# Quantum Scheduling for Millimeter-Wave Observation Satellite Constellation

Joongheon Kim  Korea University
Seoul, Korea  Yunseok Kwak  Korea University
Seoul, Korea  Soyi Jung  Korea University
Seoul, Korea  Jae-Hyun Kim  Ajou University
Suwon, Korea

###### Abstract

In beyond 5G and 6G network scenarios, the use of satellites has been actively
discussed for extending target monitoring areas, even for extreme
circumstances, where the monitoring functionalities can be realized due to the
usage of millimeter-wave wireless links. This paper designs an efficient
scheduling algorithm which minimizes overlapping monitoring areas among
observation satellite constellation. In order to achieve this objective, a
quantum optimization based algorithm is used because the overlapping can be
mathematically modelled via a max-weight independent set (MWIS) problem which
is one of well-known NP-hard problems.

###### Index Terms:

Satellite Constellation, Quantum Optimization, Scheduling, Maximum Weight
Independent Set (MWIS)

##  I Introduction

The use of satellite constellation is widely and actively used in next
generation wireless network system design and implementation [1, 2, 3, 4].
Especially, low earth orbit (LEO) satellites are getting a lot of attentions
for various 6G applications such as target area observations [5] and
flexible/robust network coverage extensions [2]. Both of them require high-
capacity satellite communications.

In order to realize the high-capacity satellite communications, (mmWave)
frequencies are used in order to take care of the huge traffic demands and the
service continuity requirements of next-generation 6G applications [1, 6, 7,
8, 9, 10]. Thus, large-scale surveillance and target area observation can be
realized [11]. In the observation satellite systems, having server
duplicated/overlapped monitoring areas among satellites is not efficient even
though millimeter-wave high-capacity communications can be realized. Thus,
scheduling algorithms in order to minimize the overlapping monitoring areas
have been actively studied and proposed, e.g., [12].

The modeling of the overlapping area scheduling for observation satellite
constellation can be realized with maximum weight independent set (MWIS)
formulation [11, 13, 14], which is one of the well-known NP-hard problems
[13]. In order to approximately solve the NP-hard problems, many algorithms
have been investigated. Among them, the use of message-passing algorithms is
one of the well-known solutions [14, 13], whereas this paper proposes a new
novel algorithm that utilizes quantum optimization and approximation
methodologies [15, 16].

Based on the advances in quantum optimization methodologies, many algorithms
have been investigated for approximating combinatorial problems (e.g., MWIS
[17], max-flow/min-cut [18] and graph cut segmentation [19]) and deep learning
training/inference problems (e.g., Quantum Convolutional Neural Network (QCNN)
[20], Quantum Random Access Memory (QRAM) [21], and Quantum Graph Recurrent
Neural Network (QGRNN) [22]). In this paper, we design a quantum-based
approximation algorithm for MWIS-based overlapping monitoring area scheduling
in observation satellite constellation.

The rest of this paper is organized as follows. Sec. II presents the
formulation of MWIS scheduling in satellite observation modeling. Sec. III and
Sec. IV describe the preliminaries of QAOA and QAOA-based MWIS scheduling for
observation satellite constellation. Sec. V concludes this paper and presents
future research directions.

##  II Maximum Weight Independent Set (MWIS) Formulation for Satellite
Observation Scheduling

We consider a network which consists of a set of observation areas [14].
According to the high data transmission rate in millimeter-wave wireless links
among satellites, the transmission queue backlog in satellites can be filled
in an instant, with the observation image data via synthetic aperture radar
(SAR) [5]. For the scheduling of observation satellite constellation, a
conflict graph is constructed with the set of nodes (physically, observation
areas) and edges where two nodes are connected by an edge if the corresponding
observation areas are overlapped more than threshold among adjacent
observation satellites. The edges between node $s_{i}$ (observation area in
satellite $i$) and node $s_{j}$ (observation area in satellite $j$) of the
conflict graph, i.e., $E_{(i,j)}$, can be modelled as,

$E_{(i,j)}=\left\\{\begin{array}[]{ll}1,&\text{if $s_{i}$ is overlapped with
$s_{j}$ where }\\\ &\text{$s_{i}\in S$, $s_{j}\in S$, and $i\neq j$},\\\
0,&\text{otherwise},\end{array}\right.$ (1)

where $S$ stands for the set of nodes (i.e., observation areas of satellites).

For scheduling problems, the main objective is to find the set of nodes (i.e.,
observation areas in satellite constellation) where two adjacent nodes those
are connected via an edge cannot be simultaneously selected because it is not
allowed to have huge overlapping monitoring areas among observation
satellites. This situation is obviously equivalent to the case which maximizes
the summation of weights of all independent sets in a given conflict graph.
Note that the weight is defined as the degree of overlapping or the number of
observation data in satellite constellation. Thus, it is obvious that this
scheduling problem can be modelled with the form of MWIS as [14],

$\displaystyle\max:$ $\displaystyle\sum_{\forall s_{k}\in
S}w_{k}\cdot\mathcal{I}_{k},$ (2) s.t.
$\displaystyle\mathcal{I}_{i}+\mathcal{I}_{j}+E_{(i,j)}\leq 2,\forall s_{i}\in
S,\forall s_{j}\in S,$ (4) $\displaystyle\mathcal{I}_{i}\in\\{0,1\\},\forall
s_{i}\in S,$

where $w_{k}$ stands for the weight of satellite $k$ (a positive integer), and

$\mathcal{I}_{i}=\left\\{\begin{array}[]{ll}1,&\text{if $s_{i}$ is scheduled
where $s_{i}\in S$},\\\ 0,&\text{otherwise},\end{array}\right.$ (5)

where this formulation aims at the case where conflicting links are not
scheduled simultaneously. $\mathcal{I}_{i}+\mathcal{I}_{j}\leq 2$ when
$E_{(i,j)}=0$ (no edge between $s_{i}$ and $s_{j}$), i.e., both of
$\mathcal{I}_{i}$ and $\mathcal{I}_{j}$ can be $1$. On the other hand,
$\mathcal{I}_{i}+\mathcal{I}_{j}\leq 1$ when $E_{(i,j)}=1$, i.e., both of
$\mathcal{I}_{i}$ and $\mathcal{I}_{j}$ can not be $1$. Thus, one of them will
be selected or both of them will not be selected.

##  III Preliminaries of Quantum Optimization

This section presents the preliminaries of quantum optimization, i.e.,bra-ket
notation (refer to Sec. III-A), quantum gates (refer to Sec. III-B), and
quantum approximate optimization algorithm (QAOA) (refer to Sec. III-C).

###  III-A Bra–Ket Notation

In quantum computing research, a bra–ket notation is widely and generally used
for mathematically presenting quantum states or qubit states [17]. The ket and
bra in this bra-ket notation can be represented as column vectors and row
vectors. As a result, single qubit states (i.e., $\ket{0}$ and $\ket{1}$), can
be mathematically presented as,

$\displaystyle\ket{0}$ $\displaystyle=$ $\displaystyle\begin{bmatrix}1\\\ 0\\\
\end{bmatrix},$ (6) $\displaystyle\ket{1}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0\\\ 1\\\ \end{bmatrix},$ (7)

and therefore,

$\displaystyle\ket{0}$ $\displaystyle=$
$\displaystyle{\bra{0}}^{\dagger}=\begin{bmatrix}1&0\end{bmatrix}^{\dagger},$
(8) $\displaystyle\ket{1}$ $\displaystyle=$
$\displaystyle{\bra{1}}^{\dagger}\begin{bmatrix}0&1\end{bmatrix}^{\dagger}.$
(9)

where $\dagger$ stands for Hermitian transpose. Therefore, the superposition
state of a single qubit state can be presented as,

$c_{1}\ket{0}+c_{2}\ket{1}=\begin{bmatrix}c_{1}\\\ c_{2}\\\ \end{bmatrix},$
(10)

where $c_{1}$ and $c_{2}$ are probability amplitudes, and note that the
$c_{1}$ and $c_{2}$ are complex numbers [23].

