{\mathbb{Z}})$. It follows from the last
paragraph’s discussion that each member of $\mathbf{G}_{\pm}$ is a sequence of
connected graphs in which graphs $G$ with even $\deg^{\pm}(G)$ may appear any
number of times and graphs with odd $\deg^{\pm}(G)$ may appear at most once.
So, the following product expands to a sum with one term for each
$G^{\prime}\in\mathbf{G}_{\pm}$ weighted by
$(-1)^{\deg^{\pm}(G^{\prime})}z^{\chi(G^{\prime})}$:

$\displaystyle\prod_{\begin{subarray}{c}G\in\mathbf{cG}_{\pm}\\\
\deg^{\pm}(G)\text{ is
even}\end{subarray}}\left(1+z^{\chi(G)}+z^{2\chi(G)}+\cdots\right)\cdot\prod_{\begin{subarray}{c}G\in\mathbf{cG}_{\pm}\\\
\deg^{\pm}(G)\text{ is odd}\end{subarray}}\left(1-z^{\chi(G)}\right).$

Also, use Proposition 3.1, $\chi(G)=1-\operatorname{{rank}}(G)$ if $G$ is
connected, and $\sum_{n\geq 0}x^{n}=\frac{1}{1-x}$. ∎

##  4\. Half-edge labeled graphs

A _set partition_ of a finite set $S$ is a set $P=\\{B_{1},\ldots,B_{r}\\}$ of
non-empty and mutually disjoint subsets of $S$ such that $B_{1}\cup\ldots\cup
B_{r}=S$. The elements $B_{i}\in P$ are called _blocks_. A bijection
$\alpha:S\rightarrow S$ fixes the partition if there is an _induced
permutation of the blocks_ $\alpha_{P}\in\operatorname{{\mathbb{S}}}_{r}$ such
that $\alpha(B_{i})=B_{\alpha_{P}(i)}$ for all $i$. For a given $\alpha$ that
fixes $P$, the induced permutation $\alpha_{P}$ on the blocks is unique. We
use the convention that the empty set has one (empty) set partition, which is
fixed by the unique bijection between empty sets.

For given $s\geq 0$ and $\alpha\in\operatorname{{\mathbb{S}}}_{2s}$, let
$\mathcal{E}(\alpha)$ be the set of set partitions of $[2s]$ that are fixed by
$\alpha$ and have $s$ blocks of size $2$. Given $s,r\geq 0$ and
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$, let $\mathcal{V}_{r}(\alpha)$ be
the set of set partitions of $[2s]$ that are fixed by $\alpha$ and have
exactly $r$ blocks of size $\geq 3$. Further, for given
$E\in\mathcal{E}(\alpha)$ and $V\in\mathcal{V}_{r}(\alpha)$, we write
$\alpha_{E}$ and $\alpha_{V}$ for the permutations that $\alpha$ induces on
the blocks of $E$ and $V$.

This terminology allows us to write the number of isomorphism classes of
even/odd-orientable graphs with a fixed number of edges and vertices as signed
sums over set partitions.

######  Theorem 4.1.

For all $s,r\geq 0$, we have

$\displaystyle\left|\\{G\in\mathbf{G}_{+}:|E_{G}|=s,|V_{G}|=r\\}\right|$
$\displaystyle=\frac{1}{(2s)!}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\sum_{(E,V)\in\mathcal{E}(\alpha)\times\mathcal{V}_{r}(\alpha)}{\operatorname{sign}}(\alpha_{E})$
$\displaystyle\left|\\{G\in\mathbf{G}_{-}:|E_{G}|=s,|V_{G}|=r\\}\right|$
$\displaystyle=\frac{1}{(2s)!}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\sum_{(E,V)\in\mathcal{E}(\alpha)\times\mathcal{V}_{r}(\alpha)}{\operatorname{sign}}(\alpha)\cdot{\operatorname{sign}}(\alpha_{V}).$

The proof of this theorem will occupy the remainder of this section. The first
step is to interpret the Cartesian product
$\mathcal{E}(\alpha)\times\mathcal{V}_{r}(\alpha)$ in the sums above as the
set of _half-edge labeled graphs_ with $r$ vertices that have $\alpha$ as an
automorphism. We do so in the following definition.

######  Definition 4.2.

Given a set $S$, an $S$-_labeled graph_ is a pair $\Gamma=(E,V)$ of two set
partitions $E,V$ of $S$ such that all blocks of $E$ have size $2$ and all
blocks of $V$ have size $\geq 3$. The blocks of $E$ and $V$ are the _edges_
and _vertices_ of $\Gamma$, respectively. An automorphism of an $S$-labeled
graph is a bijection $S\rightarrow S$, which fixes the partitions $E$ and $V$.
These bijections form a group ${\operatorname{Aut}}(\Gamma)$.

It is easy to see that giving a graph with $s$ edges together with a labeling
of its edges’ end-points (or half-edges) with integers from the set
$[2s]=\\{1,\ldots,2s\\}$ is equivalent to giving a $[2s]$-labeled graph. The
following lemma quantifies the number of $[2s]$-labeled graphs we can
construct for a given isomorphism class of graphs.

######  Lemma 4.3.

Each isomorphism class of graphs with $s$ edges can be represented by exactly
$(2s)!/|{\operatorname{Aut}}(\Gamma)|$ $[2s]$-labeled graphs where $\Gamma$
can be any such latter graph.

###### Proof.

Let $\mathrm{lab}(G)$ be the set of $[2s]$-labeled representatives of the
isomorphism class $G$. The symmetric group $\operatorname{{\mathbb{S}}}_{2s}$
acts transitively on $\mathrm{lab}(G)$ by permuting the labels. By Definition
4.2, the stabilizer of an element $\Gamma\in\mathrm{lab}(G)$ is the group
${\operatorname{Aut}}(\Gamma)$. Hence,
$|\operatorname{{\mathbb{S}}}_{2m}|=|\mathrm{lab}(G)||{\operatorname{Aut}}(\Gamma)|$.
∎

The sign factors in the statement of Theorem 4.1 make the non-orientable
graphs cancel out. In the even-orientability case, this mechanism is
straightforward:

######  Lemma 4.4.

For an $[2s]$-labeled graph $\Gamma$, the sum
$\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha_{E})$
evaluates to $|{\operatorname{Aut}}(\Gamma)|$ if $\Gamma$ is even-orientable
and to $0$ otherwise.

###### Proof.

The graph $\Gamma$ is even-orientable if it has no automorphism $\alpha$ for
which ${\operatorname{sign}}(\alpha_{E})=-1$. In that case,
$\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha_{E})=|{\operatorname{Aut}}(\Gamma)|$.
The map
$\xi:{\operatorname{Aut}}(\Gamma)\rightarrow{\operatorname{GL}}_{1}({\mathbb{Z}})$
given by $\xi:\alpha\mapsto{\operatorname{sign}}(\alpha_{E})$ is a group
homomorphism. If $\Gamma$ is not even-orientable, then $\xi$ is surjective and
$\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha_{E})=\sum_{\alpha\in\xi^{-1}(1)}1-\sum_{\alpha\in\xi^{-1}(-1)}1=|\ker(\xi)|-|\ker(\xi)|=0.$
∎

For the odd-orientation we need an additional lemma, which for given
$[2s]$-labeled graph $\Gamma$ and automorphism
$\alpha\in{\operatorname{Aut}}(\Gamma)\subset\operatorname{{\mathbb{S}}}_{2s}$
relates the signs of the induced permutations $\alpha_{E}$ and $\alpha_{V}$ on
the edge and vertex set of $\Gamma$ with the determinants of the automorphisms
that $\alpha$ induces on $\Gamma$’s homology,
$\alpha_{H_{k}}:H_{k}(\Gamma;{\mathbb{Z}})\rightarrow
H_{k}(\Gamma;{\mathbb{Z}})$ with $k\in\\{0,1\\}$.

######  Lemma 4.5.

For a pair $(\Gamma,\alpha)$ with the induced permutations/automorphisms as
above, we have

${\operatorname{sign}}(\alpha)\cdot{\operatorname{sign}}(\alpha_{V})={\operatorname{sign}}(\alpha_{E})\cdot\det(\alpha_{H_{0}})\cdot\det(\alpha_{H_{1}}).$

###### Proof.

Let $\widetilde{G}$ be the barycentric subdivision of $\Gamma$. That means
$\widetilde{G}$ is the graph, homeomorphic to $\Gamma$, that we construct from
$\Gamma$ by adding a new two-valent vertex in the middle of each edge of
$\Gamma$’s topological realization. The edges $\widetilde{E}$ of
$\widetilde{G}$ correspond to the half-edges of $\Gamma$ and $\widetilde{G}$’s
vertex set, $\widetilde{V}$, corresponds to the disjoint union of $\Gamma$’s
edges and vertices. The graph $\widetilde{G}$ is a simplicial complex with a
canonical orientation on its edges: we can point each edge towards the
adjacent two-valent vertex. Hence, we have a canonical boundary map,
$\partial:{\mathbb{Q}}\widetilde{E}\rightarrow{\mathbb{Q}}\widetilde{V}$,
between the chain spaces ${\mathbb{Q}}\widetilde{E}$ and
${\mathbb{Q}}\widetilde{V}$. The vector spaces ${\mathbb{Q}}\widetilde{E}$ and
${\mathbb{Q}}\widetilde{V}$ come with canonical non-degenerate bilinear forms
given, e.g., by $\langle e,e\rangle=1$ and $\langle e,e^{\prime}\rangle=0$ for
generators $e\neq e^{\prime}\in\widetilde{E}$. So, we can decompose the chain
spaces using the orthogonal complement, giving canonical isomorphisms
${\mathbb{Q}}\widetilde{E}\rightarrow\ker\partial\oplus(\ker\partial)^{\perp}$
and
${\mathbb{Q}}\widetilde{V}\rightarrow(\operatorname{im}\partial)\oplus(\operatorname{im}\partial)^{\perp}$.
We get a chain of isomorphisms,

$H_{1}(\widetilde{G};{\mathbb{Q}})\oplus{\mathbb{Q}}\widetilde{V}\rightarrow
H_{1}(\widetilde{G};{\mathbb{Q}})\oplus(\operatorname{im}\partial)\oplus(\operatorname{im}\partial)^{\perp}\rightarrow\ker\partial\oplus(\ker\partial)^{\perp}\oplus
H_{0}(\widetilde{G};{\mathbb{Q}})\rightarrow{\mathbb{Q}}\widetilde{E}\oplus
H_{0}(\widetilde{G};{\mathbb{Q}}),$

where we also used the identification of $\ker\partial$ with
$H_{1}(\widetilde{G};{\mathbb{Q}})$ and the natural isomorphism
$(\operatorname{im}\partial)^{\perp}\rightarrow
H_{0}(\widetilde{G};{\mathbb{Q}}).$ Every
$\alpha\in{\operatorname{Aut}}(\Gamma)$ induces an automorphism of
$\widetilde{G}$ and thereby automorphisms on any of the above vector spaces,
which we denote with the respective space in the subscript, e.g.,
$\alpha_{{\mathbb{Q}}\widetilde{V}}:{\mathbb{Q}}\widetilde{V}\rightarrow{\mathbb{Q}}\widetilde{V}$.
Due to the chain of isomorphisms above, we have
$\det(\alpha_{H_{1}(\widetilde{G};{\mathbb{Q}})\oplus{\mathbb{Q}}\widetilde{V}})=\det(\alpha_{{\mathbb{Q}}\widetilde{E}\oplus
H_{0}(\widetilde{G};{\mathbb{Q}})}).$ The determinants factor over direct
sums, so

$\det(\alpha_{H_{1}(\widetilde{G};{\mathbb{Q}})})\det(\alpha_{{\mathbb{Q}}\widetilde{V}})=\det(\alpha_{{\mathbb{Q}}\widetilde{E}})\det(\alpha_{H_{0}(\widetilde{G};{\mathbb{Q}})}).$

Each automorphism fixes $\widetilde{G}$’s edges’ orientation, so
$\alpha_{{\mathbb{Q}}\widetilde{E}}$ acts by permuting the basis vectors of
${\mathbb{Q}}\widetilde{E}$ corresponding to half-edges of $\Gamma$. Hence,
$\det(\alpha_{{\mathbb{Q}}\widetilde{E}})={\operatorname{sign}}(\alpha)$.
Also, $\alpha_{{\mathbb{Q}}\widetilde{V}}$ permutes the vertices of
$\widetilde{G}$ corresponding to edges and vertices of $\Gamma$, so
$\det(\alpha_{{\mathbb{Q}}\widetilde{V}})={\operatorname{sign}}(\alpha_{E})\cdot{\operatorname{sign}}(\alpha_{V})$.
The automorphism $\alpha_{H_{0}(\widetilde{G};{\mathbb{Q}})}$ permutes the
connected components, and $\alpha_{H_{1}(\widetilde{G};{\mathbb{Q}})}$ maps
cycles to cycles, possibly flipping their orientation. Hence, both can be
identified with the respective automorphisms of $H_{0}(\Gamma;{\mathbb{Z}})$
and $H_{1}(\Gamma;{\mathbb{Z}})$. The statement follows from
${\operatorname{sign}}(\alpha_{V})^{2}=1$. ∎

######  Corollary 4.6.

For an $[2s]$-labeled graph $\Gamma$, the sum
$\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha){\operatorname{sign}}(\alpha_{V})$
evaluates to $|{\operatorname{Aut}}(\Gamma)|$ if $\Gamma$ is odd-orientable
and to $0$ otherwise.

###### Proof.

After using Lemma 4.5, the proof works analogously to the one of Lemma 4.4. ∎

###### Proof of Theorem 4.1.

For given $s,r\geq 0$, let $\mathcal{L}_{s,r}$ be the set of $[2s]$-labeled
graphs with $r$ vertices and
$\mathcal{L}_{s,r}^{\mathrm{unlab}}\subset\mathcal{L}_{s,r}$ a subset that
contains one representative for each isomorphism class of graphs in
$\mathcal{L}_{s,r}$. By Definition 4.2 and Lemma 4.3,

$\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\sum_{(E,V)\in\mathcal{E}(\alpha)\times\mathcal{V}_{r}(\alpha)}{\operatorname{sign}}(\alpha_{E})=\sum_{\Gamma\in\mathcal{L}_{s,r}}\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha_{E})=\sum_{\Gamma\in\mathcal{L}_{s,r}^{\mathrm{unlab}}}\frac{(2s)!}{|{\operatorname{Aut}}(\Gamma)|}\sum_{\alpha\in{\operatorname{Aut}}(\Gamma)}{\operatorname{sign}}(\alpha_{E}).$

The first equation in Theorem 4.1 follows after using Lemma 4.4. The second
equation is proven analogously using Corollary 4.6. ∎

##  5\. Generating functions for set partitions

We will use generating functions to count the number of set partitions in
$\mathcal{E}(\alpha)$ and $\mathcal{V}_{r}(\alpha)$ for any
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$ and to get further analytic
control over the expressions in Theorem 4.1. For a permutation
$\alpha\in\operatorname{{\mathbb{S}}}_{n}$, let $c_{k}(\alpha)$ be the number
of $k$-cycles of $\alpha$ and $c(\alpha)$ its total number of cycles,
$c(\alpha)=\sum_{k}c_{k}(\alpha)$. To such an $\alpha$, we can associate the
_cycle index monomial_ in the variables $x_{1},x_{2},\ldots$ defined as
$x^{\alpha}=\prod_{k=1}^{n}x_{k}^{c_{k}(\alpha)}$. Recall that for a
permutation $\alpha\in\operatorname{{\mathbb{S}}}_{2s}$ and a set partition
$P$ of $[2s]$ that is fixed by $\alpha$; we write $\alpha_{P}$ for the
permutation induced on the blocks of $P$. We will use power series in
infinitely many variables. For instance, we define

(3) $\displaystyle V(x_{1},x_{2},\ldots)=\exp\left(\sum_{k\geq
1}\frac{x_{k}}{k}\right)-1-x_{1}-\frac{x_{1}^{2}}{2}-\frac{x_{2}}{2}=\frac{x_{1}^{3}}{6}+\frac{x_{1}x_{2}}{2}+\frac{x_{3}}{3}+\frac{x_{1}^{4}}{24}+\ldots.$

######  Proposition 5.1.

We have the following identities of power series in $\eta,\lambda,\phi$ and
$x_{1},x_{2},\ldots$:

$\displaystyle\sum_{s=0}^{\infty}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}}{(2s)!}\sum_{E\in\mathcal{E}(\alpha)}\eta^{c(\alpha_{E})}$
$\displaystyle=\exp\left(\eta\sum_{k\geq
1}\frac{x_{k}^{2}+x_{2k}}{2k}\right)\text{ and }$ $\displaystyle\sum_{s,r\geq
0}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}\lambda^{r}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}\phi^{c(\alpha_{V})}$
$\displaystyle=\exp\left(\phi\sum_{k\geq
1}\frac{\lambda^{k}}{k}V(x_{k},x_{2k},\ldots)\right).$

To prove this proposition, we first need the following lemma on the number of
permutations that fix a given set partition while acting transitively on its
set of blocks. It is a variant of a classic statement going back to Pólya. It
can also be interpreted as a statement on relations of character polynomials
of representations of the symmetric group under the _wreath_ product (see,
e.g., [2, Chapter 4.3]).