###  III-B Quantum Gates

This section presents quantum gates or operators which mathematically
represent single-qubit or $2$-qubit operations [23]. First, Hadamard gate $H$,
Pauli-$X$ gate $X$, Pauli-$Y$ gate $Y$, and Pauli-$Z$ gate $Z$ can be
formulated as,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\\\
\end{bmatrix},\text{ }$ (11) $\displaystyle X$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&1\\\ 1&0\\\ \end{bmatrix},\text{ }$ (12)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle\begin{bmatrix}0&-i\\\ i&0\\\
\end{bmatrix},\text{ and }$ (13) $\displaystyle Z$ $\displaystyle=$
$\displaystyle\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}.$ (14)

Based on this, the rotation-$X$ gate $R_{X}(\theta)$, the rotation-$Y$ gate
$R_{Y}(\theta)$, and the rotation-$Z$ gate $R_{Z}(\theta)$ are as,

$\displaystyle R_{X}(\theta)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\cos{\frac{\theta}{2}}&-i\sin{\frac{\theta}{2}}\\\
-i\sin{\frac{\theta}{2}}&\cos{\frac{\theta}{2}}\\\ \end{bmatrix},$ (15)
$\displaystyle R_{Y}(\theta)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\cos{\frac{\theta}{2}}&-\sin{\frac{\theta}{2}}\\\
\sin{\frac{\theta}{2}}&\cos{\frac{\theta}{2}}\\\ \end{bmatrix},\text{ and }$
(16) $\displaystyle R_{Z}(\theta)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}e^{-i\frac{\theta}{2}}&0\\\
0&e^{i\frac{\theta}{2}}\\\ \end{bmatrix},$ (17)

where $\theta$ is an angular value.

$\displaystyle f(y)$ $\displaystyle\triangleq$ $\displaystyle
f(y_{1},y_{2},...,y_{n}),$ (18) $\displaystyle H_{P}\ket{y}$
$\displaystyle\triangleq$ $\displaystyle f(y)\ket{y},$ (19) $\displaystyle
H_{M}$ $\displaystyle\triangleq$ $\displaystyle\sum_{k=1}^{n}X_{k},$ (20)
$\displaystyle\ket{\gamma,\beta}$ $\displaystyle\triangleq$
$\displaystyle{e^{-i\beta_{p}H_{M}}e^{-i\gamma_{p}H_{P}}\cdots
e^{-i\beta_{2}H_{M}}e^{-i\gamma_{2}H_{P}}e^{-i\beta_{1}H_{M}}e^{-i\gamma_{1}H_{P}}\ket{s},}$
(21)

###  III-C Quantum Approximate Optimization Algorithm (QAOA)

QAOA is one of the widely known noisy intermediate-scale quantum (NISQ)
optimization algorithms for approximating combinatorial problems [16, 24, 25].
This QAOA is used for formulating $H_{P}$ (i.e., problem Hamiltonian) and
$H_{M}$ (i.e., mixing Hamiltonian) based on the objective function $f(y)$.
Then, the QAOS generates the parameterized states $\ket{\gamma,\beta}$ by
alternately and iteratively applying the $H_{P}$ and $H_{M}$ on initial state
$\ket{s}$. Here, $f(y)$, $H_{P}\ket{y}$, $H_{M}$, and $\ket{\gamma,\beta}$ are
defined as (18), (19), (20), and (21), where $n\in\mathbb{Z}^{+}$,
$p\in\mathbb{Z}^{+}$, and $X_{k}$ is the Pauli-$X$ operator applying on the
$k$-th qubit; $\gamma$ and $\beta$ are the hyper-parameters those can be
computed via approximation. Here, $H_{P}$ encodes $f(y)$ in (19), operating
diagonally in $n$-qubit quantum basis states [26]. In the computation
procedure of QAOA, via the iterative measurement of $\ket{\gamma,\beta}$, the
expectation of $H_{P}$ should be obtained. Finally, the samples of $f(y)$ can
be obtained as [16],

$\expectationvalue{f(y)}_{\gamma,\beta}=\expectationvalue{H_{P}}{\gamma,\beta}.$
(22)

The near-optimal or optimal approximation values of the hyper-parameters
$\gamma$ and $\beta$ are obtained using conventional optimization, e.g.,
stochastic gradient descent [27, 28]. Thus, the solution can be computed from
(22) via the obtained hyper-parameters $\gamma$ and $\beta$. Finally, it can
be shown that the QAOA-based approximation is one of widely known hybrid
quantum-classical optimization algorithms where the efficient Hamiltonian
design (for quantum approach) and the approximation of efficient hyper-
parameters (for conventional optimization approach) are correlated [29, 30].

##  IV Quantum Scheduling for MWIS-based Formation in Satellite Constellation

This section consists of the design of Hamiltonian, i.e., Problem Hamiltonian,
i.e., $H_{P}$ (refer to Sec. IV-A) and Mixing Hamiltonian, i.e., $H_{M}$
(refer to Sec. IV-B). Lastly, QAOA iterative computation procedure is
described in Sec. IV-C.

###  IV-A Problem Hamiltonian, $H_{P}$

The problem Hamiltonian $H_{P}$ is designed via the linear combination of the
objective Hamiltonian $H_{O}$ and the constraint Hamiltonian $H_{C}$. The
objective function and constraints in the mathematical problem for solving the
considering MWIS-based scheduling problem are in $H_{O}$ (refer to Sec. IV-A1)
and $H_{C}$ (refer to Sec. IV-A2).

####  IV-A1 Hamiltonian for MWIS Objective, $H_{O}$

Suppose that a basic Boolean function $B_{1}(x)=x$ exists where
$x\in\\{0,1\\}$. According to the quantum Fourier expansion of this
$B_{1}(x)=x$, it can be mapped to Boolean Hamiltonian $H_{B_{1}}$ where $I$
and $Z$ are identity operator and Pauli-$Z$ operator [31], i.e.,

$H_{B_{1}}=\frac{1}{2}(I-Z),$ (23)

therefore, the objective function (2) can be mapped into the following
Hamiltonian,

$H_{O^{\prime}}=\sum_{\forall s_{k}\in S}\frac{1}{2}w_{k}(I-Z_{k}),$ (24)

where $Z_{k}$ is the Pauli-$Z$ operator applied to $\mathcal{I}_{k}$. Because
the objective function (2) is mapped to $H_{O^{\prime}}$, it should be
maximized via the main objective of MWIS. Thus, it is obvious that this
$H_{O^{\prime}}$ should be maximized as well. Therefore, the objective
Hamiltonian $H_{O}$ should be minimized is as,

$\boxed{H_{O}=\sum_{\forall s_{k}\in S}\frac{1}{2}w_{k}Z_{k}}.$ (25)

####  IV-A2 Hamiltonian for MWIS Constraints

In the MWIS-based scheduling problem, we should avoid the case where both
adjacent nodes of the conflict graph are selected. The scheduled and
unscheduled nodes have states are denoted as $\ket{1}$ and $\ket{0}$. Here,
$N_{i}$ and $N_{j}$ are defined as the arbitrary nodes, and
$E_{A}(N_{i},N_{j})$, $E_{B}(N_{i},N_{j})$, and $E_{C}(N_{i},N_{j})$ stands
for the edge notations for three cases where,

  * •

$E_{A}(N_{i},N_{j})$ for Case A: $s_{i}$ and $s_{j}$ are not scheduled,

  * •

$E_{B}(N_{i},N_{j})$ for Case B: One of $s_{i}$ and $s_{j}$ is scheduled,

  * •

$E_{C}(N_{i},N_{j})$ for Case C: Both of $s_{i}$ and $s_{j}$ are scheduled
(impossible situation).

Suppose that the weights of $N_{i}$ and $N_{j}$ in Case C are defined as
$W_{N_{i}}$ and $W_{N_{j}}$. Under this definition, the constraint function
$C^{\prime}(i,j)$, which counts the impossible situations, can be represented
as,

$C^{\prime}(i,j)=\sum_{i=1}^{n}\sum_{j=1}^{n}(W_{N_{i}}+W_{N_{j}})\absolutevalue{E_{C}(N_{i},N_{j})}$
(26)

where $i>j$; and $n$ and $\absolutevalue{E_{C}(N_{i},N_{j})}$ are the number
of nodes and the number of $E_{C}(N_{i},N_{j})$ where $i>j$. This is a primary
condition for avoiding impossible situations.

Based on the mathematical program of MWIS problem formulation (i.e., (1)–(5)),
$C^{\prime}(i,j)$ can be re-formed as $C(i,j)$ as,

$\displaystyle C(i,j)$ $\displaystyle=$ $\displaystyle\sum_{{\forall s_{i}\in
S}}\sum_{{\forall s_{j}\in S}}(w_{i}+w_{j})E_{(i,j)}$ (27) $\displaystyle=$
$\displaystyle\sum_{{\forall s_{i}\in S}}\sum_{{\forall s_{j}\in
S}}(w_{i}+w_{j})(\mathcal{I}_{i}\land\mathcal{I}_{j}),$

where $i>j$; and $\land$ stands for an AND gate. In (27), $C(i,j)$ should be
$0$ because it stands for the number of impossible situations. If making this
$C(i,j)$ be not possible, this $C(i,j)$ should be minimized as much as
possible. According to the quantum Fourier expansion of AND gate
$B_{2}(x_{1},x_{2})$, it can be mapped to the following Boolean Hamiltonian
$H_{B_{2}}$ where the $Z_{1}$ and $Z_{2}$ in this equation are the Pauli-$Z$
operators applying on $x_{1}$ and $x_{2}$, respectively [31],

$\displaystyle B_{2}(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle x_{1}\land
x_{2}\text{ where }$ (28) $\displaystyle x_{1}\in\\{0,1\\}\text{ and
}x_{2}\in\\{0,1\\},$ $\displaystyle H_{B_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(I-Z_{1}-Z_{2}+Z_{1}Z_{2}).$ (29)

Based on this result, the constraints (27) can be mapped into following
Hamiltonian,

$H_{C^{\prime}}=\sum_{{\forall s_{i}\in S}}\sum_{{\forall s_{j}\in
S}}\frac{1}{4}(w_{i}+w_{j})(I-Z_{i}-Z_{j}+Z_{i}Z_{j}),$ (30)

where $i>j$; and $Z_{i}$ and $Z_{j}$ are the Pauli-$Z$ operators applied to
$\mathcal{I}_{i}$ and $\mathcal{I}_{j}$, respectively. Because $C(i,j)$ should
be zero (or it should be minimized, as explained before), the $H_{C^{\prime}}$
which is mapped from $C(i,j)$ should be minimized, as well. Thus, the
constraint Hamiltonian $H_{C}$ is as,

$\boxed{H_{C}=\sum_{{\forall s_{i}\in S}}\sum_{{\forall s_{j}\in
S}}-\frac{1}{4}(w_{i}+w_{j})(Z_{i}+Z_{j}-Z_{i}Z_{j})},$ (31)

where $i>j$.

Based on the obtained $H_{O}$ and $H_{C}$ in (25) and (31), the problem
Hamiltonian $H_{P}$ is defined as,

$\boxed{H_{P}=H_{O}+\rho H_{C}},$ (32)

where $\rho$ is a hyper-parameter that represents the penalty rate which means
the ratio at which $H_{C}$ (constraints) affects $H_{P}$ compared to $H_{O}$
(objective) ($\rho\geq 1$).

###  IV-B Mixing Hamiltonian, $H_{M}$

The mixing Hamiltonian $H_{M}$ produces various cases which can appear in the
given MWIS-formulated combinatorial problem. Our considering MWIS-based
observation scheduling problem can be formulated by a binary bit string which
presents a set of nodes. Therefore, various cases can be created by flipping
the state of each node, mathematically modelled as $\ket{0}$ or $\ket{1}$. The
bit-flip can be handled by the Pauli-$X$ operator. Therefore, $H_{M}$ can be
formed as,

$\boxed{H_{M}=\sum_{{\forall s_{k}\in S}}X_{k}}.$ (33)

###  IV-C QAOA Iterative Computation

The application of the designed Hamiltonian to the QAOA iterative optimization
computation sequence starts when the design of Hamiltonian functions, i.e.,
$H_{P}$ and $H_{M}$ in (32) and (33), are completed. Then the iterative
optimization computation procedure is as follows.

  * •

First of all, the parameterized state $\ket{\gamma,\beta}$ can be generated by
applying $H_{P}$ and $H_{M}$ to (21), as defined in (25), (31), (32), and
(33). Note that the initial state $\ket{s}$ is set to the equivalent
superposition state using the Hadamard gates in (11).

  * •

The expectation of $H_{P}$ can be measured on the generated parameterized
state $\ket{\gamma,\beta}$. Here, The parameters $\gamma$ and $\beta$ are
iteratively updated with traditional optimization computation procedure.

  * •

When the QAOA iterative computation sequence terminates, the optimal (or
approximated) parameters $\gamma_{\textsf{OPT}}$ and $\beta_{\textsf{OPT}}$
are finally obtained.

Therefore, the MWIS-based monitoring area scheduling solution in observation
satellite constellation can be obtained by the measurement of the expectation
of $H_{P}$ on the optimal state
$\ket{\gamma_{\textsf{OPT}},\beta_{\textsf{OPT}}}$ as,

$\boxed{\expectationvalue{F}=\expectationvalue{H_{P}}{\gamma_{\textsf{OPT}},\beta_{\textsf{OPT}}}},$
(34)

where $\expectationvalue{F}$ is the expectation of the MWIS objective function
(2) for the obtained solution samples.

##  V Concluding Remarks and Future Work

In beyond 5G and 6G communication networks, satellites has been actively used
in many applications such as seamless monitoring target areas, even for
extreme circumstances, thanks to the use of high-capacity millimeter-wave
wireless links in satellites. This paper proposes a scheduling algorithm which
aims at the minimization of overlapping monitoring areas among observation
satellite constellation. To achieve this goal, a quantum optimization based
algorithm is used because the our considering overlapping formulation can be
mathematically modelled via a max-weight independent set (MWIS) problem.

As a future research direction, the proposed algorithm can be evaluated with
various realistic satellite scenarios and TensorFlow-Quantum based software
implementation [32].

## Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (2019M3E4A1080391,
2021R1A4A1030775). J. Kim, S. Jung, and J.-H. Kim are the corresponding
authors of this paper (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>jkim@ajou.ac.kr).

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# Power-saving error terms for the number of $D_{4}$-quartic extensions over a
number field ordered by discriminant

Alina Bucur, Alexandra Florea, Allechar Serrano López, Ila Varma

###### Abstract

We study the asymptotic count of dihedral quartic extensions over a fixed
number field with bounded norm of the relative discriminant. The main term of
this count (including a summation formula for the constant) can be found in
the literature (see [18] for the statement without proof and see [32] for a
proof), but a power-saving for the error term has not been explicitly
determined except in the case that the base field is ${\mathbb{Q}}$. In this
article, we describe the argument for obtaining both the explicit main term
and a power-saving error term for the number of $D_{4}$-quartic extensions
over a general base number field ordered by the norms of their relative
discriminants. We also give an extensive overview of the history and
development of number field asymptotics.

##  1 Introduction

Obtaining asymptotics on the count of number fields ordered by discriminant
has been a guiding force in the subfield of number theory that has been coined
(originally by Mazur) as “arithmetic statistics.”

The simplest case of this question arises from the study of quadratic fields:
How many quadratic fields are there of bounded discriminant? In this lowest-
degree case, it is fortunate that quadratic fields are uniquely determined by
their discriminant, and so answering the above question boils down to
understanding the possible discriminants for quadratic fields. Algebraic
methods lead to the conclusion that quadratic (or fundamental) discriminants
are exactly the integers of the form $D$, where $D\equiv 1\bmod 4$ and $D$ is
square-free, or $4D$, where $D\equiv 2,3\bmod 4$ and $D$ is square-free.
Analytic methods can then be used to compute that the number of such integers
with absolute value bounded by $X$ is

$\frac{6}{\pi^{2}}X+O(\sqrt{X}).$

The above asymptotic can be proved by using contour integration and the fact
that the Riemann zeta function $\zeta(s)$ does not vanish on $\Re(s)=1$.111
This result goes back many centuries, and was at least known to Gauss, who in
[28] provides evidence that that class number of imaginary quadratic rings
grows like $|D|^{1/2}$. In particular, Gauss claimed (without proof) that the
sum of class numbers of quadratic fields of discriminant up to $T$ is
$\sum_{|D|<T}H(D)\sim\frac{\pi}{18}T^{3/2},$ but Lagrange may have indexed
these rings by their discriminant earlier. One can also think of this count as
essentially reducing to counting squarefree integers up to $X$, which might
have even been done by Euler.

In the 20th century, two main paths of generalizing this question on quadratic
discriminants were studied: changing the degree of the number fields in
question and changing the base field from ${\mathbb{Q}}$ to a general number
field (or function field). Altogether, we can pose the general question that
is studied to understand number field asymptotics.

######  Question.

Let $F$ be a base number field and fix an algebraic closure $\overline{F}$ of
$F$. Fix a degree $n$ and a Galois group $G$ equipped with an embedding into
$S_{n}$. What is the number of ($F$-isomorphism classes of) degree $n$ field
extensions $L$ of $F$ such that the normal closure $\tilde{L}$ of $L$ over $F$
has Galois group isomorphic $G$ and the absolute norm of the relative
discriminant ${\rm Disc}(L/F)$ is bounded by $X$?

Here, we say two extensions $L$ and $L^{\prime}$ of a number field $F$ are
$F$-isomorphic if there exists a field isomorphism between $L$ and
$L^{\prime}$ that restricts to the identity on $F$.