######  Lemma 5.2.

For a set partition $P$ of $S$ into $r$ blocks each of size $d$, let
$\mathrm{Cyc}(P)$ be the set of permutations of $S$ that fix $P$ and induce a
cyclic permutation on its blocks. There is a surjection
$f:\mathrm{Cyc}(P)\rightarrow\operatorname{{\mathbb{S}}}_{d}$. For every pair
$\alpha,\beta$ with $\beta\in\operatorname{{\mathbb{S}}}_{d}$ and $\alpha\in
f^{-1}(\beta)$, $\alpha$ and $\beta$ have the same number of cycles such that
$\alpha$ has a $(k\cdot r)$-cycle for each $k$-cycle of $\beta$. Moreover,
$|f^{-1}(\beta)|=(r-1)!(d!)^{r-1}.$

###### Proof.

Without loss of generality, we may assume that $S$ is well ordered and
$P=\\{B_{1},\ldots,B_{r}\\}$ such that $B_{1}$ contains the smallest element
of $S$. We will construct a bijection between $\mathrm{Cyc}(P)$ and the set of
all tuples $(\beta,\alpha_{P},\gamma_{1},\ldots,\gamma_{r-1})$ where
$\alpha_{P}$ is a cyclic permutation in $\operatorname{{\mathbb{S}}}_{r}$ and
$\beta,\gamma_{1},\ldots,\gamma_{r-1}\in\left(\operatorname{{\mathbb{S}}}_{d}\right)^{r}$.
As $\alpha\in\mathrm{Cyc}(P)$ induces a cyclic permutation $\alpha_{P}$ on the
blocks of $P$, its $r$-th power, $\alpha^{r}$, fixes each block. Restricting
$\alpha^{r}$ to $B_{1}$ gives a permutation of $B_{1}$’s elements, which we
can identify canonically with an element
$\beta\in\operatorname{{\mathbb{S}}}_{d}$. As $\alpha$ permutes the $r$ blocks
of $P$ cyclically, the $k$-cycles of $\beta$ are in bijection with the $k\cdot
r$-cycles of $\alpha$. Moreover, $\alpha$ gives bijections
$B_{t(i-1)}\rightarrow B_{t(i)}$ where $t(i)=\alpha_{P}^{i}(1)$ for
$i=1,\ldots,r-1$. By identifying each block with $[d]$ in the unique order-
preserving way, we get permutations $\gamma_{1},\ldots,\gamma_{r-1}$. The
construction is invertible. The statement follows as there are $(r-1)!$ cyclic
permutations in $\operatorname{{\mathbb{S}}}_{r}$. ∎

######  Corollary 5.3.

Let $\Pi_{d}(S)$ be the set of set partitions of $S$ such that each
$P\in\Pi_{d}(S)$ only has blocks of size $d$. If $S$ has cardinality $r\cdot
d$, then

$\displaystyle\sum_{P\in\Pi_{d}(S)}\sum_{\alpha\in\mathrm{Cyc}(P)}x^{\alpha}=\frac{(r\cdot
d)!}{r\cdot(d!)}\sum_{\beta\in\operatorname{{\mathbb{S}}}_{d}}x^{[r\cdot\beta]},$

where $x^{[r\cdot\beta]}$ means replacing each instance of $x_{k}$ in
$x^{\beta}$ with $x_{r\cdot k}$.

###### Proof.

Use the surjection from Lemma 5.2 to resolve the sum over $\mathrm{Cyc}(P)$
and the fact that the number of such set partitions of $S$ into $r$ blocks of
size $d$ is $|\Pi_{d}(S)|=(r\cdot d)!/(d!)^{r}/(r!)$. ∎

In the proof of Proposition 5.1, we need to apply the last lemma with the
_exponential formula_ , which is a standard statement (see, e.g., [20, §5.1]):

######  Lemma 5.4.

Let $\Pi^{k}(S)$ be the set of all set partitions of $S$ into exactly $k$
blocks. Given a function $f:{\mathbb{Z}}_{>0}\rightarrow R$ into some ring
$R$, we define a new function $h:{\mathbb{Z}}_{\geq 0}\rightarrow R$ such that
$h(0)=1$ and for a non-empty finite set $S$,

$h(|S|)=\sum_{k\geq
1}\sum_{\\{B_{1},\ldots,B_{k}\\}\in\Pi^{k}(S)}\prod_{i=1}^{k}f(|B_{i}|),$

then, providing convergence of all involved sums, we have the identity

$\displaystyle\sum_{n=0}^{\infty}\frac{h(n)}{n!}=\exp\left(\sum_{k=1}^{\infty}\frac{f(k)}{k!}\right).$

###### Proof.

For $n\geq 1$ and integers $m_{1},m_{2},\ldots,m_{n}\geq 0$ such that
$\sum_{k}km_{k}=n$, there are exactly $n!/(\prod_{k}m_{k}!(k!)^{m_{k}})$
partitions of $[n]$ into $m_{1}$ blocks of size $1$, $m_{2}$ blocks of size
$2$ and so on. Hence,

$\displaystyle h(n)=\sum_{\begin{subarray}{c}m_{1},\ldots,m_{n}\geq 0\\\
\sum_{k}km_{k}=n\end{subarray}}\frac{n!}{\prod_{k}m_{k}!(k!)^{m_{k}}}\prod_{k=1}^{n}f(k)^{m_{k}}.\qed$

###### Proof of Proposition 5.1.

Let $EA_{s}$ be the set of all pairs $(E,\alpha)$ of a permutation
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$ and an element
$E\in\mathcal{E}(\alpha)$. Every permutation can be uniquely decomposed into
its cycle representation. So analogously, each pair $(E,\alpha)\in EA_{s}$ can
be decomposed into a set partition $\\{B_{1},\ldots,B_{k}\\}$ of $[2s]$
together with a set of pairs
$\\{(E_{1},\alpha^{(1)}),\ldots,(E_{k},\alpha^{(k)})\\}$ one for each block
$B_{i}$, where $E_{i}$ is a set partition of $B_{i}$ into blocks of size $2$,
$\alpha^{(i)}$ is a permutation of $B_{i}$’s elements that fixes the set
partition $E_{i}$ and induces a cyclic permutation on its blocks.

We want to translate this last paragraph into a formula. For $(E,\alpha)\in
EA_{s}$, we write $\alpha_{E}$ for the permutation that $\alpha$ induces on
the blocks of $E$ and $c(\alpha_{E})$ for the number of its cycles. Hence,

$\displaystyle\sum_{(E,\alpha)\in
EA_{s}}x^{\alpha}\eta^{c(\alpha_{E})}=\sum_{k\geq
0}\eta^{k}\sum_{\\{B_{1},\ldots,B_{k}\\}\in\Pi^{k}([2s])}\prod_{i=1}^{k}\left(\sum_{\widetilde{E}\subset\Pi_{2}(B_{i})}\sum_{\widetilde{\alpha}\in\mathrm{Cyc}(\widetilde{E})}x^{\widetilde{\alpha}}\right).$

As $\operatorname{{\mathbb{S}}}_{2}$ only has two elements, $(1)(2)$ and
$(12)$, we learn from Corollary 5.3 that

$\displaystyle\sum_{\widetilde{E}\subset\Pi_{2}(B_{i})}\sum_{\widetilde{\alpha}\in\mathrm{Cyc}(\widetilde{E})}x^{\widetilde{\alpha}}=\begin{cases}\frac{(2m)!}{m/2\cdot
2!}(x_{m}^{2}+x_{2m})&\text{ for even $|B_{i}|=2m$ }\\\ 0&\text{ for odd
$|B_{i}|$ }\end{cases}$

Applying Lemma 5.4 gives the first identity for Proposition 5.1 as convergence
is obvious in the power series topology.

Let $VA_{s}$ be the set of all pairs $(V,\alpha)$ of a permutation
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$ and a set partition
$V\in\mathcal{V}_{r}(\alpha)$ for some $r\geq 0$. For any $(V,\alpha)\in
VA_{s}$, we write $r(V)$ for the number of blocks of $V$ and $c(\alpha_{V})$
for the number of cycles of the induced permutation $\alpha_{V}$. By the
argument from above, we get

(4) $\displaystyle\sum_{(V,\alpha)\in
VA_{s}}\lambda^{r(V)}x^{\alpha}\phi^{c(\alpha_{V})}=\sum_{k\geq
0}\phi^{k}\sum_{\\{B_{1},\ldots,B_{k}\\}\in\Pi^{k}([2s])}\prod_{i=1}^{k}\left(\sum_{\begin{subarray}{c}d\geq
3\end{subarray}}\sum_{\widetilde{V}\subset\Pi_{d}(B_{i})}\lambda^{r(\widetilde{V})}\sum_{\widetilde{\alpha}\in\mathrm{Cyc}(\widetilde{V})}x^{\widetilde{\alpha}}\right).$

Corollary 5.3 implies that

(5) $\displaystyle\frac{1}{|B_{i}|!}\sum_{\begin{subarray}{c}d\geq
3\end{subarray}}\sum_{\widetilde{V}\subset\Pi_{d}(B_{i})}\lambda^{r(\widetilde{V})}\sum_{\widetilde{\alpha}\in\mathrm{Cyc}(\widetilde{V})}x^{\widetilde{\alpha}}=\sum_{\begin{subarray}{c}d\geq
3,r\geq 1\\\ \text{s.t. }d\cdot
r=|B_{i}|\end{subarray}}\lambda^{r}\frac{1}{r\cdot(d!)}\sum_{\beta\in\operatorname{{\mathbb{S}}}_{d}}x^{[r\cdot\beta]}.$

For $d\geq 3$ and integers $c_{1},\ldots,c_{d}\geq 0$ such that
$c_{1}+2c_{2}+3c_{3}+\ldots=d$, there are precisely
$d!/(\prod_{k}k^{c_{k}}c_{k}!)$ permutations with exactly $c_{1}$ 1-cycles,
$c_{2}$ 2-cycles, and so on. Hence,

$\sum_{d\geq
3}\frac{1}{d!}\sum_{\beta\in\operatorname{{\mathbb{S}}}_{d}}x^{[r\cdot\beta]}=\sum_{\begin{subarray}{c}c_{1},c_{2},\ldots\geq
0\\\ \text{s.t. }\sum_{k}kc_{k}\geq
3\end{subarray}}\prod_{k=1}^{d}\frac{x_{r\cdot
k}^{c_{k}}}{k^{c_{k}}c_{k}!}=\exp\left(\sum_{\ell\geq 1}\frac{x_{\ell\cdot
r}}{\ell}\right)-1-x_{r}-\frac{x_{r}^{2}}{2}-\frac{x_{2r}}{2}.$

Using this together with Eqs. (4), (5) and Lemma 5.4 gives the statement. ∎

We have now finished the purely combinatorial part of the argument that will
lead to the proof of Theorem 1.5. The next three sections will relate the
mainly _discrete_ objects discussed above with _analytic_ ones. Thanks to the
additional flexibility provided by the analytic tools, we can prove the
relatively simple formula in Theorem 1.5.

##  6\. Partition functions for graph complex Euler characteristics

In this section, we will define families of integrals whose asymptotic
expansions in a specific limit encode the numbers $\chi_{n}^{\pm}$ from Eq.
(2). For fixed $n\geq 1$, we define the following functions in
$z,x_{1},\ldots,x_{n}$. By analogy to quantum mechanical systems, we call them
_action_ functions:

(6) $\displaystyle\begin{aligned}
\mathcal{S}^{+}_{n}(z,x_{1},\ldots,x_{n})&=\phantom{-}\sum_{k=1}^{n}\frac{z^{k}}{k}\left(\exp\left(i\sum_{\ell=1}^{\lfloor
n/k\rfloor}\frac{x_{k\ell}}{\ell}\right)-1-ix_{k}\right)\\\
\mathcal{S}^{-}_{n}(z,x_{1},\ldots,x_{n})&=-\sum_{k=1}^{n}\frac{z^{k}}{k}\left(\exp\left(\phantom{i}\sum_{\ell=1}^{\lfloor
n/k\rfloor}\frac{x_{k\ell}}{\ell}\right)-1-\phantom{i}x_{k}\right),\end{aligned}$

where $\lfloor x\rfloor$ denotes the largest integer $\leq x$.

Each of these actions gives a _partition function_ , which is defined by the
following integral expression that, for each $n\geq 1$, depends on
$z\in{\mathbb{R}}_{>0}$ and a compact subset $D\subset{\mathbb{R}}^{n}$:

(7) $\displaystyle
I^{\pm}_{n}(z,D)=\left(\prod_{k=1}^{n}\frac{\exp\left(\pm\frac{\delta_{2|k}}{2k}\right)}{\sqrt{2\pi
k/z^{k}}}\right)\int_{D}\exp\left(\mathcal{S}^{\pm}_{n}(z,x_{1},\ldots,x_{n})\right)\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$

The integral exists as $D$ is compact and the integrand is an entire function.

We will use the big-$\mathcal{O}$ notation to formulate asymptotic statements.
Let $h:{\mathbb{R}}_{>0}\rightarrow{\mathbb{R}}_{>0}$ be a function defined on
sufficiently large positive reals. The notation $`\mathcal{O}(h(z))$ as
$z\rightarrow\infty$’ denotes the set of all functions $f(z)$, for which
$\limsup_{z\rightarrow\infty}|f(z)/h(z)|<\infty$. The notation
‘$f(z)=g(z)+\mathcal{O}(h(z))\text{ as }z\rightarrow\infty$’ means that
$f(z)-g(z)\in\mathcal{O}(h(z))$ as $z\rightarrow\infty$.

Let
$B_{n}(\varepsilon)=\\{(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}:\sum_{k=1}^{n}|x_{k}|/(k\varepsilon^{k})\leq
1\\}$ and recall the definition of the sequences $\chi^{\pm}_{n}$ in Eq. (2).
The main result of this section is that for a specific set of integration
domains, the large $z$ asymptotic expansion of the integrals
$I^{\pm}_{n}(z,D)$ encodes the integers $\chi^{\pm}_{n}$.

######  Theorem 6.1.

For fixed $n\geq 1$, and any $z$-dependent family, $D(z)$, of subsets of
${\mathbb{R}}^{n}$, which fulfills $B_{n}(z^{-\frac{5}{12}}/8)\subset
D(z)\subset B_{n}(z^{-\frac{5}{12}}/2),$ the integral $I^{\pm}_{n}(z,D(z))$
behaves asymptotically as

$\displaystyle I^{\pm}_{n}(z,D(z))=\sum_{k=0}^{\lfloor
n/12\rfloor}\chi^{\pm}_{k}~{}z^{-k}+\mathcal{O}(z^{-\frac{n+1}{12}})\text{ as
}z\rightarrow\infty.$

The proof of this theorem will occupy the remainder of this section. It will
follow a well-established theoretical physics stategy: We will ensure that,
for large $z$, the integrals are well-approximated by specific Gaussian
integrals. After establishing bounds on this approximation’s accuracy, we can
compute the $z\rightarrow\infty$ asymptotic expansion via slight perturbations
of the original Gaussian integral. In the finite-dimensional case, this
strategy is called _Laplace method_. Here, a major technical challenge is that
the dimension of the integral increases with $n$.

For any $z\in{\mathbb{R}}_{>0}$, we define the following complex-valued
Gaussian measures on ${\mathbb{R}}^{n}$,

(8)
$\displaystyle\mu_{n}^{\pm}(z)=\left(\prod_{k=1}^{n}\frac{\exp\left(-\frac{z^{k}}{2k}\left(x_{k}-\rho^{\pm}\cdot\delta_{2|k}\cdot
z^{-\frac{k}{2}}\right)^{2}\right)}{\sqrt{2\pi
k/z^{k}}}\right)\mathrm{d}x_{1}\cdots\mathrm{d}x_{n},$

where $\rho^{+}=i$ and $\rho^{-}=-1$ and we recall that
$\delta_{2|k}=(1+(-1)^{k})/2$.

The following proposition can be interpreted as an equivariant version of
_Wick_ ’s theorem, a standard statement in quantum field theory. Getzler and
Kapranov obtained similar statements in an entirely formal context [12,
§8.16].

######  Proposition 6.2.

Fix $z>0$ and integers $n,s$ such that $n\geq 1$ and $s\geq 0$. If $n\geq
2s+1$ and $\alpha\in\operatorname{{\mathbb{S}}}_{2s+1}$, then
$\int_{{\mathbb{R}}^{n}}(ix)^{\alpha}\mu_{n}^{+}(z)=\int_{{\mathbb{R}}^{n}}x^{\alpha}\mu_{n}^{-}(z)=0.$
If $n\geq 2s$ and $\alpha\in\operatorname{{\mathbb{S}}}_{2s}$, then

$\displaystyle\int_{{\mathbb{R}}^{n}}(ix)^{\alpha}\mu_{n}^{+}(z)$
$\displaystyle=z^{-s}\sum_{E\in\mathcal{E}(\alpha)}(-1)^{c(\alpha_{E})}\text{
and }$ $\displaystyle\int_{{\mathbb{R}}^{n}}x^{\alpha}\mu_{n}^{-}(z)$
$\displaystyle=z^{-s}{\operatorname{sign}}(\alpha)|\mathcal{E}(\alpha)|,$

where $(ix)^{\alpha}$ is the monomial $x^{\alpha}$ with each instance of
$x_{k}$ replaced by $ix_{k}$.

###### Proof.

We will first prove that for any $(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}$, we
have

(9) $\displaystyle\begin{aligned}
f^{+}&:=\int_{{\mathbb{R}}^{n}}\exp\left(i\sum_{k=1}^{n}z^{\frac{k}{2}}\frac{x_{k}y_{k}}{k}\right)\mu_{n}^{+}(z)=\exp\left(-\sum_{k=1}^{n}\frac{y_{k}^{2}+y_{2k}}{2k}\right)\\\
f^{-}&:=\int_{{\mathbb{R}}^{n}}\exp\left(\phantom{i}\sum_{k=1}^{n}z^{\frac{k}{2}}\frac{x_{k}y_{k}}{k}\right)\mu_{n}^{-}(z)=\exp\left(\phantom{-}\sum_{k=1}^{n}\frac{y_{k}^{2}-y_{2k}}{2k}\right),\end{aligned}$

where we agree that $y_{k}=0$ for all $k>n$. From the definition of
$\mu^{\pm}_{n}$ in Eq. (8), we get

$\displaystyle f^{\pm}$
$\displaystyle=\prod_{k=1}^{n}\left(\int_{-\infty}^{\infty}\frac{\exp\left(A_{k}^{\pm}\right)}{\sqrt{2\pi
k/z^{k}}}\mathrm{d}x_{k}\right),\quad\text{where, }$ $\displaystyle A_{k}^{+}$
$\displaystyle=-\frac{z^{k}}{2k}\left(x_{k}-i\delta_{2|k}~{}z^{-\frac{k}{2}}\right)^{2}+iz^{\frac{k}{2}}\frac{x_{k}y_{k}}{k}=-\frac{z^{k}}{2k}\left(x_{k}-i\delta_{2|k}~{}z^{-\frac{k}{2}}-iy_{k}z^{-\frac{k}{2}}\right)^{2}-\frac{y_{k}^{2}}{2k}-\frac{\delta_{2|k}y_{k}}{k}$
$\displaystyle A_{k}^{-}$
$\displaystyle=-\frac{z^{k}}{2k}\left(x_{k}+\delta_{2|k}~{}z^{-\frac{k}{2}}\right)^{2}+z^{\frac{k}{2}}\frac{x_{k}y_{k}}{k}=-\frac{z^{k}}{2k}\left(x_{k}+\delta_{2|k}~{}z^{-\frac{k}{2}}-y_{k}z^{-\frac{k}{2}}\right)^{2}+\frac{y_{k}^{2}}{2k}-\frac{\delta_{2|k}y_{k}}{k}$

Eq. (9) follows from the formula
$\int_{-\infty}^{\infty}e^{-a(x+c)^{2}}\mathrm{d}x=\sqrt{\pi/a}$, which holds
for all $c\in{\mathbb{C}}$ and $a>0$.

If $c_{1},\ldots,c_{m}\geq 0$ with $\sum_{k}kc_{k}=m$, there are
$m!/(\prod_{k\geq 1}k^{c_{k}}c_{k}!)$ permutations with exactly $c_{1}$
$1$-cycles, $c_{2}$ $2$-cycles, and so on. Recall our convention that
$y_{k}=0$ for $k>n$, we find

$\displaystyle\sum_{m=0}^{\infty}z^{\frac{m}{2}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{m}}\frac{(ix)^{\alpha}y^{\alpha}}{m!}=\sum_{m=0}^{\infty}z^{\frac{m}{2}}\sum_{\begin{subarray}{c}c_{1},\ldots,c_{m}\geq
0\\\ c_{1}+2c_{2}+\ldots=m\end{subarray}}\prod_{k\geq
1}\frac{(ix_{k}y_{k})^{c_{k}}}{k^{c_{k}}c_{k}!}=\exp\left(i\sum_{k=1}^{n}z^{\frac{k}{2}}\frac{x_{k}y_{k}}{k}\right).$

Using this together with Eq. (9) and Proposition 5.1, we find that

$\displaystyle\int_{{\mathbb{R}}^{n}}\left(\sum_{m=0}^{\infty}z^{\frac{m}{2}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{m}}\frac{(ix)^{\alpha}y^{\alpha}}{m!}\right)\mu_{n}^{+}(z)=\sum_{s=0}^{\infty}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{y^{\alpha}}{(2s)!}\sum_{E\in\mathcal{E}(\alpha)}(-1)^{c(\alpha_{E})}.$

The integrand on the left-hand side is an entire function in the
$y$-variables. So, both sides agree coefficient-wise in $y_{1},\ldots,y_{n}$.
Also, each coefficient in front of $y^{\alpha}$ only depends on the cycle type
of $\alpha$. The first stated equation follows. For the second one, observe
that due to Proposition 5.1,

$\displaystyle\exp\left(\sum_{k=1}^{n}\frac{y_{k}^{2}-y_{2k}}{2k}\right)=\sum_{s=0}^{\infty}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}(-1)^{\sum_{k}c_{2k}(\alpha)}\frac{y^{\alpha}}{(2s)!}|\mathcal{E}(\alpha)|=\sum_{s=0}^{\infty}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}{\operatorname{sign}}(\alpha)\frac{y^{\alpha}}{(2s)!}|\mathcal{E}(\alpha)|,$

where $c_{2k}(\alpha)$ is the number of $2k$-cycles of $\alpha$. ∎

The following corollary tells us how to extract the numbers of graphs out of
Gaussian integrals.

######  Corollary 6.3.

Fix $z>0$ and integers $n,s,r$ such that $n\geq 1$ and $s\geq 0$. For all
$n\geq 2s$,

$\displaystyle\int_{{\mathbb{R}}^{n}}\left(\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{(ix)^{\alpha}}{(2s)!}|\mathcal{V}_{r}(\alpha)|\right)\mu_{n}^{+}(z)$
$\displaystyle=(-1)^{s}z^{-s}\left|\\{G\in\mathbf{G}_{+}:|E_{G}|=s,|V_{G}|=r\\}\right|,\text{
and }$
$\displaystyle\int_{{\mathbb{R}}^{n}}\left(\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}(-1)^{c(\alpha_{V})}\right)\mu_{n}^{-}(z)$
$\displaystyle=(-1)^{r}z^{-s}\left|\\{G\in\mathbf{G}_{-}:|E_{G}|=s,|V_{G}|=r\\}\right|.$

###### Proof.

Using Proposition 6.2, we find that the left-hand sides are equal to

$\displaystyle\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{1}{(2s)!}|\mathcal{V}_{r}(\alpha)|\sum_{E\in\mathcal{E}(\alpha)}(-1)^{c(\alpha_{E})},\text{
and
}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{1}{(2s)!}{\operatorname{sign}}(\alpha)|\mathcal{E}(\alpha)|\sum_{V\in\mathcal{V}_{r}(\alpha)}(-1)^{c(\alpha_{V})}\text{
respectively.}$

The statement follows from Theorem 4.1 because for any permutation
$\alpha\in\operatorname{{\mathbb{S}}}_{n}$, the sign and the parity of its
number of cycles are related by
$(-1)^{c(\alpha)}=(-1)^{n}{\operatorname{sign}}(\alpha)$. ∎

We need bounds on the accuracy of expansions of their integrands to apply
these methods to the integral expressions in Eq. (7). The following three
technical statements, which give inequalities required for the proof of
Theorem 6.1, provide these bounds.

######  Lemma 6.4.

If $R\geq 0$, $z\geq 2^{12}=4096$, $\phi=\pm 1$, $|\lambda|\leq z$ and
$|x_{k}|\leq z^{-\frac{5}{12}k}$ for all $k\geq 1$, then

$\displaystyle\left|\exp\left(\phi\sum_{k\geq
1}\frac{\lambda^{k}}{k}V(x_{k},x_{2k},\ldots)\right)-\sum_{\begin{subarray}{c}s,r\geq
0\\\
2s<R\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}\lambda^{r}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}\phi^{c(\alpha_{V})}\right|\leq\left(\frac{8}{7}\right)^{2}\left(\frac{2^{12}}{z}\right)^{R/12}.$

###### Proof.

By Proposition 5.1, the expression inside the absolute function on the left-
hand side is

$\displaystyle\mathcal{R}_{R}:=\sum_{\begin{subarray}{c}s,r\geq 0\\\ 2s\geq
R\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}\lambda^{r}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}\phi^{c(\alpha_{V})},$
which we can estimate using the bounds on $\lambda$ and $x_{1},x_{2},\ldots$
$\displaystyle|\mathcal{R}_{R}|\leq\sum_{\begin{subarray}{c}s,r\geq 0\\\
2s\geq
R\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{z^{-\frac{5}{6}s}z^{r}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}|\phi|.$

Recall that $\mathcal{V}_{r}(\alpha)$ only contains partitions of $[2s]$ into
$r$ blocks, each of size $\geq 3$. Thus, the sum only has support if $r\leq
2s/3$. Hence, in the sum $z^{-\frac{5}{6}s+r}\leq
z^{-\frac{5}{6}s+\frac{2}{3}s}=z^{-\frac{1}{6}s}$. Moreover, it follows from
$2s\geq R$ and $z\geq 2^{12}$ that
$z^{-\frac{1}{6}s}\leq\left({2^{12}/z}\right)^{R/12}2^{-2s}$. Therefore,

$\displaystyle|\mathcal{R}_{R}|\leq\left(\frac{2^{12}}{z}\right)^{R/12}\sum_{\begin{subarray}{c}s,r\geq
0\\\ 2s\geq
R\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{2^{-2s}}{(2s)!}\sum_{V\in\mathcal{V}_{r}(\alpha)}|\phi|.$
After dropping the requirement that $2s\geq R$ in the sum, we may apply
Proposition 5.1 to get
$\displaystyle|\mathcal{R}_{R}|\leq\left(\frac{2^{12}}{z}\right)^{{R}/{12}}\exp\left(|\phi|\sum_{k\geq
1}\frac{1}{k}V\left(2^{-k},2^{-2k},\ldots\right)\right)=\left(\frac{2^{12}}{z}\right)^{{R}/{12}}\exp\left(\sum_{k\geq
1}\frac{1}{k}\frac{2^{-3k}}{1-2^{-k}}\right),$

where we used Eq. (3). The statement follows as $\sum_{k\geq
1}\frac{1}{k}\frac{2^{-3k}}{1-2^{-k}}\leq\frac{1}{1-2^{-1}}\sum_{k\geq
1}\frac{2^{-3k}}{k}=2\log\frac{8}{7}$. ∎

Recall that for $\varepsilon>0$, we defined
$B_{n}(\varepsilon)=\\{(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}:\sum_{k=1}^{n}|x_{k}|/(k\varepsilon^{k})\leq
1\\}$.

######  Lemma 6.5.

For each $n\geq 0$, there is a constant $C_{n}>0$ such that for all
$z,\varepsilon\in{\mathbb{R}}_{>0}$, $s\in{\mathbb{Z}}_{\geq 0}$, and
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$, which fulfill
$\varepsilon^{2}z\geq 1$ and $s\leq n/2$, the following inequality holds

$\displaystyle\int_{{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)}\left|x^{\alpha}\mu^{\pm}(z)\right|\leq
C_{n}z^{-s}e^{-\varepsilon\sqrt{z}}.$

Here, the vertical bars mean that we take the modulus of the complex-valued
integrand. For instance with
$\mu^{\pm}(z)=f^{\pm}(z,x_{1},\ldots,x_{n})\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$,
$|\mu^{\pm}(z)|=|f^{\pm}(z,x_{1},\ldots,x_{n})|\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$.

###### Proof.

We start by proving three inequalities. For all integers $n\geq k\geq 1$,
$(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$, $z\in{\mathbb{R}}_{>0}$, and a
complex number $\rho^{\pm}\in{\mathbb{C}}$ of modulus $\leq 1$, it follows
directly that

(10)
$\displaystyle\left|\exp\left(-\frac{z^{k}}{2k}\left(x_{k}-\rho^{\pm}\cdot\delta_{2|k}\cdot
z^{-\frac{k}{2}}\right)^{2}\right)\right|\leq\exp\left(-\frac{z^{k}}{2k}x_{k}^{2}+\frac{z^{\frac{k}{2}}}{k}|x_{k}|+\frac{1}{2k}\right).$

For any given non-negative integer $s$ with $2s\leq n$ and
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$, we have,

(11)
$\displaystyle|x^{\alpha}|\leq(2s)!\left(\frac{e^{2}}{z}\right)^{s}\exp\left(\sum_{k=1}^{n}\frac{z^{\frac{k}{2}}|x_{k}|}{k}\right),$

To prove this second inequality observe that $k\geq 1\Rightarrow\log k\leq k$,
$\frac{|x|^{k}}{k!}\leq e^{|x|}$ and therefore

$|x_{k}|^{c_{k}}=z^{-\frac{kc_{k}}{2}}e^{c_{k}\log
k}\left(\frac{z^{\frac{k}{2}}|x_{k}|}{k}\right)^{c_{k}}\leq\left(\frac{e^{2}}{z}\right)^{\frac{kc_{k}}{2}}c_{k}!\exp\left(\frac{z^{\frac{k}{2}}|x_{k}|}{k}\right).$

Eq. (11) follows by writing $|x^{\alpha}|=\prod_{k=1}^{n}|x_{k}|^{c_{k}}$,
where $c_{k}$ is the number of $k$-cycles of $\alpha$, because
$\sum_{k}c_{k}=c(\alpha)$ is the total number of cycles of $\alpha$,
$\sum_{k}kc_{k}=2s$, and $a!b!\leq(a+b)!$ for all non-negative integers $a,b$.

For all $\varepsilon,z>0$ with $\varepsilon^{2}z\geq 1$ and restricting the
$x$-values to lie outside $B_{n}(\varepsilon)\subset{\mathbb{R}}^{n}$, i.e.
$(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}\setminus B_{n}(\varepsilon)$, we have
the third inequalitity

(12) $\displaystyle
0<\sum_{k=1}^{n}\frac{z^{\frac{k}{2}}|x_{k}|}{k}-\varepsilon\sqrt{z},$

which holds, because $(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)$ implies
$1<\sum_{k=1}^{n}\frac{|x_{k}|}{k\varepsilon^{k}}=\frac{1}{\varepsilon\sqrt{z}}\sum_{k=1}^{n}\frac{z^{\frac{k}{2}}|x_{k}|}{k(\varepsilon\sqrt{z})^{k-1}},$
as we require $\varepsilon^{2}z\geq 1$, Eq. (12) follows.