Let $N_{F,n}(G,X)$ denote the number of $F$-isomorphism classes of degree $n$
extensions $L$ over $F$ such that ${\rm Gal}(\tilde{L}/F)\cong G$ and
$\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm Disc}(L/F))\leq X$. We also require
that the Galois group ${\rm Gal}(\tilde{L}/F)$ of the Galois closure
$\tilde{L}$ over $F$, viewed as permuting the set of embeddings of $L$ into
$\tilde{L}$, is isomorphic to the fixed choice of image of $G$ in $S_{n}$. Our
focus is then on determining the constants of the main term of the asymptotic
formula for $N_{F,n}(G,X)$; that is, finding $a_{n}(G),b_{F,n}(G)$ and
$c_{F,n}(G)$ such that

$\lim_{X\rightarrow\infty}\frac{N_{F,n}(G,X)}{X^{a_{n}(G)}\log(X)^{b_{F,n}(G)}}=c_{F,n}(G).$
(1)

The study of the positive rational constants $a_{n}(G)=a_{F,n}(G)$ and
$b_{F,n}(G)$ is the focus of Malle’s Conjecture [39, 40]. We first discuss the
development of results and techniques for number field asymptotics predating
the statement of Malle’s Conjecture. After introducing our main result, we
discuss the evolution of the study of number field asymptotics after Malle’s
statement of their conjecture.

###  1.1 Number field asymptotics predating Malle’s Conjecture

Much of the work predating the statement of Malle’s conjecture focused on the
case when $F={\mathbb{Q}}$. Past the $n=2$ case described above, Cohn [19]
determined in 1954 using class field theory and generating functions that
equation (1) holds when $F={\mathbb{Q}}$, $n=3$, and $G=C_{3}$ with

$a_{{\mathbb{Q}},3}(C_{3})=1/2,\quad b_{{\mathbb{Q}},3}(C_{3})=0,\quad
c_{{\mathbb{Q}},3}(C_{3})=\frac{11\sqrt{3}}{36\pi}\cdot\prod_{p\equiv 1\bmod
6}\frac{(p+2)(p-1)}{p(p+1)}.$

In 1971, Davenport–Heilbronn [21] complemented this work by determining that
the asymptotic in (1) holds for $F={\mathbb{Q}}$, $n=3$ and $G=S_{3}$ with

$a_{{\mathbb{Q}},3}(S_{3})=1,\quad b_{{\mathbb{Q}},3}({S_{3}})=0,\quad
c_{{\mathbb{Q}},3}(S_{3})=\frac{1}{3\zeta(3)}.$

Their results utilized an algebraic parametrization of cubic rings in terms of
${\rm GL}_{2}({\mathbb{Z}})$-orbits of integer binary cubic forms; geometry-
of-numbers techniques to relate the count of certain orbits to a volume of a
body in the fundamental domain for ${\rm GL}_{2}({\mathbb{Z}})$ on real-
coefficient binary cubic forms; and finally sieving methods to restrict to
maximal rings and thus obtain an asymptotic count for $S_{3}$-cubic fields.

In 1979, Baily [6] proved the asymptotic (1) in the abelian degree 4 cases
($F={\mathbb{Q}}$, $n=4$, and $G=C_{4}$ or $V_{4}$) with

$a_{{\mathbb{Q}},4}(C_{4})=1/2,\,\,b_{{\mathbb{Q}},4}(C_{4})=0,\,\,c_{{\mathbb{Q}},4}(C_{4})=\frac{3}{\pi^{2}}\left(-1+\frac{25}{24}\prod_{p\equiv
1\bmod 4}\left(1+\frac{2}{p^{3/2}+p^{1/2}}\right)\right),$

and

$a_{{\mathbb{Q}},4}(V_{4})=1/2,\,\,b_{{\mathbb{Q}},4}(V_{4})=2,\,\,c_{{\mathbb{Q}},4}(V_{4})=\frac{1}{48\zeta(2)^{3}}\prod_{p}\left(1+\frac{3p+1}{p^{3}+3p^{2}}\right)^{-1}.$

Baily used class field theory and generating functions in a similar manner as
the cyclic cubic case for the above results.

In addition, Baily proved that $cX<N_{{\mathbb{Q}},4}(D_{4},X)<c^{\prime}X$
for two constants $c$ and $c^{\prime},$ where $D_{4}$ denotes the symmetries
of the square. The upper bound is obtained using bounds on the number of
quadratic ray class characters of quadratic fields which correspond under
class field theory to $D_{4}$-quartic fields and summing these bounds over all
quadratic fields with bounded discriminant. The lower bound was obtained by
studying quadratic ray class characters of certain imaginary quadratic fields
with even class number.

In the 1970s and early 1980s, a variety of authors worked on asymptotics for
various families of $p$-groups, such as the case $G=C_{p^{k}}$ for $p\geq 3$,
$k\in{\mathbb{Z}}_{\geq 2}$, $F={\mathbb{Q}}$ and $n=p^{k}$, as well as
$G=(C_{p})^{k}$ and direct products of $p$-groups with $p^{\prime}$-groups,
where $p,p^{\prime}\geq 3$ are distinct primes (see [55, 61, 56, 46]. For more
detailed lists, see Pg. 7 of [38]).

In the late 1980s, a lot of different avenues saw progress, mostly surrounding
work of Mäki and Wright. In 1985, Mäki [38] determined that for any abelian
group $G$ of order $n=|G|$,

$a_{{\mathbb{Q}},|G|}(G)=\frac{1}{n}\left(1-\frac{1}{p}\right),\quad
b_{{\mathbb{Q}},|G|}(G)=\begin{cases}\frac{n_{p}}{p-1}-1&\mbox{if }p\geq 3\\\
n_{p}-1&\mbox{if }p=2\end{cases}$

where $p$ is the smallest prime dividing $n$ and $n_{p}$ denotes the number of
elements of $G$ of order $p$. In 1988, Datskovsky–Wright [20] determined for
any number field $F$,

$a_{F,3}(S_{3})=1,\,\,b_{F,3}(S_{3})=0,\,\,c_{F,3}(S_{3})=\left(\frac{2}{3}\right)^{r_{1}(F)-1}\left(\frac{1}{6}\right)^{r_{2}(F)}\frac{\zeta_{F}(1)}{\zeta_{F}(3)},$

where $\zeta_{F}(1)$ denotes the residue at $s=1$ of $\zeta_{F}(s)$, the
Dedekind zeta function associated to $F$, $r_{1}(F)$ denotes the number of
real embeddings of $F$ into $\overline{F}$, and $r_{2}(F)$ denotes the number
of pairs of complex embeddings of $F$ into $\overline{F}$. They also obtained
analogous results for any global field of characteristic not equal to 2 or 3;
see [20, Theorem I.1]. As a byproduct, they determined

$a_{F,2}(S_{2})=1,\,\,b_{F,2}(S_{2})=0,\,\,c_{F,2}(S_{2})=2^{-r_{2}(F)-1}\frac{\zeta_{F}(1)}{\zeta_{F}(2)}.$

The authors accomplished this work building on the work of Davenport and
Heilbronn [21] and through the use of the adelic Shintani zeta functions [52,
53].

Building on work of Mäki, Wright [60] proved that for any finite abelian group
$G$ of cardinality $n$ and any number field $F$, there exists a constant
$c_{F,n}(G)>0$ such that the asymptotic (1) holds with the exponents

$a_{n}(G)=\frac{1}{n}\left(1-\frac{1}{p}\right),\,\,b_{F,n}(G)=\frac{n_{p}}{[F(\zeta_{p}):F]}-1,$

where $p$ is the smallest prime dividing $n$, and $n_{p}$ denotes the number
of elements of $G$ of order $p$. Wright developed general techniques in the
vein of [19] and [6], i.e. incorporating class field theory with the use of
generating functions that happen to be related to Dirichlet series. Standard
Tauberian theorems (most notably, Perron’s formula) then allow for the
extraction of number field asymptotics.

In the the early 2000s, Cohen–Diaz y Diaz–Olivier [18] determined that, for
$F={\mathbb{Q}}$, $n=4$, and $G=D_{4}$,

$a_{{\mathbb{Q}},4}(D_{4})=1\quad b_{{\mathbb{Q}},4}(D_{4})=0\quad
c_{{\mathbb{Q}},4}(D_{4})=\sum_{[K:{\mathbb{Q}}]=2}\frac{2^{-r_{2}(K)-1}}{{\rm
Disc}(K)^{2}}\frac{\zeta_{K}(1)}{\zeta_{K}(2)}.$ (2)

They utilized Kummer theory in conjunction with the strategy of Baily [6] to
relate the count of $D_{4}$-quartic extensions to the count of quadratic
extensions of the quadratic extensions. The authors of [18] define a Dirichlet
series associated to counting quadratic extensions of quadratic extensions and
relate it to the generating function for $D_{4}$-quartic fields. They also
remark that it is immediate to extend this result to the case of relative
$D_{4}$-extensions over a general base field.