We can apply the inequalities Eq. (10), (11), and (12) to the definition of
$\mu^{\pm}(z)$ in Eq. (8) and find that for $n\geq 2s\geq 2$,
$\alpha\in\operatorname{{\mathbb{S}}}_{2s}$,
$z,\varepsilon\in{\mathbb{R}}_{>0}$ with $\varepsilon^{2}z\geq 1$ and
$\gamma\in{\mathbb{R}}_{\geq 0}$, we have

$\displaystyle\left|\int_{{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)}x^{\alpha}\mu^{\pm}(z)\right|\leq(2s)!\left(\frac{e^{2}}{z}\right)^{s}e^{-\gamma\varepsilon\sqrt{z}}\int_{{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)}\frac{\exp\left(\sum_{k=1}^{n}f_{k}(|x_{k}|,z,\gamma)\right)}{\prod_{k=1}^{n}\sqrt{2\pi
k/z^{k}}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n},$

where
$f_{k}(x_{k},z,\gamma)=-\frac{z^{k}}{2k}x_{k}^{2}+(2+\gamma)\frac{z^{\frac{k}{2}}}{k}x_{k}+\frac{1}{2k}=-\frac{z^{k}}{2k}(x_{k}+(2+\gamma)z^{-\frac{k}{2}})^{2}+\frac{(2+\gamma)^{2}+1}{2k}$
and we multiplied Eq. (12) with the arbitrary non-negative $\gamma$ before its
application under the integral sign. We can write the integrand on the right-
hand side as an integral over the positive orthant, which we can extend to a
purely Gaussian integral over ${\mathbb{R}}^{n}$:

$\displaystyle\int_{{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)}\frac{\exp\left(\sum_{k=1}^{n}f_{k}(|x_{k}|,z,\gamma)\right)}{\prod_{k=1}^{n}\sqrt{2\pi
k/z^{k}}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$ $\displaystyle\leq
2^{n}\int_{{\mathbb{R}}_{\geq 0}^{n}\setminus
B_{n}(\varepsilon)}\frac{\exp\left(\sum_{k=1}^{n}f_{k}(x_{k},z,\gamma)\right)}{\prod_{k=1}^{n}\sqrt{2\pi
k/z^{k}}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$ $\displaystyle\leq
2^{n}\int_{{\mathbb{R}}^{n}}\frac{\exp\left(\sum_{k=1}^{n}f_{k}(x_{k},z,\gamma)\right)}{\prod_{k=1}^{n}\sqrt{2\pi
k/z^{k}}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$
$\displaystyle=2^{n}\exp\left(\sum_{k=1}^{n}\frac{(2+\gamma)^{2}+1}{2k}\right),$

where we used the formula
$\sqrt{\pi/a}=\int_{{\mathbb{R}}}e^{-a(x+b)^{2}}\mathrm{d}x$ that holds for
all $a>0$ and $b\in{\mathbb{R}}$ in the last step. The statement follows as
$\sum_{k=1}^{n}\frac{1}{k}\leq n$ and by choosing $\gamma=1$ and
$C_{n}=n!(2e^{6})^{n}$. ∎

######  Corollary 6.6.

For each $n\geq 0$, there is a constant $C_{n}^{\prime}>0$ such that for all
$z,\varepsilon\in{\mathbb{R}}_{>0}$ which fulfill $z\geq 1$ and
$\varepsilon^{2}z\geq 1$, the following inequality holds

$\displaystyle\int_{{\mathbb{R}}^{n}\setminus
B_{n}(\varepsilon)}\left|\left(\sum_{\begin{subarray}{c}r,s\geq 0\\\ 2s\leq
n\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{x^{\alpha}z^{r}}{(2s)!}|\mathcal{V}_{r}(\alpha)|\right)\mu_{n}^{\pm}(z)\right|\leq
C_{n}^{\prime}e^{-\varepsilon\sqrt{z}}.$

###### Proof.

Use Lemma 6.5, that $\mathcal{V}_{r}(\alpha)$ is only non-empty if
$r\leq\frac{2s}{3}$ and $z\geq 1$. ∎

###### Proof of Theorem 6.1.

We will only discuss the even case in detail. The odd case follows
analogously. We fix $n\geq 1$ and a $z$-dependent family of subsets
$D(z)\in{\mathbb{R}}^{n}$ that fulfills $B_{n}(z^{-\frac{5}{12}}/8)\subset
D(z)\subset B_{n}(z^{-\frac{5}{12}}/2).$ We define $A^{+}_{n}(z)$ to be the
generating polynomial of signed even-orientable graphs with Euler
characteristic bounded from below by $-n/6$, i.e., with $\chi^{+}_{n}$ from
Eq. (2),

$\displaystyle
A^{+}_{n}(z)=\sum_{k=0}^{\lfloor\frac{n}{6}\rfloor}\chi^{+}_{k}z^{-k}=\sum_{\begin{subarray}{c}s,r\geq
0\\\
r-s\geq-n/6\end{subarray}}(-1)^{s}\left|\\{G\in\mathbf{G}_{+}:|E_{G}|=s,|V_{G}|=r\\}\right|z^{r-s}.$

Recall that an (admissible) graph $G$ fulfills $2|E_{G}|\geq 3|V_{G}|$. Hence,
the Euler characteristic $\chi(G)=|V_{G}|-|E_{G}|$ is negative for every non-
empty $G$, and the right-most sum above only has support if $2s\geq 3r$. In
the sum, we therefore have $s-r\leq\frac{1}{3}s\leq n/6\Rightarrow 2s\leq n$.
Hence,

$\displaystyle A^{+}_{n}(z)=\sum_{\begin{subarray}{c}s,r\geq 0\\\ 2s\leq
n\end{subarray}}(-1)^{s}\left|\\{G\in\mathbf{G}_{+}:|E_{G}|=s,|V_{G}|=r\\}\right|z^{r-s}+\mathcal{R}^{(1)}_{n}(z),$

where the error term is a (finite) sum over graphs that have at most $n/2$
edges but whose Euler characteristic is smaller than $-n/6$. Hence,
$\mathcal{R}^{(1)}_{n}(z)\in z^{-\lceil n/6\rceil-1}{\mathbb{Z}}[z^{-1}]$, and
$\mathcal{R}^{(1)}_{n}(z)\in\mathcal{O}(z^{-\lceil n/6\rceil-1})$ as
$z\rightarrow\infty$. Using Corollary 6.3, we write the expression above as an
integral,

$\displaystyle
A^{+}_{n}(z)=\int_{{\mathbb{R}}^{n}}\left(\sum_{\begin{subarray}{c}r,s\geq
0\\\ 2s\leq
n\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{(ix)^{\alpha}z^{r}}{(2s)!}|\mathcal{V}_{r}(\alpha)|\right)\mu_{n}^{+}(z)+\mathcal{R}^{(1)}_{n}(z),$

whose integration to a domain we may restrict to $D(z)$,

$\displaystyle A^{+}_{n}(z)=\int_{D(z)}\left(\sum_{\begin{subarray}{c}r,s\geq
0\\\ 2s\leq
n\end{subarray}}\sum_{\alpha\in\operatorname{{\mathbb{S}}}_{2s}}\frac{(ix)^{\alpha}z^{r}}{(2s)!}|\mathcal{V}_{r}(\alpha)|\right)\mu_{n}^{+}(z)+\mathcal{R}^{(1)}_{n}(z)+\mathcal{R}^{(2)}_{n}(z).$

Observe that if $\varepsilon(z)=z^{-\frac{5}{12}}/8$ and $z\geq 2^{36}$, then
$\varepsilon(z)^{2}z=z^{\frac{1}{6}}/8^{2}\geq 2^{6}/8^{2}=1$. So, as
$D(z)\supset B_{n}(z^{-\frac{5}{12}}/8)$, we may use Corollary 6.6 to estimate
the error term, $|\mathcal{R}^{(2)}_{n}(z)|\leq
C^{\prime}_{n}\exp(-z^{\frac{1}{12}}/8)$.

Recall the definition of $V$ in Eq. (3) and its relation to the sum under the
integral sign above due to Proposition 5.1. For any $(x_{1},\ldots,x_{n})\in
B_{n}(z^{-\frac{5}{12}}/2)$, we have $|x_{k}|\leq
k2^{-k}z^{-\frac{5}{12}k}\leq z^{-\frac{5}{12}k}$ for all
$k\in\\{1,\ldots,n\\}$. So, if $(x_{1},\ldots,x_{n})\in D(z)\subset
B_{n}(z^{-\frac{5}{12}}/2)$ and $z\geq 2^{12}$, then the conditions for Lemma
6.4 are fulfilled, and it is reasonable to write

$\displaystyle
A^{+}_{n}(z)=\int_{D(z)}\exp\left(\sum_{k=1}^{n}\frac{z^{k}}{k}V(ix_{k},ix_{2k},\ldots,ix_{\lfloor
n\rfloor_{k}},0,\ldots)\right)\mu_{n}^{+}(z)+\mathcal{R}^{(1)}_{n}(z)+\mathcal{R}^{(2)}_{n}(z)+\mathcal{R}^{(3)}_{n}(z),$

where $\lfloor n\rfloor_{k}$ denotes the largest multiple of $k$ that does not
exceed $n$. Lemma 6.4 provides an estimate of the third error term
$|\mathcal{R}^{(3)}_{n}(z)|\leq\left(\frac{8}{7}\right)^{2}\left({2^{12}/z}\right)^{\frac{n+1}{12}}$
for $z\geq 2^{12}$.

By expanding the exponents in the definitions of $V$ in Eq. (3) and the
Gaussian measure $\mu_{n}^{+}(z)$ in Eq. (8) while using the definition of the
action function $\mathcal{S}_{n}^{+}$ in Eq. (6), we recover the partition
function $I^{+}_{n}(z,D)$ as defined in Eq. (7), in the integral above. Hence,

$\displaystyle
A^{+}_{n}(z)=I^{+}_{n}(z,D(z))+\mathcal{R}^{(1)}_{n}(z)+\mathcal{R}^{(2)}_{n}(z)+\mathcal{R}^{(3)}_{n}(z).$

The statements follow as the dominant error term for $z\rightarrow\infty$ is
$\mathcal{R}^{(3)}_{n}(z)$, which falls off as $z^{-\frac{n+1}{12}}$ for large
$z$, and by definition,
$A^{+}_{n}(z)=\sum_{k=0}^{\lfloor\frac{n}{6}\rfloor}\chi^{+}_{k}z^{-k}=\sum_{k=0}^{\lfloor\frac{n}{12}\rfloor}\chi^{+}_{k}z^{-k}+\mathcal{O}(z^{-\frac{n+1}{12}})$
as $z\rightarrow\infty$. ∎

##  7\. A variable transformation

This section will show that a specific choice for the integration domain $D$
for the partition function (7) results in a convenient simplification. In this
simple case, we find families of functions, $J_{n}^{\pm}(z,\varepsilon)$,
indexed by $n$ and dependent on two parameters, $z$ and $\varepsilon$.

Recall that we defined $M_{k}(z)=\frac{1}{k}\sum_{d|k}\mu(d)z^{k/d}$ in
Section 1.3. Again, we will define two _action_ -type functions, which depend
on an integer index $k\geq 1$ and variables $z,y$,

(13) $\displaystyle\begin{aligned}
s_{k}^{+}(z,y)&=\phantom{-}\frac{z^{k}}{k}\left(e^{iy}-1\right)-iyM_{k}(z)\\\
s_{k}^{-}(z,y)&=-\frac{z^{k}}{k}\left(e^{\phantom{i}y}-1\right)+\phantom{i}yM_{k}(z).\end{aligned}$

The functions $J_{n}^{\pm}$ are defined as

(14) $\displaystyle
J_{n}^{\pm}(z,\varepsilon)=\prod_{k=1}^{n}\left(\exp\left(\pm\frac{\delta_{2|k}}{2k}\right)\int_{-k\varepsilon^{k}}^{k\varepsilon^{k}}\exp\left(s_{k}^{\pm}(z,y)\right)\frac{\mathrm{d}y}{\sqrt{2\pi
k/z^{k}}}\right).$

To relate the integrals $I^{\pm}_{n}$ from Eq. (7) with the ones above, we
define a linear map $\iota_{n}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n}$
such that $y=\iota_{n}(x)$ is expressed coordinate-wise as
$y_{k}=\sum_{\ell=1}^{\lfloor n/k\rfloor}\frac{x_{k\cdot\ell}}{\ell}$. The map
$\iota_{n}$ is bijective, and the inverse transformation $x=\iota_{n}^{-1}(y)$
is given coordinate-wise by $x_{k}=\sum_{\ell=1}^{\lfloor
n/k\rfloor}\mu(\ell)\frac{y_{k\cdot\ell}}{\ell}$.

For each $\varepsilon>0$, let
$K_{n}(\varepsilon)=\\{(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}:|y_{k}|\leq
k\varepsilon^{k}\text{ for }k\in\\{1,\ldots n\\}\\}$. As $K_{n}(\varepsilon)$
is a convex set, its preimage, $\iota_{n}^{-1}(K_{n}(\varepsilon))$, is too.
Further, it is easy to see that triangular matrices with $1$’s on the diagonal
represent $\iota_{n}$ and $\iota_{n}^{-1}$. Therefore,
$\det(\iota_{n})=\det(\iota_{n}^{-1})=1$.

######  Proposition 7.1.

Fix $n\geq 1$, and let
$D(\varepsilon)=\iota_{n}^{-1}(K_{n}(\varepsilon))\subset{\mathbb{R}}^{n}$,
then $J^{\pm}_{n}(z,\varepsilon)=I^{\pm}_{n}(z,D(\varepsilon))$.

###### Proof.

To recover Eq. (14), change variables to $y_{k}=\sum_{\ell=1}^{\lfloor
n/k\rfloor}\frac{x_{\ell k}}{\ell}$ in the partition integral Eq. (7). As
$\det(\iota_{n})=1$, the Jacobian is trivial. The only nontrivial computation
is
$\sum_{k=1}^{n}\frac{z^{k}}{k}x_{k}=\sum_{k=1}^{n}\frac{z^{k}}{k}\sum_{\ell=1}^{\lfloor
n/k\rfloor}\mu(\ell)\frac{y_{k\ell}}{\ell}=\sum_{k=1}^{n}y_{k}M_{k}(z)$, where
we interchanged the order of summation. ∎

The main technical result of this section is that the choice of integration
domain $D(\varepsilon)=\iota_{n}^{-1}(K_{n}(\varepsilon))$ also fulfills the
requirements of Theorem 6.1 with a specific choice for $\varepsilon$.

######  Theorem 7.2.

For all $n\geq 1$ and $\varepsilon\leq\frac{1}{3}$, we have
$B_{n}(\frac{3}{4}\varepsilon)\subset\iota_{n}^{-1}(K_{n}(\varepsilon))\subset
B_{n}(3\varepsilon)$.

######  Corollary 7.3.

For fixed $n\geq 1$,

$\displaystyle J^{\pm}_{n}(z,z^{-\frac{5}{12}}/6)=\sum_{k=0}^{\lfloor
n/12\rfloor}\chi^{\pm}_{k}~{}z^{-k}+\mathcal{O}(z^{-\frac{n+1}{12}})\text{ as
}z\rightarrow\infty.$

###### Proof.

Let $\varepsilon(z)=z^{-\frac{5}{12}}/6$ and
$D(z)=\iota_{n}^{-1}(K_{n}(\varepsilon(z)))$. Then, by Theorem 7.2,
$B_{n}(z^{-\frac{5}{12}}/8)\subset D(z)\subset B_{n}(z^{-\frac{5}{12}}/2)$.
So, the statement follows from Proposition 7.1 and Theorem 6.1. ∎

The proof of Theorem 7.2 will occupy the remainder of this section.

######  Lemma 7.4.

If $n\geq 1$, $\varepsilon\leq\frac{1}{3}$ and
$(x_{1},\ldots,x_{n})\in\iota^{-1}(K_{n}(\varepsilon))$, then $|x_{k}|\leq
2k\varepsilon^{k}$ for $k\in\\{1,\ldots,n\\}$.

###### Proof.

Suppose $n$, $\varepsilon$ and $(x_{1},\ldots,x_{n})$ are given as required.
The definitions of $\iota_{n}$ and $K_{n}(\varepsilon)$ imply that
$\left|\sum_{\ell=1}^{\lfloor
n/k\rfloor}\frac{x_{k\cdot\ell}}{\ell}\right|\leq k\varepsilon^{k}$ for all
$k\in\\{1,\ldots,n\\}$. It follows that

(15) $\displaystyle|x_{k}|\leq k\varepsilon^{k}+\sum_{\ell=2}^{\lfloor
n/k\rfloor}\frac{|x_{k\cdot\ell}|}{\ell}\text{ for all }k\in\\{1,\ldots,n\\}.$

We will use this inequality to prove the lemma via induction. First, observe
that Eq. (15) implies $|x_{n}|\leq n\varepsilon^{n}\leq 2n\varepsilon^{n}$.
Assume that for given $m\in\\{1,\ldots,n\\}$, we have $|x_{k}|\leq
2k\varepsilon^{k}$ for all $k\in\\{m,\ldots,n\\}$. It follows from Eq. (15)
that for all $k$ with $\frac{m}{2}\leq k\leq n$,

$|x_{k}|\leq k\varepsilon^{k}+\sum_{\ell=2}^{\lfloor
n/k\rfloor}\frac{|x_{k\cdot\ell}|}{\ell}\leq
k\varepsilon^{k}+2k\sum_{\ell=2}^{\lfloor
n/k\rfloor}\varepsilon^{k\ell}=k\varepsilon^{k}\left(1+2\sum_{\ell=2}^{\lfloor
n/k\rfloor}\varepsilon^{k(\ell-1)}\right)\leq
k\varepsilon^{k}\left(1+2\sum_{\ell=1}^{\infty}\varepsilon^{k\ell}\right),$

where we used the assumption in the second step. Using
$\varepsilon^{k}\leq\frac{1}{3}$ and
$\sum_{\ell=1}^{\infty}\frac{1}{3^{\ell}}=\frac{1}{2}$ proves that
$|x_{k}|\leq 2k\varepsilon^{k}$ for all $k$ with $\frac{m}{2}\leq k\leq n$,
which inductively implies the statement. ∎

###### Proof of Theorem 7.2.

We start by proving
$B_{n}(\frac{3}{4}\varepsilon)\subset\iota_{n}^{-1}(K_{n}(\varepsilon))$. Fix
$(x_{1},\ldots,x_{n})\in B_{n}(\frac{3}{4}\varepsilon)$ which means that
$\sum_{k=1}^{n}\frac{|x_{k}|}{k(3\varepsilon/4)^{k}}\leq 1$ implying
$|x_{k}|\leq k(3\varepsilon/4)^{k}$ for $k\in\\{1,\ldots,n\\}$. Hence,
$\left|\sum_{\ell=1}^{\lfloor
n/k\rfloor}\frac{x_{k\cdot\ell}}{\ell}\right|\leq k\sum_{\ell=1}^{\lfloor
n/k\rfloor}(3\varepsilon/4)^{k\ell}\leq
k\sum_{\ell=1}^{\infty}(3\varepsilon/4)^{k\ell}=k\frac{(3\varepsilon/4)^{k}}{1-(3\varepsilon/4)^{k}}$
for all $k\in\\{1,\ldots,n\\}$. The inclusion
$B_{n}(3\varepsilon/4)\subset\iota_{n}^{-1}(K_{n}(\varepsilon))$ follows, as
$\varepsilon\leq\frac{1}{3}$ implies
$(\frac{3}{4})^{k}/(1-(3\varepsilon/4)^{k})\leq(\frac{3}{4})^{k}\frac{4}{3}\leq
1.$

To prove $\iota_{n}^{-1}(K_{n}(\varepsilon))\subset B_{n}(3\varepsilon)$, fix
$(x_{1},\ldots,x_{n})\in\iota_{n}^{-1}(K_{n}(\varepsilon))$, which, by Lemma
7.4, implies $|x_{k}|\leq 2k\varepsilon^{k}$ for all $k\in\\{1,\ldots,n\\}$
and therefore $\sum_{k=1}^{n}\frac{|x_{k}|}{k(3\varepsilon)^{k}}\leq
2\sum_{k=1}^{n}\frac{1}{3^{k}}\leq 2\sum_{k=1}^{\infty}\frac{1}{3^{k}}=1.$ ∎

##  8\. Evaluating the partition functions

In this section, we will prove Theorem 1.5 using the previously established
asymptotic expansions. The strategy is to use a known asymptotic expansion for
the $\Gamma$ function to find a more explicit expression for the
$z\rightarrow\infty$ asymptotic expansion of the integrals in Eq. (14). To do
so, it is convenient to define the following notation for the part of the
expression in Eq. (1) that does not involve the Bernoulli numbers:

(16) $\displaystyle
G_{k}^{\pm}(z)=\pm\left((1-L_{k}(z))M_{k}(z)-\frac{z^{k}}{k}+\frac{\delta_{2|k}}{2k}\right)-\frac{1}{2}L_{k}(z)$

We will first prove the first part of Theorem 1.5, which states that
$\Psi_{k}^{\pm}(z)\in
z^{-\lceil\frac{k}{6}\rceil}{\mathbb{Q}}[[\frac{1}{z}]]$. The following two
lemmas imply it.

######  Lemma 8.1.

If $k\geq 1$, $u\in{\mathbb{C}}$ such that $|u|\leq\frac{1}{4}$, then
$|ku^{k}M_{k}(\frac{1}{u})|\geq\frac{1}{2}$.

###### Proof.

Recall that $M_{k}(z)=\frac{1}{k}\sum_{d|k}\mu(d)z^{\frac{k}{d}}$ and hence
$ku^{k}M_{k}(1/u)=1+\sum_{d|k,d\neq 1}\mu(d)u^{k(1-\frac{1}{d})}$. Because
$\mu(1)=1$, $|\mu(d)|\leq 1$, and $|u|\leq 1$, we have $|ku^{k}M_{k}(1/u)|\geq
1-k|u|^{\frac{k}{2}}$. For all $k\geq 1$, the inequality $k\leq 2^{k-1}$
holds. Hence, if $|u|\leq\frac{1}{4}$, then
$k|u|^{\frac{k}{2}}\leq\frac{k}{2^{k}}\leq\frac{1}{2}$. ∎

######  Lemma 8.2.

For $k\geq 1$, the function $u\mapsto G_{k}^{\pm}(1/u)$ is holomorphic in the
disc $\\{u\in{\mathbb{C}}:|u|\leq\frac{1}{8}\\}$. Moreover, there is a
constant $C$ such that $|G_{k}^{\pm}(1/u)|\leq C|u|^{\lceil\frac{k}{6}\rceil}$
for all $u$ inside that domain.

###### Proof.

By Lemma 8.1, $u\mapsto L_{k}(1/u)=\log(ku^{k}M_{k}(1/u))$ defines an analytic
function for all $|u|\leq\frac{1}{4}$. It follows from Eq. (16) that $u\mapsto
G_{k}^{\pm}(1/u)$ is also analytic for all $|u|\leq\frac{1}{4}$ with the
possible exception of $u=0$ due to the $M_{k}(1/u)$ term, which has a pole at
$u=0$. We will prove that $G_{k}^{\pm}(1/u)$ is regular at $u=0$. Let
$W_{k}=1-{1}/{(ku^{k}M_{k}(1/u))}$ and observe that

$W_{k}=-\delta_{2|k}u^{\frac{k}{2}}+\frac{(ku^{k}M_{k}(\frac{1}{u})-1+\delta_{2|k}u^{\frac{k}{2}})+\delta_{2|k}u^{\frac{k}{2}}\left(ku^{k}M_{k}(\frac{1}{u})-1\right)}{ku^{k}M_{k}(\frac{1}{u})}.$

The denominator is bounded from below by $\frac{1}{2}$ for all
$|u|\leq\frac{1}{8}\leq\frac{1}{4}$ due to Lemma 8.1. The numerator is a sum
of two polynomials in $u^{\lceil\frac{2k}{3}\rceil}{\mathbb{Q}}[u]$ whose
coefficients are $\pm 1$. Hence,

$|W_{k}+\delta_{2|k}u^{\frac{k}{2}}|\leq
2u^{\lceil\frac{2k}{3}\rceil}(2+2u+2u^{2}+\ldots)\leq
4u^{\lceil\frac{2k}{3}\rceil}\frac{1}{1-u}\leq\frac{4}{1-2^{-3}}u^{\lceil\frac{2k}{3}\rceil}\leq\frac{32}{7}u^{\lceil\frac{2k}{3}\rceil},$

and similarly $|W_{k}|\leq
u^{\lceil\frac{k}{2}\rceil}(1+4u+4u^{2}\ldots)\leq\frac{11}{7}u^{\lceil\frac{k}{2}\rceil}.$

Writing
$\ell_{m}(W_{k})=\frac{1}{W_{k}^{m}}\left(\log\frac{1}{1-W_{k}}-\sum_{n=1}^{m-1}\frac{W_{k}^{n}}{n}\right)=\sum_{n=0}^{\infty}\frac{W_{k}^{n}}{n+m}$,
gives

$\displaystyle
G_{k}^{\pm}(1/u)=\mp\frac{u^{-k}}{k}W_{k}^{3}\frac{\ell_{3}(W_{k})+\frac{1}{2}}{1-W_{k}}\mp\frac{u^{-k}}{2k}\left(W_{k}+\delta_{2|k}u^{\frac{k}{2}}\right)\left(W_{k}-\delta_{2|k}u^{\frac{k}{2}}\right)-\frac{1}{2}W_{k}\ell_{1}(W_{k}).$

Each summand is regular at $u=0$ and falls off at least like
$u^{\lceil\frac{k}{6}\rceil}$ for $u\rightarrow 0$ by the estimates for
$W_{k}$ above. Moreover, it follows from $k\geq 1,|u|\leq\frac{1}{8}$ and
$|W_{k}|\leq\frac{11}{7}u^{\lceil\frac{k}{2}\rceil}\leq\frac{11}{7\sqrt{8}}<\frac{3}{4}$
that
$|\ell_{m}(W_{k})|\leq\sum_{n=0}^{\infty}\frac{|W_{k}|^{n}}{n+m}\leq\sum_{n=0}^{\infty}|W_{k}|^{n}=\frac{1}{1-|W_{k}|}\leq
4$ for all $m,k\geq 1$. Therefore,

$\displaystyle|G_{k}^{\pm}(1/u)|\leq\frac{|u|^{-k}}{k}|W_{k}|^{3}4(4+\frac{1}{2})+\frac{|u|^{-k}}{2k}\left|W_{k}+\delta_{2|k}u^{\frac{k}{2}}\right|\left(|W_{k}|+|u|^{\lceil\frac{k}{2}\rceil}\right)+\frac{1}{2}|W_{k}|4$
$\displaystyle\leq\frac{18}{k}\left(\frac{11}{7}\right)^{3}|u|^{\lceil\frac{3k}{2}\rceil-k}+\frac{\frac{32}{7}\cdot(\frac{11}{7}+1)}{2k}|u|^{\lceil\frac{2k}{3}\rceil+\lceil\frac{k}{2}\rceil-k}+2\frac{11}{7}|u|^{\lceil\frac{k}{2}\rceil}\leq\frac{14534}{343}|u|^{\lceil\frac{k}{6}\rceil}.\qed$

To prove the expression for the Euler characteristics
$\chi(\mathcal{GC}_{\pm}^{g})$ in the statement of Theorem 1.5, we define two
integrals depending on three parameters,

(17) $\displaystyle\begin{aligned}
Q^{+}(z,\varepsilon,u)&=\int_{-\varepsilon+iu}^{\varepsilon+iu}e^{z(e^{iy}-1-iy)}\frac{\mathrm{d}y}{\sqrt{2\pi/z}},&Q^{-}(z,\varepsilon,u)&=\int_{-\varepsilon-u}^{\varepsilon-u}e^{-z(e^{y}-1-y)}\frac{\mathrm{d}y}{\sqrt{2\pi/z}},\end{aligned}$

because the integrands are entire functions in $y$, they do not depend on the
integration path and are well-defined for all
$\varepsilon\in{\mathbb{R}}_{>0}$ and $u\in{\mathbb{C}}$. The proof of Theorem
1.5 relies on the following two statements on the functions $Q^{\pm}$.

######  Proposition 8.3.

If $z,\varepsilon,\varepsilon^{\prime}\in{\mathbb{R}}_{>0}$, and
$u\in{\mathbb{C}}$ with
$|u|+|\varepsilon-\varepsilon^{\prime}|\leq\frac{\varepsilon}{4}\leq\frac{1}{5}$,
and $z\varepsilon^{3}\leq\frac{64}{125}$, then

$\displaystyle\left|Q^{\pm}(z,\varepsilon^{\prime},u)-Q^{\pm}(z,\varepsilon,0)\right|\leq\frac{2}{5}e^{e}\frac{\exp(-\frac{7}{32}z\varepsilon^{2})}{\sqrt{2\pi/z}}.$

The second proposition is the origin of the Bernoulli numbers, $B_{m}$, in
Theorem 1.5.

######  Proposition 8.4.

If $\xi\in{\mathbb{R}}_{>0}$, then for all $R\geq 0$,

$\displaystyle Q^{\pm}(z,\xi\cdot
z^{-\frac{5}{12}},0)=\exp\left(\mp\sum_{m=1}^{R-1}\frac{B_{m+1}}{m(m+1)}z^{-m}\right)+\mathcal{O}(z^{-R})\text{
as }z\rightarrow\infty.$

Before we dive into the proofs of these last two statements, we use them to
prove Theorem 1.5:

###### Proof of Theorem 1.5.

By definition of $M_{k}(z)$, $1/M_{k}(z)^{m}\in
z^{-km}{\mathbb{Q}}[[\frac{1}{z}]]$ and, by Lemma 8.2, $G^{\pm}_{k}(z)\in
z^{-\lceil\frac{k}{6}\rceil}{\mathbb{Q}}[[\frac{1}{z}]]$. Hence,
$\Psi_{k}^{\pm}(z)\in
z^{-\lceil\frac{k}{6}\rceil}{\mathbb{Q}}[[\frac{1}{z}]]$. To prove the formula
for $\chi(\mathcal{GC}_{\pm}^{g})$, we write the integrals in Eq. (14) in
terms of the functions $Q^{\pm}$. Fix $k\geq 1$ and assume that $z$ is
sufficiently large such that $M_{k}(z)>0$. For each
$\varepsilon^{\prime}\in{\mathbb{R}}_{>0}$,

(18)
$\displaystyle\exp\left(\pm\frac{\delta_{2|k}}{2k}\right)\int_{-\varepsilon^{\prime}}^{\varepsilon^{\prime}}\exp({s_{k}^{\pm}(z,y)})\frac{\mathrm{d}y}{\sqrt{2\pi
k/z^{k}}}=\exp\left({G^{\pm}_{k}(z)}\right)Q^{\pm}(M_{k}(z),\varepsilon^{\prime},L_{k}(z)),$

which follows from the definition of $s_{k}^{\pm}$ in Eq. (13), of $Q^{\pm}$
in Eq. (17), of $G^{\pm}_{k}$ in Eq. (16), and by shifting the integration
variable $y\rightarrow y-iL_{k}(z)$ in the $+$ case and $y\rightarrow
y+L_{k}(z)$ in the $-$ case.

We now specialize to
$\varepsilon^{\prime}=\varepsilon^{\prime}(z)=kz^{-\frac{5}{12}k}/6^{k}$ and
also define
$\varepsilon(z)=k^{\frac{17}{12}}{M_{k}^{-\frac{5}{12}}}(z)/6^{k}$. Because
$M_{k}(z)=z^{k}/k+\mathcal{O}(z^{k/2})$ as $z\rightarrow\infty$, we have
$\varepsilon(z)=kz^{-\frac{5}{12}k}/6^{k}+\mathcal{O}(z^{-\frac{11}{12}k})$.
Moreover, $L_{k}(z)\in\mathcal{O}(z^{-\frac{k}{2}})$ and hence
$\varepsilon^{\prime}(z)-\varepsilon(z)=\mathcal{O}(z^{-\frac{11}{12}k})$ as
$z\rightarrow\infty$. Similarly, it follows that
$M_{k}(z)\varepsilon(z)^{3}=\mathcal{O}(z^{-\frac{1}{4}k})$ and
$M_{k}(z)\varepsilon(z)^{2}=kz^{\frac{1}{6}k}/6^{2k}+\mathcal{O}(z^{-\frac{1}{3}k})$.
Therefore, for all sufficiently large $z$ we find that
$|L_{k}(z)|+|\varepsilon(z)-\varepsilon^{\prime}(z)|\leq\varepsilon(z)/4\leq
1/5$ and $z\varepsilon(z)^{3}\leq\frac{64}{125}$. Hence, Proposition 8.3
implies

(19) $\displaystyle
Q^{\pm}(M_{k}(z),\varepsilon^{\prime}(z),L_{k}(z))=Q^{\pm}(M_{k}(z),\varepsilon(z),0)+\mathcal{O}\left(z^{\frac{k}{2}}\exp\left(-\frac{7k}{32\cdot
6^{2k}}z^{\frac{1}{6}k}\right)\right)\text{ as }z\rightarrow\infty.$

Applying Proposition 8.4 with $\xi=k^{\frac{17}{12}}/6^{k}$ while substituting
$z\rightarrow M_{k}(z)$ gives for all $R\geq 0$,

(20) $\displaystyle
Q^{\pm}(M_{k}(z),\varepsilon(z),0)=\exp\left(\mp\sum_{m=1}^{R-1}\frac{B_{m+1}}{m(m+1)}M_{k}(z)^{-m}\right)+\mathcal{O}(M_{k}(z)^{-R})\text{
as }z\rightarrow\infty.$

Combining Eqs. (18), (19), and (20) with Eq. (14) establishes that for all
$R\geq 0$ and $n\geq 1$,

$\displaystyle
J_{n}^{\pm}(z,z^{-\frac{5}{12}}/6)=\prod_{k=1}^{n}\exp\left(G^{\pm}_{k}(z)\mp\sum_{m=1}^{\lfloor(R-1)/k\rfloor}\frac{B_{m+1}}{m(m+1)}M_{k}(z)^{-m}\right)+\mathcal{O}(z^{-R})\text{
as }z\rightarrow\infty,$

where we used $\mathcal{O}(M_{k}(z)^{-R})=\mathcal{O}(z^{-kR})$ as
$z\rightarrow\infty$.