###  1.2 Main Results

In this paper, we give a proof of the asymptotic (1) in the case of quartic
dihedral extensions of a number field $F$, with $a_{F,4}(D_{4})=1$,
$b_{F,4}(D_{4})=0$, and determine the constant $c_{F,4}(D_{4})$. Further, we
obtain an explicit formula for a power-saving error term in terms of the
degree of $F$. More precisely, we prove the following result.

######  Theorem 1.

Let $F$ be a number field with $[F:{\mathbb{Q}}]=d$. We have that the
asymptotic number of ($F$-isomorphism classes of) quartic $D_{4}$-extensions
of $F$ whose norm of the relative $F$-discriminant is bounded by $X$ is, as
$X$ grows,

$N_{F,4}(D_{4},X)=\left(\sum_{[K:F]=2}\frac{1}{2^{r_{2}(K)+1}}\cdot\frac{1}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(K/F))^{2}}\cdot\frac{\zeta_{K}(1)}{\zeta_{K}(2)}\right)\cdot
X+O\left(X^{\frac{3}{4}+\epsilon}\right).$ (3)

If we assume the Lindelöf hypothesis, then the error term above can be
improved to $O\left(X^{\frac{1}{4}+\epsilon}\right).$

Recall that for $\Re(s)>1$ the Dedekind zeta function of any number field $E$
is given by

$\zeta_{E}(s)=\prod_{{\mathfrak{p}}\subseteq{\mathcal{O}}_{E}}\left({1-\operatorname{Nm}_{E/{\mathbb{Q}}}({\mathfrak{p}})^{-s}}\right)^{-1},$

where ${\mathfrak{p}}$ runs over prime ideals of ${\mathcal{O}}_{E}$. It has
meromorphic continuation to the full complex plane, and is analytic everywhere
except for a simple pole at $s=1$. By abuse of notation, we denote by
$\zeta_{E}(1)$ the residue at $s=1$.

The main term in Theorem 1 is proven by Klüners [32] without a power-saving
error term. We show that the proof argument utilized by Cohen–Diaz y
Diaz–Olivier [18] to prove that $a_{{\mathbb{Q}},4}(D_{4})=1$ and
$b_{{\mathbb{Q}},4}(D_{4})=0$ holds true for general $F$, and that the formula
(3) is the same as the one stated by them in Remark (4) after [18, Corollary
6.1] with $\alpha=\frac{3}{4}$. We point out that our power-saving error term
does not depend on the degree of $F$, as the aforementioned authors suggest!

###  1.3 Malle’s conjecture, counterexamples, refinements, and proven cases

In 2004, Malle [39, 40] made a prediction for the constants
$a_{F,n}(G)=a_{n}(G)$ and $b_{F,n}(G)$ for all transitive groups
$G\hookrightarrow S_{n}$. The prediction was that

$a_{n}(G)=\frac{1}{\min_{g\in G\backslash 1}{\rm Ind}(g)},$

where the index of an element $g$ is defined as ${\rm Ind}(g):=n-\\#$ orbits
of $\\{1,2,...n\\}$ when acted upon by $g$ via the embedding into $S_{n}$, and

$b_{F,n}(G)=b^{\rm Malle}_{F,n}(G):=\\#\\{{\rm Gal}(\overline{F}/F)\text{-
orbits of }G\text{-conjugacy classes of minimal index }{\rm Ind}(C)\\}-1,$

where the notation is defined as follows: Note that if $g$ and $g^{\prime}$
are in the same conjugacy class $C$ of $G$, then ${\rm Ind}(g)={\rm
Ind}(g^{\prime})$, and furthermore, the absolute Galois group ${\rm
Gal}(\overline{F}/F)$ acts on the set of conjugacy classes $\\{C\\}$ of $G$
via the action of the cyclotomic character, i.e., $\mu:{\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})\rightarrow\hat{{\mathbb{Z}}}^{\times}$
so that $\sigma\in{\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on
roots of unity as $\sigma(\zeta_{n})=\zeta_{n}^{\mu(\sigma)}$. For $g\in G$,
we can define $\sigma(g)=g^{\mu(\sigma)}$, or in fact on conjugacy classes $C$
of $G$: $\sigma(C)=C^{\mu(\sigma)}$. We call the orbits under this action on
the $G$-conjugacy classes $\\{C\\}$ by ${\rm Gal}(\overline{F}/F)$-orbits.

Note that in the trivial case when $n=1$ and $G$ is the trivial group we have
$a_{1,F}(1)=b_{1,F}(1)=0$ and $c_{1,F}(1)=1$ for any field $F.$

It is also worth pointing out that when the image of $G$ in $S_{n}$ contains a
simple transposition $(ab)$, then ${\rm Ind}((ab))=1$ since $(ab)$ has $n-1$
orbits in $S_{n}$, hence $a_{n}(G)$ is predicted to be 1. The number of
$G$-conjugacy classes of minimal index $1$ will be $1$, since all simple
transpositions are conjugate to one another and no other permutation type can
have $n-1$ orbits, hence $b_{F,n}^{\rm Malle}(G)=0$ in this case.

In 2005, two different directions of progress arose from the development of
averaging techniques in geometry-of-numbers counting due to Bhargava and a
noteworthy counterexample and subsequent work to Malle’s Conjecture due to
Klüners.

Bhargava [9] determined for $S_{4}$-quartic fields over ${\mathbb{Q}}$ that
$a_{{\mathbb{Q}},4}(S_{4})=1$ and $b_{{\mathbb{Q}},4}(S_{4})=0$ and obtained
an Euler product formula for

$c_{{\mathbb{Q}},4}(S_{4})=\frac{5}{24}\prod_{p}\left(1+\frac{1}{p^{2}}-\frac{1}{p^{3}}-\frac{1}{p^{4}}\right).$

Bhargava’s methods built on the work of Davenport-Heilbronn [21],
Delone–Faddeev [22], and Sato-Kimura [48] using the same proof strategy of a
parametrization (this time, in terms of pairs of ternary quadratic forms)
allowing for geometry-of-numbers methods followed by sieving techniques.
However, Bhargava produced a general machine of averaging techniques for
cutting off cusps when computing a volume in the fundamental domain and
relating it to the number of integral orbits or lattice points that enumerate
quartic rings. This allowed for generalizations to the $n=5$ case [10], in
which he determined in 2010 that $a_{{\mathbb{Q}},5}(S_{5})=1$ and
$b_{{\mathbb{Q}},5}(S_{5})=0$ and obtained an Euler product formula for

$c_{{\mathbb{Q}},5}(S_{5})=\frac{13}{120}\prod_{p}\left(1+\frac{1}{p^{2}}-\frac{1}{p^{4}}-\frac{1}{p^{5}}\right).$

In addition, Bhargava and Wood [13] and independently, Belabas and Fouvry [7]
verified for $S_{3}$-sextic fields that $a_{{\mathbb{Q}},6}(S_{3})=1/3$ and
$b_{{\mathbb{Q}},6}(S_{3})=0$ and furthermore determined that

$c_{{\mathbb{Q}},6}(S_{3})=\frac{1}{3}\left(1+\frac{1}{3}+\frac{1}{3^{5/3}}+\frac{2}{3^{7/3}}\right)\left(1-\frac{1}{3}\right)\prod_{p\neq
3}\left(1+\frac{1}{p}+\frac{1}{p^{4/3}}\right)\left(1-\frac{1}{p}\right).$

While studying these questions, Bhargava [8] formulated a conjecture for the
constant in the case when $G=S_{n}$. Namely, he conjectured that

$c_{F,n}(S_{n})=\frac{1}{2}\cdot\zeta_{F}(1)\left(\frac{\\#S_{n}[2]}{n!}\right)^{r_{1}(F)}\left(\frac{1}{n!}\right)^{r_{2}(F)}\prod_{\mathfrak{p}}\left(\sum_{k=0}^{n}\frac{q(k,n-k)-q(k-1,n-k+1)}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\mathfrak{p}})}\right)$

for all $n$, where $q(k,n)$ denotes the number of partitions of $k$ into at
most $n$ parts (see conjecture A in the Appendix of [8]). A conjecture for
$c_{F,n}(G)$ for general $G$ has not been formulated as the direct analogy to
Bhargava’s heuristic is contradicted by the result of [18] in the case of
$n=4$ and $G=D_{4}$ (see [18] and [4, Section §3.3]). Bhargava’s quartic and
quintic results were generalized to all number fields $F$ by
Bhargava–Shankar–Wang [12], including formulas for the constants
$c_{F,4}(S_{4})$ and $c_{F,5}(S_{5})$ that verified Bhargava’s conjecture
along with Malle’s conjecture.