Due to Corollary 7.3, we know that the asymptotic behavior of
$J_{n}^{\pm}(z,z^{-\frac{5}{12}}/6)$ encodes the numbers $\chi^{\pm}_{k}$. By
matching both asymptotic expansions, we find that for all $R>n\geq 1$,

$\displaystyle\sum_{k=0}^{n}\chi^{\pm}_{k}~{}z^{-k}=\prod_{k=1}^{12n}\exp\left(G^{\pm}_{k}(z)\mp\sum_{m=1}^{\lfloor(R-1)/k\rfloor}\frac{B_{m+1}}{m(m+1)}\frac{1}{M_{k}(z)^{m}}\right)+\mathcal{O}(z^{-n-\frac{1}{12}})\text{
as }z\rightarrow\infty.$

We can identify the expression in Eq. (1) in the exponent. Due to the freedom
to choose $n$ and $R$, both sides must agree as power series when
$z,R\rightarrow\infty$. Therefore, as power series in $1/z$,

$\sum_{k=0}^{\infty}\chi^{\pm}_{k}~{}z^{-k}=\exp\left(\sum_{k=1}^{\infty}\Psi^{\pm}_{k}(z)\right)$

Taking the $\log$ on both sides and applying Proposition 3.2 results in

$\displaystyle\sum_{k=1}^{\infty}\Psi^{\pm}_{k}(z)=\log\left(\sum_{k=0}^{\infty}\chi^{\pm}_{k}~{}z^{-k}\right)=\sum_{g\geq
2}\chi(\mathcal{GC}_{\pm}^{g})\log\frac{1}{1-z^{1-g}}=\sum_{g\geq
2}\chi(\mathcal{GC}_{\pm}^{g})\sum_{m=1}^{\infty}\frac{z^{(1-g)m}}{m}.$

Hence,

$\displaystyle\sum_{\ell=1}^{\infty}\frac{\mu(\ell)}{\ell}\sum_{k=1}^{\infty}\Psi^{\pm}_{k}(z^{\ell})=\sum_{g\geq
2}\chi(\mathcal{GC}_{\pm}^{g})\sum_{\ell=1}^{\infty}\frac{\mu(\ell)}{\ell}\sum_{m=1}^{\infty}\frac{z^{(1-g)m\ell}}{m}=\sum_{g\geq
2}\chi(\mathcal{GC}_{\pm}^{g})\sum_{m=1}^{\infty}\frac{z^{(1-g)m}}{m}\sum_{\ell|m}{\mu(\ell)}.$

The statement follows from $\mu$’s defining property: $\mu(1)=1$ and
$\sum_{d|m}\mu(d)=0$ for $m\geq 2$. ∎

In the remainder of this section, we will prove the statements in Propositions
8.3 and 8.4. To do so, we define the function $A(z,y)$ with expansions
coefficients $a_{r,n}$ by

(21) $\displaystyle A(z,y)=\sum_{r,n}a_{r,n}z^{r}y^{n}=\exp\left(z\sum_{k\geq
3}\frac{y^{k}}{k!}\right)=\exp\left(z\left(e^{y}-1-y-\frac{y^{2}}{2}\right)\right).$

It follows that $a_{r,n}\neq 0$ only if $0\leq r,n$ and $3r\leq n$. Moreover,
we have $a_{r,n}\geq 0$ for all $r,n$.

######  Lemma 8.5.

If $R\geq 0$, $\varepsilon\in{\mathbb{R}}_{>0}$, $z\in{\mathbb{R}}$, and
$y\in{\mathbb{C}}$ with $|y|\leq\varepsilon\leq 1$ and $|z|\varepsilon^{3}\leq
1$, then

$\displaystyle\left|A(z,y)-\sum_{n=0}^{R-1}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}z^{r}y^{n}\right|\leq
e^{e}\left(\frac{|y|}{\varepsilon}\right)^{R}.$

###### Proof.

Because the coefficients $a_{r,n}$ are all non-negative, we find that for all
$|y|\leq\varepsilon$,

$\displaystyle\left|A(z,y)-\sum_{n=0}^{R-1}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}z^{r}y^{n}\right|/|y|^{R}\leq\varepsilon^{-R}\left(\sum_{n=R}^{\infty}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}|z|^{r}\varepsilon^{n}\right)\leq\varepsilon^{-R}A(|z|,\varepsilon).$

As $\sum_{k=3}^{\infty}x^{k}/k!\leq x^{3}\exp(x)$ for $x\geq 0$, the statement
follows from $A(|z|,\varepsilon)\leq\exp(|z|\varepsilon^{3}e^{\varepsilon})$.
∎

###### Proof of Proposition 8.3.

Recall the definitions of $Q^{\pm}$ in Eq. (17). We will only discuss the
proof for the $+$ case to avoid repetition and because the other case follows
analogously. We may write

$\displaystyle
Q^{+}(z,\varepsilon^{\prime},u)-Q^{+}(z,\varepsilon,0)=\int_{\varepsilon}^{\varepsilon^{\prime}+u}A(z,iy)e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}+\int_{-\varepsilon^{\prime}+u}^{-\varepsilon}A(z,iy)e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}},$

with $A(z,y)$ from Eq. (21).

If we choose a straight integration path for the integrals above, then
$|y|\leq\varepsilon+|\varepsilon-\varepsilon^{\prime}|+|u|$ inside the
integration domains. So, as
$\varepsilon+|\varepsilon-\varepsilon^{\prime}|+|u|\leq\frac{5}{4}\varepsilon\leq
1$, and $z(\varepsilon+|\varepsilon-\varepsilon^{\prime}|+|u|)^{3}\leq
z(\frac{5}{4}\varepsilon)^{3}=\frac{125}{64}z\varepsilon^{3}\leq 1$, we find
by applying Lemma 8.5 with $R=0$ and $\varepsilon$ substituted by
$\varepsilon+|\varepsilon-\varepsilon^{\prime}|+|u|$ that $|A(z,iy)|\leq
e^{e}$ under the integral signs. Therefore, it is sufficient to provide an
estimate for

$\left|\int_{\varepsilon}^{\varepsilon^{\prime}+u}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}+\int_{-\varepsilon^{\prime}+u}^{-\varepsilon}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}\right|$

For all points $y\in{\mathbb{C}}$ that lie on the line segment from
$\varepsilon$ to $\varepsilon+(\varepsilon^{\prime}-\varepsilon)+u$, we have

$-|y|^{2}\leq-\varepsilon^{2}+2\varepsilon(|\varepsilon^{\prime}-\varepsilon|+|u|)+(|\varepsilon^{\prime}-\varepsilon|+|u|)^{2}\leq-\varepsilon^{2}+\varepsilon^{2}/2+\varepsilon^{2}/16=-\frac{7}{16}\varepsilon^{2}.$

A similar bound holds on the line segment $-\varepsilon^{\prime}+u$ to
$-\varepsilon$. Hence, the integrals above are bound by
$|u|\exp(-\frac{7}{32}z\varepsilon^{2})/\sqrt{2\pi/z}$. The statement follows
as $|u|\leq\frac{1}{5}$. ∎

######  Lemma 8.6.

If $k\in{\mathbb{Z}}_{\geq 0}$ and $z,\varepsilon\in{\mathbb{R}}_{>0}$ with
$k\leq z\varepsilon^{2}$, then

$\displaystyle\int_{\varepsilon}^{\infty}y^{k}\exp\left({-z\frac{y^{2}}{2}}\right)\frac{\mathrm{d}y}{\sqrt{2\pi/z}}\leq\frac{1}{2}\varepsilon^{k}\exp\left({-z\frac{\varepsilon^{2}}{2}}\right).$

###### Proof.

The left-hand side is equal to

$\displaystyle\int_{0}^{\infty}(y+\varepsilon)^{k}\exp\left({-z\frac{y^{2}}{2}-z\varepsilon
y-z\frac{\varepsilon^{2}}{2}}\right)\frac{\mathrm{d}y}{\sqrt{2\pi/z}}.$

For fixed $z,\varepsilon$, and $k$, the function
$y\mapsto(y+\varepsilon)^{k}e^{-z\varepsilon y}$ has its unique maximum at
$y_{0}=\frac{k}{z\varepsilon}-\varepsilon$, and it is decreasing for $y\geq
y_{0}$. As $k\leq z\varepsilon^{2}$, $y_{0}\leq 0$ and
$(y+\varepsilon)^{k}e^{-z\varepsilon y}\leq\varepsilon^{k}$ for all $y\geq 0$.
Using that for all $a\in{\mathbb{R}}_{>0}$,
$\int_{0}^{\infty}e^{-ay^{2}}\mathrm{d}y=\frac{1}{2}\int_{-\infty}^{\infty}e^{-ay^{2}}\mathrm{d}y=\frac{1}{2}\sqrt{\pi/a}$,
gives the statement. ∎

######  Lemma 8.7.

If $\xi\in{\mathbb{R}}_{>0}$, then for all $R\in{\mathbb{Z}}_{\geq 0}$,

$\displaystyle Q^{\pm}(z,\xi\cdot z^{-\frac{5}{12}},0)$
$\displaystyle=\sum_{s=0}^{6R-1}\sum_{r=0}^{\lfloor
2s/3\rfloor}(-1)^{s}(2s-1)!!(\pm z)^{r-s}a_{r,2s}+\mathcal{O}(z^{-R})\text{ as
}z\rightarrow\infty,$

where $(2s-1)!!=(2s-1)\cdot(2s-3)\cdots 3\cdot 1$ is the _double factorial_.

###### Proof.

We will only prove the $+$ case in detail. The other case follows analogously
after trivial adjustments of signs and factors of $i$. We abbreviate
$\varepsilon(z)=\xi\cdot z^{-\frac{5}{12}}$ and assume that $z$ is large
enough such that $\varepsilon(z)\leq 1$,
$z\varepsilon(z)^{3}=\xi^{3}z^{-\frac{1}{4}}\leq 1$ and
$z\varepsilon(z)^{2}=\xi^{2}z^{\frac{1}{6}}\geq 12R$. First, we use the
definition of $A(z,y)$ in Eq. (21) to rewrite the integral expression for
$Q^{+}(z,\varepsilon(z),0)$ from Eq. (17) and immediately apply the
substitutions suggested by Lemmas 8.5 and 8.6:

$\displaystyle Q^{+}(z,\varepsilon(z),0)$
$\displaystyle=\int_{-\varepsilon(z)}^{\varepsilon(z)}A(z,iy)\frac{e^{-z\frac{y^{2}}{2}}\mathrm{d}y}{\sqrt{2\pi/z}}=\sum_{n=0}^{12R-1}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}z^{r}\int_{-\varepsilon(z)}^{\varepsilon(z)}(iy)^{n}\frac{e^{-z\frac{y^{2}}{2}}\mathrm{d}y}{\sqrt{2\pi/z}}+\mathcal{R}^{(1)}(z)$
$\displaystyle=\sum_{n=0}^{12R-1}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}z^{r}\int_{-\infty}^{\infty}(iy)^{n}\frac{e^{-z\frac{y^{2}}{2}}\mathrm{d}y}{\sqrt{2\pi/z}}+\mathcal{R}^{(1)}(z)+\mathcal{R}^{(2)}(z).$

The Gaussian integrals
$\int_{-\infty}^{\infty}(iy)^{2s}e^{-zy^{2}/2}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}=(2s-1)!!(-z)^{-s}$,
$\int_{-\infty}^{\infty}(iy)^{2s+1}e^{-zy^{2}/2}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}=0$,
imply the statement if we ensure that
$\mathcal{R}^{(1)}(z),\mathcal{R}^{(2)}(z)\in\mathcal{O}(z^{-R})$, which
remains to be proven. From Lemma 8.5, we get

$\displaystyle|\mathcal{R}^{(1)}(z)|\leq
e^{e}\varepsilon(z)^{-12R}\int_{-\varepsilon(z)}^{\varepsilon(z)}y^{12R}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}\leq
e^{e}\varepsilon(z)^{-12R}\int_{-\infty}^{\infty}y^{12R}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}$
$\displaystyle=e^{e}(12R-1)!!\varepsilon(z)^{-12R}z^{-12R}=e^{e}(12R-1)!!\xi^{12R}z^{-R}.$

By Lemma 8.6,

$\displaystyle\left|\int_{-\infty}^{\infty}(iy)^{n}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}-\int_{-\varepsilon(z)}^{\varepsilon(z)}(iy)^{n}e^{-z\frac{y^{2}}{2}}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}\right|\leq\varepsilon(z)^{n}\exp\left(-\frac{z\varepsilon(z)^{2}}{2}\right).$

So, also due to Lemma 8.5 applied with $R=0$, the second remainder term is
bound by

$\displaystyle|\mathcal{R}^{(2)}(z)|\leq
e^{-\frac{z\varepsilon(z)^{2}}{2}}\sum_{n=0}^{12R-1}\sum_{r=0}^{\lfloor
n/3\rfloor}a_{r,n}z^{r}\varepsilon(z)^{n}\leq
e^{-\frac{z\varepsilon(z)^{2}}{2}}A(z,\varepsilon(z))\leq
e^{e}e^{-\frac{1}{2}\xi^{2}z^{\frac{1}{6}}}.\qed$

######  Lemma 8.8.

For all $R\geq 0$, the asymptotic expansion involving the Bernoulli numbers
holds,

$\int_{-\infty}^{\infty}e^{-z(e^{y}-1-y)}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}=\exp\left(\sum_{m=1}^{R-1}\frac{B_{m+1}}{m(m+1)}z^{-m}\right)+\mathcal{O}(z^{-R})\text{
as }z\rightarrow\infty.$

###### Proof.

The $\Gamma$ function is defined by
$\Gamma(z)=\int_{0}^{\infty}t^{z}e^{-t}\frac{\mathrm{d}t}{t}$ for all $z>0$.
The statement follows from the _Stirling expansion_ of this function. Using
the variable substitution $t=ze^{y}$, we find

$\int_{-\infty}^{\infty}e^{-z(e^{y}-1-y)}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}=\frac{e^{z}z^{-z+\frac{1}{2}}}{\sqrt{2\pi}}\int_{0}^{\infty}t^{z}e^{-t}\frac{\mathrm{d}t}{t}=\frac{\Gamma(z)}{\sqrt{2\pi}z^{z-\frac{1}{2}}e^{-z}}.$

The Stirling expansion (see, e.g., [10, §3.10]) states that for all $R\geq 0$,

$\displaystyle\Gamma(z)=\exp\left((z-\frac{1}{2})\log
z-z+\frac{1}{2}\log(2\pi)+\sum_{m=1}^{R-1}\frac{B_{m+1}}{m(m+1)}z^{-m}\right)+\mathcal{O}(z^{-R})\text{
as }z\rightarrow\infty.\qed$

Only a special case of the following lemma is required for the proof of
Proposition 8.4, but we will need the general case later in Section 9.

######  Lemma 8.9.

Let $g(z,y,\alpha)=-z(e^{y}-1-y)+\alpha\sqrt{z}(e^{\frac{1}{2}y}-1)$. If
$\varepsilon,z\in{\mathbb{R}}_{>0}$ and $\alpha\in{\mathbb{C}}$ such that
$\varepsilon<2$, and $z\varepsilon^{2}\geq 16|\alpha|^{2}$, then

$\left|\int_{-\infty}^{\infty}e^{g(z,y,\alpha)}\mathrm{d}y-\int_{-\varepsilon}^{\varepsilon}e^{g(z,y,\alpha)}\mathrm{d}y\right|\leq\frac{1}{z}e^{-\frac{z}{2}}+2\sqrt{\frac{2\pi}{z}}\exp\left({-z\frac{\varepsilon^{2}}{8}}\right).$

###### Proof.

Observe that $z\varepsilon^{2}\geq 16|\alpha|^{2}$ and $\varepsilon<2$ imply
$\varepsilon\geq 4\frac{|\alpha|}{\sqrt{z}}$ and $\sqrt{z}\geq 2|\alpha|$. In
the range, $y\in(-\infty,-2]$, we have $-(e^{y}-1-y)\leq 1+y$ and
$e^{\frac{y}{2}}-1\leq 1$. So, there $\mathrm{Re}(g(z,y,\alpha))\leq
z(1+y)+|\alpha|\sqrt{z}$. In the range, $y\in[-2,-\varepsilon]$, it follows
from $e^{y/2}\geq 1+y/2$ that
$e^{y}\geq(1+y/2)^{2}\Rightarrow-(e^{y}-1-y)\leq-y^{2}/4$ and
$0<1-e^{y/2}\leq-\frac{y}{2}$. Hence, in that range
$\mathrm{Re}(g(z,y,\alpha))\leq-y^{2}/4+|\alpha|\sqrt{z}\frac{|y|}{2}$. For
$y\in[\varepsilon,\infty)$, we have $e^{y}-1-y\geq
e^{\frac{1}{2}y}-1-\frac{y}{2}$, $e^{y}-1-y=\sum_{k=2}^{\infty}y^{k}/k!\geq
y^{2}/2$ and because $\sqrt{z}\geq 2|\alpha|$ we further find that
$\mathrm{Re}(g(z,y,\alpha))\leq-\frac{z}{4}y^{2}+|\alpha|\sqrt{z}\frac{y}{2}$.
Using $\varepsilon\geq\frac{4|\alpha|}{\sqrt{z}}$, we get
$-z\frac{y^{2}}{4}+|\alpha|\sqrt{z}\frac{|y|}{2}\leq-z\frac{y^{2}}{8}$ for all
$|y|\geq\varepsilon$. Therefore, it follows from Lemma 8.6 that

$\displaystyle\left|\int_{-\infty}^{2}e^{g(z,y,\alpha)}\mathrm{d}y\right|$
$\displaystyle\leq\int_{-\infty}^{2}e^{z(1+y)+|\alpha|\sqrt{z}}\mathrm{d}y=\frac{1}{z}e^{-z+|\alpha|\sqrt{z}}\leq\frac{1}{z}e^{-\frac{z}{2}}$
$\displaystyle\left|\int_{-2}^{\varepsilon}e^{g(z,y,\alpha)}\mathrm{d}y\right|$
$\displaystyle\leq\int_{-2}^{\varepsilon}e^{-z\frac{y^{2}}{8}}\mathrm{d}y\leq\frac{1}{2}\sqrt{\frac{8\pi}{z}}e^{-z\frac{\varepsilon^{2}}{8}}$
$\displaystyle\left|\int_{\varepsilon}^{\infty}e^{g(z,y,\alpha)}\mathrm{d}y\right|$
$\displaystyle\leq\int_{\varepsilon}^{\infty}e^{-z\frac{y^{2}}{8}}\mathrm{d}y\leq\frac{1}{2}\sqrt{\frac{8\pi}{z}}e^{-z\frac{\varepsilon^{2}}{8}}.\qed$

###### Proof of Proposition 8.4.

For any fixed, $\varepsilon,z>0$ with $\varepsilon<2$, it follows from Eq.
(17) that

$\displaystyle
Q^{-}(z,\varepsilon,0)=\int_{-\infty}^{\infty}e^{-z(e^{y}-1-y)}\frac{\mathrm{d}y}{\sqrt{2\pi/z}}+\mathcal{R}(z,\varepsilon),$

where the remainder term is bound as follows due to Lemma 8.9 applied with
$\alpha=0$:
$|\mathcal{R}(z,\varepsilon)|\leq\frac{1}{z}e^{-\frac{z}{2}}+2\sqrt{\frac{2\pi}{z}}e^{-z\frac{\varepsilon^{2}}{8}}$.
Hence, for all $R\geq 0$, $\mathcal{R}(z,\xi\cdot
z^{-\frac{5}{12}})=\mathcal{O}(z^{-R})$ as $z\rightarrow\infty$ and by Lemma
8.8,

$Q^{-}(z,\xi\cdot
z^{-\frac{5}{12}},0)=\exp\left(\sum_{m=1}^{R-1}\frac{B_{m+1}}{m(m+1)}z^{-m}\right)+\mathcal{O}(z^{-R})\text{
as }z\rightarrow\infty.$

The statement follows as by Lemma 8.7 the asymptotic $z\rightarrow\infty$
expansions of $Q^{-}(z,\xi\cdot z^{-\frac{5}{12}},0)$ and $Q^{+}(z,\xi\cdot
z^{-\frac{5}{12}},0)$ only differ by a sign flip in $z$ and because
$B_{m+1}=0$ for even $m\geq 1$. ∎

##  9\. Asymptotic analysis of the generating functions

This section will prove the asymptotic statements in Theorems 1.1, 1.4, and
1.6. The proofs will be based on the generating function from Theorem 1.5.
Asymptotically, only the term involving the Bernoulli numbers in Eq. (1) will
contribute, and within that term, only summands with $\ell=1,k\in\\{1,2\\}$
and $m\rightarrow\infty$ turn out to be significant.

The super-exponential growth rate of the magnitude of the Euler
characteristics is due to the rapid growth of the magnitude of the Bernoulli
numbers. This growth rate can be read off from following classical formula due
to Euler, which relates the Bernoulli numbers to the even values of the
Riemann $\zeta$-function, $\zeta(n)=\sum_{k=1}^{\infty}\frac{1}{k^{n}}$:

(22) $\displaystyle B_{n}$
$\displaystyle=(-1)^{\frac{n}{2}+1}2\frac{n!}{(2\pi)^{n}}\zeta(n)\quad\text{
for all even }n\geq 2.$

For odd $n>1$, the numbers $B_{n}$ vanish.

We will fix a notation for the coefficients of the part of Eq. (1) that
involves the Bernoulli numbers. Let $\xi_{n,k}^{\pm}\in{\mathbb{Q}}$ denote
the coefficients defined via the following power series in
${\mathbb{Q}}[[\frac{1}{z}]]$,

(23)
$\displaystyle\sum_{n=1}^{\infty}\xi_{n,k}^{\pm}z^{-n}=\mp\sum_{m=1}\frac{B_{m+1}}{m(m+1)}\frac{1}{M_{k}(z)^{m}}\text{
for all }k\geq 1.$

Because $M_{k}(z)$ has $\frac{1}{k}z^{k}$ as the leading monomial,
$\xi_{n,k}^{\pm}\neq 0$ only if $n\geq k$.

We can estimate the coefficients $\xi_{n,k}^{\pm}$ uniformly over $n$ and $k$:

######  Lemma 9.1.

There is a constant $C$ such that for all $n\geq k\geq 1$,
$|\xi_{n,k}^{\pm}|\leq C4^{n}(\lfloor n/k\rfloor-1)!$.

###### Proof.

We define the expansion coefficients $a_{n}^{(k,m)}$ such that
$A^{(k,m)}(u)=\sum_{n=0}^{\infty}a_{n}^{(k,m)}u^{n}=(ku^{k}M_{k}(1/u))^{-m}$.
By Lemma 8.2, the functions $A^{(k,m)}:{\mathbb{C}}\rightarrow{\mathbb{C}}$
are holomorphic in the disc $|u|\leq\frac{1}{4}$, where they are bound by
$|A^{(k,m)}(u)|\leq 2^{m}$. Hence, by Cauchy’s inequality,
$|a_{n}^{(k,m)}|\leq 4^{n}\sup_{|u|=\frac{1}{4}}|A^{(k,m)}(u)|\leq 2^{2n+m}$.
Writing $\xi_{n,k}^{\pm}$ in terms of the $a_{n}^{(k,m)}$ gives

$\xi_{n,k}^{\pm}=\mp\sum_{m=1}^{\lfloor\frac{n}{k}\rfloor}\frac{B_{m+1}}{m(m+1)}k^{m}a^{(k,m)}_{n-km}.$

As $\zeta(n)$ is decreasing with $n$, it follows immediately from Eq. (22)
that
$\left|\frac{B_{n+1}}{n(n+1)}\right|\leq\frac{2\zeta(2)}{(2\pi)^{n+1}}(n-1)!$
for all $n\geq 1$. After using this fact and the bound on $a_{n}^{(k,m)}$, we
get for all $n\geq 0$ and $k\geq 1$,

$|\xi_{n,k}^{\pm}|\leq
4^{n}\frac{\zeta(2)}{\pi}\sum_{m=1}^{\lfloor\frac{n}{k}\rfloor}\left(\frac{k}{2^{2k}\pi}\right)^{m}(m-1)!\leq
4^{n}\left(\left\lfloor\frac{n}{k}\right\rfloor-1\right)!\frac{\zeta(2)}{\pi}\frac{1}{\pi-1},$

where we used $k\leq 2^{2k}$ for all $k\geq 1.$ ∎

Thanks to this estimate and our knowledge of the coefficients of the non-
Bernoulli terms of Eq. (1) from Lemma 8.2, we can prove that the dominant
asymptotic behavior of the numbers $\chi(\mathcal{GC}_{\pm}^{g})$ comes from
only three terms in the double sum over $\ell$ and $k$ in Theorem 1.5:

######  Proposition 9.2.

$\chi(\mathcal{GC}_{\pm}^{n+1})=\xi^{\pm}_{n,1}+\xi^{\pm}_{n,2}-\frac{1}{2}\delta_{2|n}\xi^{\pm}_{\frac{n}{2},1}+\mathcal{O}(5^{n}(\lfloor
n/3\rfloor-1)!)$.

###### Proof.

Let $h^{\pm}_{n}$ denote the expansion coefficients of
$G^{\pm}(z)=\sum_{k,\ell\geq
1}\frac{\mu(\ell)}{\ell}G^{\pm}_{k}(z^{\ell})=\sum_{n=1}^{\infty}h^{\pm}_{n}z^{-n}.$
By Lemma 8.2, $G_{k}^{\pm}(1/u)$ is holomorphic in $u$ and bound by
$|G_{k}^{\pm}(1/u)|\leq\frac{14534}{343}|u|^{\lceil\frac{k}{6}\rceil}$ in a
disc of size $\frac{1}{8}$ around the origin. Due to the bound on
$|G_{k}^{\pm}(1/u)|$ the double sum, $G^{\pm}(1/u)=\sum_{k,\ell\geq
1}\frac{\mu(\ell)}{\ell}G^{\pm}_{k}(u^{-\ell})$ converges uniformly for all
$|u|\leq\frac{1}{8}$. So, $G^{\pm}(1/u)$ is also holomorphic in this domain.
Hence, there is a constant $C$ such that the upper bound $|h^{\pm}_{n}|\leq
C8^{n}$ holds for all $n\geq 1$.

From Theorem 1.5, the definitions in Eqs. (1), (16), and (23), we get by
extracting coefficients

$\chi(\mathcal{GC}_{\pm}^{n+1})=h^{\pm}_{n}+\sum_{\ell|n}\frac{\mu(\ell)}{\ell}\sum_{k=1}^{n/\ell}\xi^{\pm}_{\frac{n}{\ell},k},\quad\text{for
all }n\geq 1.$

Due to Lemma 9.1, there is a constant $C^{\prime}$ such that the coefficients
$|\xi^{\pm}_{\frac{n}{\ell},k}|\leq
C^{\prime}4^{n}(\lfloor\frac{n}{3}\rfloor-1)!$ if $n\geq k\cdot\ell>2$. The
sum above has at most $n^{2}$ terms, $\mu(1)=1$, $\mu(2)=-1$, and
$|\mu(\ell)|\leq 1$, so

$\displaystyle\left|\chi(\mathcal{GC}_{\pm}^{n+1})-\xi^{\pm}_{n,1}-\xi^{\pm}_{n,2}+\frac{1}{2}\delta_{2|n}\xi^{\pm}_{\frac{n}{2},1}\right|\leq
C8^{n}+C^{\prime}4^{n}n^{2}\left(\left\lfloor\frac{n}{3}\right\rfloor-1\right)!\in\mathcal{O}(5^{n}(\lfloor
n/3\rfloor-1)!).\qed$

The following two propositions compute the explicit asymptotic behavior of the
coefficients $\xi^{\pm}_{n,1}$ for all $n$ and the one of $\xi^{\pm}_{n,2}$
for all even $n$. For odd $n$, the large-$n$ asymptotic growth rate of
$\xi^{\pm}_{n,1}$ overshadows one of $\xi^{\pm}_{n,2}$ and
$\xi^{\pm}_{n/2,1}$. For even $n$, $\xi^{\pm}_{n,1}$ vanishes and the
$\xi^{\pm}_{n,2}$ term dominates:

######  Proposition 9.3.

For all $n\geq 0$, $\xi^{\pm}_{2n,1}=0$ and
$\xi^{\pm}_{2n-1,1}\sim\pm(-1)^{n}\sqrt{\frac{\pi}{n^{3}}}\left(\frac{n}{e\pi}\right)^{2n}$
as $n\rightarrow\infty$.

###### Proof.

Eq. (23), $M_{1}(z)=z$, and the vanishing of $B_{m}$ for all odd $m$ imply
$\xi^{\pm}_{2n,1}=0$ for all $n\geq 1$. We get from Eq. (22) that
$\xi^{\pm}_{2n-1,1}=\pm 2(-1)^{n}\frac{(2n-2)!}{(2\pi)^{2n}}\zeta(2n).$ The
statement follows from Stirling’s expansion $n!=\sqrt{2\pi
n}n^{n}e^{-n}(1+\mathcal{O}(1/n))$ and $\zeta(n)=1+\mathcal{O}({1}/{2^{n}})$
as $n\rightarrow\infty$. ∎

######  Proposition 9.4.

$\xi^{\pm}_{2n,2}\sim\pm\frac{1}{\sqrt{2n\pi}}\left(\frac{n}{e\pi}\right)^{n}e^{\sqrt{\frac{\pi
n}{2}}}\cos\left(\sqrt{\frac{\pi
n}{2}}-\pi\frac{n+\frac{3}{4}}{2}\right)\text{ as }n\rightarrow\infty.$

The proof of this proposition relies on another integral expansion. We
postpone this proof and first prove Theorems 1.1 and Theorem 1.4.

###### Proof of Theorems 1.1, 1.2, and 1.4.

By Proposition 9.3,
$\xi^{\pm}_{n,1}\in\mathcal{O}\left(\left({n}/{(2e\pi)}\right)^{n-\frac{1}{2}}\right),$
and similarly
$\xi^{\pm}_{\frac{n}{2},1}\in\mathcal{O}\left(\left({n}/{(4e\pi)}\right)^{\frac{n-1}{2}}\right)$.
From Proposition 9.4, we get
$\xi^{\pm}_{n,2}\in\mathcal{O}\left(e^{\sqrt{\frac{\pi
n}{4}}}\left({n}/{(2e\pi)}\right)^{\frac{n-1}{2}}\right)$.