Also in 2005, Klüners [30] provided a counterexample to Malle’s Conjecture by
proving a lower bound for $N_{{\mathbb{Q}},6}(C_{3},\wr C_{2},X)$ that implies
that $b_{{\mathbb{Q}},6}(C_{3}\wr C_{2})>0$, contradicting the formula given
by Malle, which would have predicted that $b_{{\mathbb{Q}},6}({C_{3}\wr
C_{2}})=0$. The contradiction seems to arise from the fact that while there
aren’t third roots of unity in the base field, third roots of unity can arise
in an intermediate extension, and $C_{3}\wr C_{2}$-extensions of
${\mathbb{Q}}$ that contain ${\mathbb{Q}}(\zeta_{3})$ and have discriminant
bounded by $X$ are themselves at least $c\cdot X^{1/2}\log(x)$ for some
positive constant $c$. Nevertheless, Klüners [32] did prove that Malle’s
conjecture holds for $G=C_{2}\wr H$ where $H$ is a transitive subgroup of
$S_{n/2}$ such that over $F$, there is at least one extension of $F$ with
Galois group $H$ and $N_{F,n/2}(H,X)=O_{F,H,\epsilon}(X^{1+\epsilon})$ for
every $\epsilon>0$. Building on [18], Klüners [32] computed

$c_{F,n}(C_{2}\wr H)=\sum_{{[K:F]=n/2\atop{\rm
Gal}(\tilde{K}/F)=H}}\frac{1}{2^{r_{2}(K)}\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(K)^{2})}\cdot\frac{\zeta_{K}(1)}{\zeta_{K}(2)}.$

In particular, when $H=C_{2}$, Klüners [32] proved the generalization of (2)
in the $D_{4}$-quartic case from over ${\mathbb{Q}}$ to over a general base
number field $F$ without a power-saving error term. Furthermore, Klüners [33]
verified Malle’s Conjecture for generalized quaternion groups of order $4m$,
determining that $a_{F,4m}(Q_{4m})=1/2m$ and $b_{F,4m}(Q_{4m})=0$.

In 2015, Türkelli [54] formulated function field heuristics for $N_{F,n}(G,X)$
(i.e., when $F$ is taken to be a function field ${\mathbb{F}}_{q}(t)$ instead
of a number field), and came up with a way to salvage Malle’s conjecture that
is consistent with Klüners’s counterexample. Namely, consider first for a
group $G$ the set ${\mathcal{N}}(G)$ of normal subgroups $N$ in $G$ such that
$G/N$ is abelian and $a_{n}(N)=a_{n}(G)$ (and where $a_{n}(\\{1\\})=0$).
Define

$b_{F,n}(G,N)=\max_{\varphi\in{\rm Sur}({\rm
Gal}(F(\mu_{\infty})/F),G/N))}b_{F,n}(G,N,\varphi),$

where $b_{F,n}(G,N,\varphi)$ is the number of twisted $F$-orbits of
$N$-conjugacy classes of minimal index, where the action of ${\rm
Gal}(\overline{F}/F)$ is defined now with a twist: for a conjugacy class $C$
of $N$, $\sigma(C)=\varphi(\operatorname{Res}(\sigma))^{-1}\cdot
C^{\mu(\sigma)}$, where $\operatorname{Res}:{\rm
Gal}(\overline{F}/F)\rightarrow{\rm Gal}(F(\mu_{\infty})/F)$ and
$\varphi(\operatorname{Res}(\sigma))^{-1}$ acts by conjugation on the
$N$-conjugacy class $C^{\mu(\sigma)}$.222It is worth pointing out that in
Malle’s original formulation, he only considered the case of $N=G$, and so
$b^{\rm Malle}_{F,n}(G)=b_{F,n}(G,G)$. This only coincides with $b_{F,n}(G)$
when $b_{F,n}(G,G)$ is the maximum over all $b_{F,n}(G,N)$ for
$N\in{\mathcal{N}}(G)$. Then, we can define

$b_{F,n}(G)=-1+\max_{N\in{\mathcal{N}}(G)}b_{F,n}(G,N)$

as the salvage to $b_{F,n}^{\rm Malle}(G)$.

Türkelli’s heuristics [54] seemed to contradict earlier predictions made by
Ellenberg–Venkatesh [24] for counting extensions of the function field
${\mathbb{F}}_{q}(t)$ which agree with Malle’s original conjecture. However,
Ellenberg–Venkatesh focused on extensions that do not contain any nontrivial
constant intermediate extensions, while Türkelli [54] allowed for those to be
counted in their heuristic, which has a stronger parallel to Malle’s
conjecture where extensions containing cyclotomic subfields are included in
the asymptotics. (On the geometric side, Türkelli [54] includes connected, but
not geometrically connected, covers of the projective line over
${\mathbb{F}}_{q}$ while Ellenberg-Venkatesh [24] require covers to be
geometrically connected as well.)

More recently, Wang [58] has made substantial progress on techniques for
proving Malle’s conjecture in compositums of number fields. The first
application of these methods was to prove Malle’s Conjecture for certain
direct product groups of the form $G=S_{n}\times A$, including the case when
$n=3$ and $A$ is an odd abelian group; the case when $n=4$ and $A$ is an
abelian group of order coprime to 6; and the case when $n=5$ and $A$ is an
abelian group of order relatively prime to 30 (see [57]). Later
generalizations came from joint work of Masri–Thorne–Tsai–Wang [42] to proving
Malle’s conjecture for $G=S_{n}\times A$ where $A$ is any abelian group and
$n=3,4,5$.

Most recently, Fouvry–Koymans [27] proved Malle’s conjecture for nonic
Heisenberg extensions by estimating a certain character sum. They determined
that there exists a constant $c_{{\mathbb{Q}},9}(\mathrm{Heis}_{3})>0$, which
they determine explicitly (see Pgs. 17-18 of [27]), such that (1) holds with

$a_{{\mathbb{Q}},9}(\mathrm{Heis}_{3})=1/4,\,\,b_{{\mathbb{Q}},9}(\mathrm{Heis}_{3})=0.$

Additionally, Koymans–Pagano [36] proved that for any number field $F$ and any
finite nilpotent group $G$ of cardinality $n$ such that all elements of order
$p$ are central, where $p$ is the smallest prime dividing $n$, there exists a
constant $c_{F,n}(G)>0$, and determined

$a_{n}(G)=\frac{1}{n}\left(1-\frac{1}{p}\right),\,\,b_{F,n}(G)=\frac{n_{p}}{[F(\zeta_{p}):F]}-1,$

where $n_{p}$ denotes the number of elements of $G$ of order $p$.

At the time of writing this article, the authors have heard of forthcoming
work of Alberts–Lemke Oliver–Wang–Wood [3] in which the authors develop
inductive counting methods to prove Malle’s conjecture for groups $G$ for
which all elements in $G$ of sufficiently small index are contained in a
normal subgroup which is either abelian or generated by conjugates of a single
subgroup of $G$ isomorphic to $S_{3}$. Additionally, work-in-progress at the
time of writing this article includes Shankar–Varma [50] for verifying Malle’s
conjecture in the Galois octic case by determining
$a_{{\mathbb{Q}},8}(D_{4})=1/4$ and $b_{{\mathbb{Q}},8}(D_{4})=2$ and
computing an Euler product for $c_{{\mathbb{Q}},8}(D_{4})$.

We do not claim that this is a full history of the study of number field
asymptotics; we tried to focus on the results that specifically proved (the
strong form of) Malle’s conjecture and study of the constants $c_{F,n}(G)$ in
those cases. Much work has been developed in proving the weak form of Malle’s
conjecture [39], i.e., that there exist positive real constants
$c^{(1)}_{F,n}(G)$ and $c^{(2)}_{F,n}(G,\epsilon)$ such that

$c^{(1)}_{F,n}(G)X^{a(G)}\leq
N_{F,n}(G,X)<c^{(2)}_{F,n}(G,\epsilon)X^{a(G)+\epsilon}$

for large enough $X$ (see [34], [2], [1]). Many analogous results have been
obtained over function fields, the case of quartic $D_{4}$-extensions by
Keliher [29], the case of cubic $S_{3}$-extensions studied by Wood [59], work
of Ellenberg–Venkatesh–Westerland [26] on degree $p$ $D_{p}$-extensions for
odd primes $p$, the case of $C_{2}$-extensions (with added local conditions)
studied by Kurlberg–Rudnick [37], the case of degree $p$ $C_{p}$-extensions
(with local conditions) for odd primes $p$ studied by Bucur–David–Feigon–Lalín
[15] and Bucur–David–Feigon–Kaplan–Lalín–Ozman–Wood [14].

In addition, much work has been done in proving and improving upper and lower
bounds for $N_{F,n}(G,X)$ (see [49], [51], [31], [25], [23], [47], [35], [41],
[43], [11], [45], [5]) as well as other important work on building methods and
techniques for answering these questions.

###  1.4 Summary of our methods

Fix a number field $F$ of degree $d$, and let $\overline{F}$ denote a fixed
algebraic closure of $F$. In the sequel, all extensions of $F$ will be assumed
to be contained in $\overline{F}$.

Recall that $N_{F,n}(G,X)$ denotes the number of $F$-isomorphism classes of
degree $n$ extensions $L$ over $F$ such that the normal closure $\tilde{L}$ of
$L$ over $F$ has Galois group ${\rm Gal}(\tilde{L}/F)\cong G$ and
$\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm Disc}(L/F))\leq X$, and we say two
extensions $L$ and $L^{\prime}$ of a number field $F$ are $F$-isomorphic if
there exists a field isomorphism between $L$ and $L^{\prime}$ that restricts
to the identity on $F$.

We are interested in counting the number of ($F$-isomorphism classes of)
quartic extensions $L$ whose normal closure over $F$ has Galois group $D_{4}$,
the symmetries of a square, when such extensions are ordered by the norms of
their relative discriminants over $F$. We will call such an $L$ a
$D_{4}$-quartic extension over $F$, and let ${\rm Disc}(L/F)$ denote the
relative discriminant. Denote by ${\mathcal{F}}_{F,4}(D_{4})$ the set of
$F$-isomorphism classes of $D_{4}$-quartic extensions over $F$.

To obtain the order of magnitude for $D_{4}$-quartic extensions over a general
base number field $F$ with discriminant norm bounded by $X$, we must
understand the poles of the Dirichlet series $\Phi_{F,4}(D_{4},s)$ associated
to the count $N_{F,4}(D_{4},X)$:

$\Phi_{F,4}(D_{4},s)=\sum_{L\in\mathcal{F}_{F,4}(D_{4})}\frac{1}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(L/F))^{s}},$

where ${\mathcal{F}}_{F,4}(D_{4})$ denotes the set of $F$-isomorphism classes
of $D_{4}$-quartic extensions of $F$. Perron’s formula implies that

$N_{F,4}(D_{4},X)=\frac{1}{2\pi
i}\int_{\Re(s)=2}\Phi_{F,4}(D_{4},s)\frac{X^{s}}{s}\,ds.$ (4)

Determining the residue of the poles of the above integral after shifting the
contours of integration gives the main term contribution for
$N_{F,4}(D_{4},X)$, and we additionally obtain (a power-saving) error term
from bounding the contribution of the integrals on the other sides of the
contour of integration.

To study the residue, we follow the same strategy as Cohen–Diaz y Diaz–Olivier
utilized in [18, Corollary 1.4] to prove the case $F={\mathbb{Q}}$.
Specifically, we will relate the count $N_{F,4}(D_{4},X)$ to the count of
quadratic extensions of quadratic extensions of $F$, and therefore relate the
Dirichlet series $\Phi_{F,4}(D_{4},s)$ to a Dirichlet series

$\frac{1}{2}\Phi(s)=\frac{1}{2}\sum_{[K:F]=2}\frac{1}{\operatorname{Nm}({\rm
Disc}(K/F))^{2s}}\left(1+\sum_{[L:K]=2}\frac{1}{\operatorname{Nm}({\rm
Disc}(L/K))^{2s}}\right),$

which has the same residue at the simple pole at $s=1$ as
$\Phi_{F,4}(D_{4},s)$. We can then use Cohen–Diaz y Diaz–Olivier’s results on
counting on quadratic extensions of a number field to give a summation formula
for the constant that verifies [18, Remark 4].

##  2 Notation and Background

In this section, we introduce the notation and background information we will
be using throughout the article.

For any number field $E$, let ${\mathcal{O}}_{E}$ denote its ring of integers,
let $r_{1}(E)$ denote the number of real embeddings of $E$, and let $r_{2}(E)$
denote the number of pairs of complex conjugate embeddings.

###  2.1 $L$-functions associated to a number field

For $s\in{\mathbb{C}}$ such that $\Re(s)>1$, the Dedekind zeta function of a
number field $E$ is given by the Euler product running over nonzero primes
ideals of ${\mathcal{O}}_{E}$

$\zeta_{E}(s)=\prod_{{\mathfrak{p}}\subseteq{\mathcal{O}}_{E}}\frac{1}{1-\operatorname{Nm}_{E/{\mathbb{Q}}}({\mathfrak{p}})^{-s}}.$

Hecke proved that $\zeta_{E}(s)$ has meromorphic continuation to the entire
complex plane with only a simple pole at $s=1$.

For a character $\chi$ of a ray class group
$\mathcal{C}\ell_{\mathfrak{m}}(F)$ of conductor ${\mathfrak{m}}$ of $E$, let
$L_{E}(s,\chi)$ denote the $L$-function associated to the character $\chi$,
namely for $\Re(s)>1$,

$L_{E}(s,\chi)=\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathcal{O}_{E}\\\
(\mathfrak{a},\mathfrak{m})=1\end{subarray}}\frac{\chi(\mathfrak{a})}{\operatorname{Nm}_{E/{\mathbb{Q}}}(\mathfrak{a})^{s}}.$

######  Lemma 2.1.1.

If $\chi=\chi_{0}$ is the trivial character, then $L_{E}(s,\chi_{0})$ has a
simple pole at $s=1$. Otherwise, $L_{E}(s,\chi)$ is analytic everywhere.

###### Proof.

We see that

$L_{E}(s,\chi_{0})=\zeta_{E}(s)\prod_{{\mathfrak{p}}\mid{\mathfrak{m}}}\left(1-\operatorname{Nm}_{E/{\mathbb{Q}}}({\mathfrak{p}})^{-s}\right),$
(5)

and the conclusion follows from the properties of $\zeta_{E}(s)$ as the
product over ${\mathfrak{p}}\mid{\mathfrak{m}}$ is finite. For nontrivial
$\chi$, the result follows from [44, Chapter VII, Theorem 8.5]. ∎

If there is a pole at $s=s_{0}$, and we write $L_{E}(s_{0},\chi)$ or
$\zeta_{E}(s_{0})$, we are abusing notation and referring to the residue at
the pole.

The Lindelöf hypothesis would give that

$\zeta_{E}\left(\frac{1}{2}+it\right)\ll t^{\epsilon},$ (6)

and a similar bound holds for $s=\sigma+it$ and $\sigma>\frac{1}{2}$.

###  2.2 Counting $G$-extensions of a number field and associated Dirichlet
series

Fix a finite transitive subgroup $G$ of $S_{n}$ for some $n\geq 2$. Let
$\mathcal{F}_{E,m}(G)$ denote the set of $E$-isomorphism classes of extensions
$E^{\prime}/E$ of degree $m$ (in a fixed algebraic closure of $E$) such that
the Galois group of the Galois closure of $E^{\prime}/E$ is isomorphic to $G$.

Denote by $N_{E,m}(G,X)$ the number of elements of $\mathcal{F}_{E,m}(G)$
whose discriminant is bounded by $X$; namely

$N_{E,m}(G,X)=\\#\\{E^{\prime}\in\mathcal{F}_{E,m}(G);\,\operatorname{Nm}_{E/{\mathbb{Q}}}({\rm
Disc}(E^{\prime}/E))\leq X\\}.$

In order to obtain asymptotic formulas for $N_{E,m}(G,X)$, we need to
understand the analytic properties of the function $\Phi_{E,m}(G,s)$ defined
by

$\Phi_{E,m}(G,s)=\sum_{E^{\prime}\in\mathcal{F}_{E,m}(G)}\frac{1}{\operatorname{Nm}_{E/{\mathbb{Q}}}({\rm
Disc}(E^{\prime}/E))^{s}}.$

Using Perron’s formula, we will obtain the main term in the asymptotic formula
for $N_{E,m}(G,X)$ from the right-most pole of $\Phi_{E,m}(G,s)$, while the
error term will be dictated by how much information we have about the
meromorphic continuation of $\Phi_{E,m}(G,s)$.

##  3 Counting quadratic extensions of quadratic extensions of a number field

For the rest of the paper, we fix a number field $F$ and set
$d=[F:{\mathbb{Q}}].$ We are interested in computing an asymptotic formula for
$N_{F,4}(D_{4},X)$, and our strategy is to study $\Phi_{F,4}(D_{4},s)$ by
relating it to the Dirichlet series associated to counting quadratic
extensions of quadratic extensions of $F$ of bounded discriminant norm. The
equivalent of $\Phi_{F,m}(G,s)$ for quadratic extensions of quadratic
extensions is the function

$\tilde{\Phi}(s)=\sum_{[K:F]=2}\sum_{[L:K]=2}\frac{1}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(L/F))^{s}}.$ (7)

We will first study the relationship of $\Phi_{F,4}(D_{4},s)$ and
$\tilde{\Phi}(s)$, showing that they only differ by a holomorphic function. We
will then study the poles and residues of $\tilde{\Phi}(s)$ by instead
considering a slightly altered Dirichlet series $\Phi(s)$ that again only
differs from $\tilde{\Phi}(s)$ (and therefore also from $\Phi_{F,4}(D_{4},s)$)
by a holomorphic function.