So, if $g$ is even and large, then by Proposition 9.2,
$\chi(\mathcal{GC}^{g}_{\pm})\sim\xi_{g-1}^{\pm}$, as
$\xi^{\pm}_{\frac{n}{2},1}$ and $\xi^{\pm}_{n,2}$ are negligible in comparison
to $\xi^{\pm}_{n,1}$ which is non-zero if $n=g-1$ is odd. If $g$ is odd and
large then, $\xi^{\pm}_{g-1,1}$ vanishes, and the asymptotic contribution from
$\xi^{\pm}_{n,2}$ dominates. The precise formula for the large $g$ behavior in
Theorems 1.1 and 1.4 follows from Proposition 9.3 and
$\lim_{g\rightarrow\infty}(1-1/g)^{g/2}=e^{-1/2}$. Theorem 1.2 and the
estimate in Theorem 1.4 follow then from Proposition 2.1. ∎

Proposition 9.4, which remains to be proven, requires two technical lemmas.
These contain the last key step to the proof of the asymptotic formula of
Theorem 1.1, which is to rephrase a heavily oscillating sum as an integral
that can be evaluated in the desired limit. We start by proving an asymptotic
formula for the integral expression that already appeared in Lemma 8.9:

######  Lemma 9.5.

Let $g(z,y,\alpha)=-z(e^{y}-1-y)+\alpha\sqrt{z}(e^{\frac{1}{2}y}-1).$ For all
$\alpha\in{\mathbb{C}}$,

$\int_{-\infty}^{\infty}e^{g(z,y,\alpha)}\mathrm{d}y=\sqrt{\frac{2\pi}{z}}e^{\frac{1}{8}\alpha^{2}}+\mathcal{O}(z^{-\frac{3}{4}})\text{
as }z\rightarrow\infty.$

###### Proof.

From Lemma 8.9, we get

$\int_{-\infty}^{\infty}e^{g(z,y,\alpha)}\mathrm{d}y=\int_{-z^{-\frac{5}{12}}}^{z^{-\frac{5}{12}}}e^{g(z,y,\alpha)}\mathrm{d}y+\mathcal{O}(z^{-\frac{1}{2}}\exp(-z^{\frac{1}{6}}))\text{
as }z\rightarrow\infty.$

For $y\in[-z^{-\frac{5}{12}},z^{-\frac{5}{12}}]$ and large $z$, we can
approximate the exponent $g(z,y,\alpha)$ by a shifted quadric, i.e.,
$g(z,y,\alpha)=-z\frac{y^{2}}{2}+\frac{1}{2}\alpha\sqrt{z}y+\mathcal{R}^{\prime}(z,y,\alpha)$,
where due to $\sum_{n\geq k}\frac{|y|^{n}}{n!}\leq\frac{1}{k!}|y|^{k}e^{|y|}$,

$|\mathcal{R}^{\prime}(z,y,\alpha)|\leq\frac{z|y|^{3}}{3!}e^{|y|}+|\alpha|\sqrt{z}\frac{(|y|/2)^{2}}{2!}e^{|y|/2}\leq
e\frac{z^{-\frac{1}{4}}}{6}+\sqrt{e}|\alpha|\frac{z^{-\frac{1}{3}}}{8}\text{
if }|y|\leq z^{-\frac{5}{12}}\leq 1.$

Using also
$|\exp(\mathcal{R}^{\prime}(z,y,\alpha))-1|\leq\frac{|\mathcal{R}^{\prime}(z,y,\alpha)|}{1-|\mathcal{R}^{\prime}(z,y,\alpha)|}\in\mathcal{O}(z^{-\frac{1}{4}})$
as $z\rightarrow\infty$ results in

$\int_{-\infty}^{\infty}e^{g(z,y,\alpha)}\mathrm{d}y=\int_{-z^{-\frac{5}{12}}}^{z^{-\frac{5}{12}}}e^{-z\frac{y^{2}}{2}+\frac{1}{2}\alpha\sqrt{z}y}\mathrm{d}y(1+\mathcal{O}(z^{-\frac{1}{4}})).$

If $z$ is large enough such that $z^{\frac{1}{12}}\geq 2|\alpha|$ and hence
$\frac{z\cdot y^{2}}{4}\geq\frac{1}{2}|\alpha|\sqrt{z}|y|$ for all $|y|\geq
z^{-\frac{5}{12}}$, then

$\int_{z^{-\frac{5}{12}}}^{\infty}e^{-z\frac{y^{2}}{2}+\frac{1}{2}\alpha\sqrt{z}|y|}\mathrm{d}y\leq\int_{z^{-\frac{5}{12}}}^{\infty}e^{-z\frac{y^{2}}{4}+\frac{1}{2}|\alpha|\sqrt{z}|y|-z\frac{y^{2}}{4}}\mathrm{d}y\leq\int_{z^{-\frac{5}{12}}}^{\infty}e^{-z\frac{y^{2}}{4}}\mathrm{d}y\in\mathcal{O}\left(z^{-\frac{1}{2}}e^{-z^{\frac{1}{6}}/4}\right),$

by Lemma 8.6. Hence, we may complete the Gaussian integral, evaluate it, and
obtain,

$\displaystyle\int_{-\infty}^{\infty}e^{-z\frac{y^{2}}{2}+\frac{1}{2}\alpha\sqrt{z}y}\mathrm{d}y=\int_{-\infty}^{\infty}e^{-\frac{z}{2}\left(y-\frac{\alpha}{2\sqrt{z}}\right)^{2}+\frac{1}{8}\alpha^{2}}\mathrm{d}y=\sqrt{\frac{2\pi}{z}}e^{\frac{1}{8}\alpha^{2}}.\quad\qed$

We will use this integral expression to evaluate the following sum, which
relates closely to $\xi^{\pm}_{n,2}$:

######  Lemma 9.6.

For all $A\in{\mathbb{C}}$ that do not fall on the negative real axis,

$\displaystyle\sum_{\ell=0}^{n-1}A^{\ell}\frac{(n+\ell-1)!}{(2\ell)!}=\sqrt{\frac{\pi}{2n}}n^{n}e^{-n+\sqrt{An}+\frac{A}{8}}(1+\mathcal{O}(n^{-\frac{1}{4}})),$

where we take the square root of $A$ with a positive real part.

###### Proof.

We denote the left-hand side by $f_{n}(A)$. Using
$(n-1)!=\int_{0}^{\infty}t^{n}e^{-t}\frac{\mathrm{d}t}{t}$ gives

$f_{n}(A)=\sum_{\ell=0}^{\infty}A^{\ell}\frac{(n+\ell-1)!}{(2\ell)!}+\mathcal{R}_{n}(A)=\sum_{\ell=0}^{\infty}\frac{A^{\ell}}{(2\ell)!}\int_{0}^{\infty}t^{n+\ell}e^{-t}\frac{\mathrm{d}t}{t}+\mathcal{R}_{n}(A).$

Because $(2n+2\ell)!\geq(2n+\ell)!\ell!$, we can see that the remainder term
is small,

$\displaystyle|\mathcal{R}_{n}(A)|\leq\sum_{\ell=n}^{\infty}|A|^{\ell}\frac{(n+\ell-1)!}{(2\ell)!}=\sum_{\ell=0}^{\infty}|A|^{\ell}\frac{(2n+\ell-1)!}{(2n+2\ell)!}\leq\sum_{\ell=0}^{\infty}|A|^{\ell}\frac{(2n+\ell-1)!}{(2n+\ell)!}\frac{1}{\ell!}\leq
e^{|A|}.$

We use the identity
$\frac{1}{2}(e^{\sqrt{x}}+e^{-\sqrt{x}})=\frac{1}{2}\sum_{\ell\geq
0}(1-(-1)^{\ell})\frac{\sqrt{x}^{\ell}}{\ell!}=\sum_{\ell\geq
0}\frac{x^{\ell}}{(2\ell)!}$ to perform the sum in our expression for
$f_{n}(A)$ and get

$\displaystyle
f_{n}(A)=\int_{0}^{\infty}t^{n}\frac{1}{2}\left(e^{\sqrt{At}}+e^{-\sqrt{At}}\right)e^{-t}\frac{\mathrm{d}t}{t}+\mathcal{R}_{n}(A).$

The integral above is absolutely convergent for all positive values of $n$ and
all $A\in{\mathbb{C}}$. So, exchanging sum and integral was justified. We will
find the large-$n$ asymptotic expansion of each summand above. With the change
of variables $t=ne^{y}$, the integrals in the sum become

$\int_{0}^{\infty}t^{n}e^{\pm\sqrt{At}-t}\frac{\mathrm{d}t}{t}=n^{n}e^{-n\pm\sqrt{An}}\int_{-\infty}^{\infty}e^{g(n,y,\pm\sqrt{A})}\mathrm{d}y=\sqrt{\frac{2\pi}{n}}n^{n}e^{-n\pm\sqrt{An}+\frac{A}{8}}(1+\mathcal{O}(n^{-\frac{1}{4}})),$

where $g(n,y,\pm\sqrt{A})=-n(e^{y}-1-y)\pm\sqrt{An}(e^{\frac{1}{2}y}-1)$ and
we used Lemma 9.5 in the last step. As we assume that $\sqrt{A}$ is not purely
imaginary, only the square root of $A$ with a positive real part significantly
contributes in the limit $n\rightarrow\infty$. ∎

###### Proof of Proposition 9.4.

Using $1/(1-x)^{m}=\sum_{\ell\geq 0}\binom{m+\ell-1}{\ell}x^{\ell}$ gives

$\displaystyle\mp\sum_{m=1}^{\infty}\frac{B_{m+1}}{m(m+1)}\frac{2^{m}}{(z^{2}-z)^{m}}$
$\displaystyle=\mp\sum_{n\geq 2}z^{-n}\sum_{\begin{subarray}{c}m\geq
1,\ell\geq 0\\\
2m+\ell=n\end{subarray}}\frac{B_{m+1}}{m(m+1)}2^{m}\binom{m+\ell-1}{\ell}.$

If $n$ is even, the last sum has only support for even $\ell$. Hence, by Eq.
(23) and $M_{2}(z)=\frac{1}{2}(z^{2}-z)$,

$\xi^{\pm}_{2n,2}=\mp\sum_{\begin{subarray}{c}m\geq 1,\ell\geq 0\\\
m+\ell=n\end{subarray}}\frac{B_{m+1}}{m(m+1)}2^{m}\binom{m+2\ell-1}{2\ell}.$

Observe that $\mathrm{Re}\left(i^{m+1}\right)=0$ for even $m$ and
$\mathrm{Re}\left(i^{m+1}\right)=(-1)^{(m+1)/2}$ for odd $m$. By Eq. (22),

$\frac{B_{m+1}}{m(m+1)}=-2\mathrm{Re}\left(i^{m+1}\right)\frac{(m-1)!}{(2\pi)^{m+1}}\zeta(m+1)\text{
for all }m\geq 1.$

We hence get the following expression for $\xi^{\pm}_{2n,2}$ by also using
$\binom{a}{b}={a!/(b!(a-b)!)}$:

$\displaystyle\xi^{\pm}_{2n,2}$
$\displaystyle=\pm\frac{1}{\pi}\sum_{\ell=0}^{n-1}\mathrm{Re}\left(i^{n-\ell+1}\right)\frac{(n+\ell-1)!}{\pi^{n-\ell}(2\ell)!}\zeta(n-\ell+1).$

Recall that $\zeta(m)=\sum_{k\geq 1}\frac{1}{k^{m}}$ and therefore for all
$m\geq 2$,

$|\zeta(m)-1|=2^{-m}\sum_{k\geq
1}\frac{2^{m}}{(k+1)^{m}}=2^{-m}\left(\sum_{k\geq
1}\frac{1}{k^{m}}+\sum_{k\geq 1}\frac{1}{(k+\frac{1}{2})^{m}}\right)\leq
2^{-m}2\zeta(m)\leq 2^{1-m}\zeta(2).$

Thus, with the notation
$f_{n}(A)=\sum_{\ell=0}^{n-1}A^{\ell}\frac{(n+\ell-1)!}{(2\ell)!},$

$\displaystyle\xi^{\pm}_{2n,2}$
$\displaystyle=\pm\mathrm{Re}\left(\frac{i^{n+1}}{\pi^{n+1}}f_{n}\left(-i\pi\right)\right)+\mathcal{R}_{n},$

where
$|\mathcal{R}_{n}|\leq\frac{2\zeta(2)}{(2\pi)^{n+1}}f_{n}\left(2\pi\right)$.
The statement follows from Lemma 9.6 and $\mathrm{Re}(e^{ix})=\cos(x)$. ∎

###  9.1. Extension to the formula of Getzler and Kapranov

Let $\phi(n)=n\sum_{d|n}\frac{\mu(d)}{d}$ denote the _Euler totient function_.
For any integer $k\geq 1$, we write
$\widetilde{M}_{k}(z)=\frac{1}{k}\sum_{d|k}\phi(d)z^{k/d}$,
$\widetilde{L}_{k}(z)=\log(k\widetilde{M}_{k}(z)/z^{k})$. Getzler and Kapranov
define the family of power series

$\displaystyle\widetilde{\Psi}_{k}^{\pm}(z)=-\left((1-\widetilde{L}_{k}(z))\widetilde{M}_{k}(z)-\frac{z^{k}}{k}+\frac{\delta_{2|k}}{2k}\right)-\frac{1}{2}\widetilde{L}_{k}(z)+\sum_{m=1}^{\infty}\frac{B_{m+1}}{m(m+1)}\frac{1}{\widetilde{M}_{k}(z)^{m}},$

and prove a generating function of the Euler characteristic of the associative
graph complex $\mathcal{AGC}$:

######  Theorem 9.7 (Theorem 9.18 of [12]).

For all $k\geq 1$, $\widetilde{\Psi}_{k}^{\pm}(z)\in z^{-\lceil
k/6\rceil}{\mathbb{Q}}[[\frac{1}{z}]]$ and

$\sum_{g\geq 2}\chi(\mathcal{AGC}^{g})z^{1-g}=\sum_{k,\ell\geq
1}\frac{\mu(\ell)}{\ell}\widetilde{\Psi}^{\pm}_{k}(z^{\ell}).$

###### Proof of Theorem 1.6.

Based on Theorem 9.7 instead of Theorem 1.5, the proof works completely
analogously to the proof of Theorems 1.1 and 1.4. The only significant
difference is that $\phi(1)=\phi(2)=1$ instead of $\mu(1)=1$ and $\mu(2)=-1$.
This modification only leads to a sign change that is not significant for the
leading asymptotic term. Specifically, in the proof of Proposition 9.4, the
quantity $1/(1+x)$ instead of $1/(1-x)$ needs to be expanded. Eventually, only
the even coefficients of these expansions are relevant in both cases. ∎

## References

  * [1] Dror Bar-Natan.  On the Vassiliev knot invariants.  Topology, 34(2):423–472, 1995.
  * [2] François Bergeron, Gilbert Labelle, and Pierre Leroux.  Combinatorial species and tree-like structures.  Cambridge University Press, 1998.
  * [3] Michael Borinsky, Benjamin Brück, and Thomas Willwacher.  Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group.  preprint arXiv:2308.16845, 2023.
  * [4] Michael Borinsky and Karen Vogtmann.  The Euler characteristic of the moduli space of graphs.  Adv. Math., 432:Paper No. 109290, 58, 2023.
  * [5] Francis Brown.  Mixed Tate motives over $\mathbb{Z}$.  Ann. of Math. (2), 175(2):949–976, 2012.
  * [6] Francis Brown.  Invariant differential forms on complexes of graphs and Feynman integrals.  SIGMA, 17:Paper No. 103, 54, 2021.
  * [7] Melody Chan, Carel Faber, Søren Galatius, and Sam Payne.  The $\operatorname{{\mathbb{S}}}_{n}$-equivariant top weight Euler characteristic of $\mathcal{M}_{g,n}$.  Amer. J. Math., 145(5):1549–1585, 2023.
  * [8] Melody Chan, Søren Galatius, and Sam Payne.  Tropical curves, graph complexes, and top weight cohomology of $\mathcal{M}_{g}$.  J. Amer. Math. Soc., 34(2):565–594, 2021.
  * [9] James Conant and Karen Vogtmann.  On a theorem of Kontsevich.  Algebr. Geom. Topol., 3:1167–1224, 2003.
  * [10] Nicolaas Govert De Bruijn.  Asymptotic methods in analysis.  Dover Publications, 1981.
  * [11] V. A. Dolgushev, C. L. Rogers, and T. H. Willwacher.  Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields.  Ann. of Math. (2), 182(3):855–943, 2015.
  * [12] Ezra Getzler and Mikhail M. Kapranov.  Modular operads.  Compos. Math., 110(1):65–125, 1998.
  * [13] Louis Hainaut and Dan Petersen.  Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies.  preprint arXiv:2305.03046, 2023.
  * [14] John Harer and Don Zagier.  The Euler characteristic of the moduli space of curves.  Invent. Math., 85(3):457–485, 1986.
  * [15] Anton Khoroshkin, Thomas Willwacher, and Marko Živković.  Differentials on graph complexes.  Adv. Math., 307:1184–1214, 2017.
  * [16] Maxim Kontsevich.  Formal (non)commutative symplectic geometry.  In The Gelfand Mathematical Seminars, 1990–1992, pages 173–187. Birkhäuser Boston, Boston, MA, 1993.
  * [17] Maxim Kontsevich.  Formality conjecture.  In Deformation theory and symplectic geometry (Ascona, 1996), volume 20 of Math. Phys. Stud., pages 139–156. Kluwer Acad. Publ., Dordrecht, 1997.
  * [18] K. Mahler.  On the approximation of $\pi$.  Indag. Math., 15:30–42, 1953.
  * [19] Paul Arnaud Songhafouo Tsopméné and Victor Turchin.  Euler characteristics for spaces of string links and the modular envelope of $\mathcal{L}_{\infty}$.  Homology Homotopy Appl., 20(2):115–144, 2018.
  * [20] Richard P. Stanley.  Enumerative Combinatorics, volume 2.  Cambridge University Press, 1997.
  * [21] Pierre Vogel.  Algebraic structures on modules of diagrams.  J. Pure Appl. Algebra, 215(6):1292–1339, 2011.
  * [22] Thomas Willwacher.  M. Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra.  Invent. Math., 200(3):671–760, 2015.
  * [23] Thomas Willwacher and Marko Živković.  Multiple edges in M. Kontsevich’s graph complexes and computations of the dimensions and Euler characteristics.  Adv. Math., 272:553–578, 2015.
  * [24] Don Zagier.  unpublished notes.  2008\.
  * [25] Doron Zeilberger and Wadim Zudilin.  The irrationality measure of $\pi$ is at most 7.103205334137….  Mosc. J. Comb. Number Theory, 9(4):407–419, 2020.