###  3.1 Quadratic extensions of quadratic extensions

For a quadratic extension $K$ of $F$, let ${\mathcal{F}}_{2}(K)$ denote the
$K$-isomorphism classes of quadratic extensions $L$ of $K$. There are three
possibilities for the extension $L/F$ in this case:

  * •

$L/F$ is a degree $4$ non-Galois extension whose normal closure has Galois
group $D_{4}$ over $F$. These are the extensions we want to count.

  * •

$L/F$ is a Galois extensions with Galois group $C_{4}$, the cyclic group of
order $4.$

  * •

$L/F$ is a Galois extension with Galois group
$V_{4}\simeq{\mathbb{Z}}/2\times{\mathbb{Z}}/2$, the Klein four-group.

By [18, Corollary 2.3(3)],

$\sum_{K\in{\mathcal{F}}_{2}(F)}\sum_{\begin{subarray}{c}L\in{\mathcal{F}}_{2}(K)\\\
\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm Disc}(L/F))\leq
X\end{subarray}}1\qquad=\quad
2N_{F,4}(D_{4},X)+N_{F,4}(C_{4},X)+3N_{F,4}(V_{4},X)$ (8)

To see this note that for a fixed quadratic extension $K$ of $F$, conjugate
$D_{4}$-extensions of $F$ will be counted separately by the sum on the left,
and every $D_{4}$-extension contains a single quadratic subextension (shared
with its unique $F$-conjugate). The sum on the left counts $C_{4}$-extensions
up to $F$-isomorphism just once since these extensions are Galois and contain
a single quadratic subextension. Each $V_{4}$-extension $L/F$ contains three
quadratic subfields, so the sum on the left is counting each $V_{4}$-extension
up to $F$-isomorphism three times.

Hence, we can express the function $\Phi_{F,4}(D_{4},s)$ we are interested in
as

$\Phi_{F,4}(D_{4},s)=\frac{1}{2}\left(\tilde{\Phi}(s)-\Phi_{F,4}(C_{4},s)-3\Phi_{F,4}(V_{4},s)\right).$
(9)

On the other hand, note that by the relative discriminant formula, if
$K\in{\mathcal{F}}_{2}(F)$ and $L\in{\mathcal{F}}_{2}(K)$, then ${\rm
Disc}(L/F)={\rm Disc}(K/F)^{2}\operatorname{Nm}_{K/F}({\rm Disc}(L/K))$. Thus,
we can write

$\displaystyle\tilde{\Phi}(s)$
$\displaystyle=\sum_{[K:F]=2}\sum_{[L:K]=2}\frac{1}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(K/F))^{2s}\operatorname{Nm}_{K/{\mathbb{Q}}}({\rm Disc}(L/K))^{s}}$
$\displaystyle=\sum_{[K:F]=2}\frac{1}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(K/F))^{2s}}\sum_{[L:K]=2}\frac{1}{\operatorname{Nm}_{K/{\mathbb{Q}}}({\rm
Disc}(L/K))^{s}}$
$\displaystyle=\sum_{[K:F]=2}\frac{\Phi_{K,2}(C_{2},s)}{\operatorname{Nm}_{F/{\mathbb{Q}}}({\rm
Disc}(K/F))^{2s}}.$ (10)

###  3.2 Counting using Dirichlet series

The previous section indicates that in order to study $\tilde{\Phi}(s)$, we
will be compelled to deal with not just $\Phi_{F,4}(D_{4},s)$, but also
$\Phi_{F,4}(C_{4},s)$ and $\Phi_{F,4}(V_{4},s)$ that appears in (9), which
were first considered by Wright [60]. By [60, Theorem I.1], it is known that
$\Phi_{F,4}(C_{4},s)$ and $\Phi_{F,4}(V_{4},s)$ are absolutely convergent for
$\Re(s)>1/2$. Using these functions, Wright proved that

$N_{F,4}(C_{4},X)=O\left(X^{1/2}\right)$

and

$N_{F,4}(V_{4},X)=O\left(X^{1/2}\log^{2}(X)\right).$

The functions $\Phi_{F,4}(C_{4},s)$ and $\Phi_{F,4}(V_{4},s)$ can also be
rewritten in terms of $L$-functions as given below.

######  Proposition 3.2.1 ([17, Theorem 4.3]).

We have that

$\Phi_{F,4}(C_{4},s)=\frac{2^{r_{1}(F)+r_{2}(F)-1}}{4^{2[F:{\mathbb{Q}}]s}\zeta_{F}(2s)}\sum_{\mathfrak{c}|4{\mathcal{O}}_{F}}\operatorname{Nm}(\mathfrak{c})^{2s}\sum_{\chi}L_{F}(2s,\chi)F_{\mathfrak{c},\chi}(s),$
(11)

where the sum over $\chi$ is over quadratic characters of ${\rm
Cl}_{\mathfrak{c}}(F)$, the sum over ${\mathfrak{c}}$ is over ideals dividing
$(4)$, and where $F_{\mathfrak{c},\chi}(s)$ is defined at the bottom of page
$499$ in [17] and converges for $\Re(s)>1/3.$ (Note that
$F_{\mathfrak{c},\chi}(s)$ is given in terms of a sum over quadratic
extensions $k/F$ which are embeddable into a $C_{4}$-extension.)

######  Proposition 3.2.2 ([16, Theorem 5.1]).

We have that

$\Phi_{F,4}(V_{4},s)=\frac{1}{6}\Psi(s)-\frac{1}{2}\Phi_{F,2}(C_{2},2s)-\frac{1}{6}$

where

$\displaystyle\Psi(s)$
$\displaystyle=\frac{2^{2r_{1}(F)+2r_{2}(F)}}{2^{6[F:{\mathbb{Q}}]s}}$
$\displaystyle\times\sum_{\begin{subarray}{c}\mathfrak{n}|2{\mathcal{O}}_{F}\\\
\mathfrak{m}|2{\mathcal{O}}_{F}\end{subarray}}\operatorname{Nm}(\mathfrak{nm})^{2s-1}\sum_{\mathfrak{c}|\mathfrak{n}}\frac{\mu_{F}(\mathfrak{n}/\mathfrak{c})}{\operatorname{Nm}(\mathfrak{n}/\mathfrak{c})}\operatorname{Nm}((\mathfrak{m},\mathfrak{c}))^{2s}\prod_{\begin{subarray}{c}{\mathfrak{p}}\text{
s.t. }\\\ 1\leq v_{\mathfrak{p}}(\mathfrak{m})\leq
v_{\mathfrak{p}}(\mathfrak{c})\end{subarray}}\left(1-\operatorname{Nm}(\mathfrak{p})^{-4s}\right)\prod_{\mathfrak{p}|\mathfrak{m}/(\mathfrak{m},\mathfrak{c})}\left(1-\operatorname{Nm}(\mathfrak{p})^{-2s}\right)$
$\displaystyle\times\sum_{\chi,\psi}\sum_{{\mathfrak{q}}\mid\mathfrak{m}/(\mathfrak{m},\mathfrak{c})}L_{F}(2s,\chi_{1,{\mathfrak{q}}}\chi_{2,{\mathfrak{q}}})L_{F}(2s,\chi_{1,{\mathfrak{q}}}\psi)L_{F}(2s,\chi_{2,{\mathfrak{q}}}\psi)H_{\mathfrak{q}}(s,\chi,\psi),$
(12)

where $\mathfrak{n},\mathfrak{m}$, and $\mathfrak{c}$ run over integral
ideals, ${\mathfrak{p}}$ and ${\mathfrak{q}}$ run over prime ideals, $\chi$
and $\psi$ run over all quadratic characters of ${\rm Cl}_{{\mathfrak{m}}}(F)$
and ${\rm Cl}_{{\mathfrak{n}}}(F)$ respectively, and
$H_{\mathfrak{q}}(\chi,\psi,s)$ is given by an Euler product that converges
for $\Re(s)>1/4$.

######  Corollary 3.2.3.

The poles and the residues of $\Phi_{F,4}(D_{4},s)$ are the same as the poles
of $\frac{1}{2}\tilde{\Phi}(s)$.

###### Proof.

This follows directly from equation (9) and work of Wright (see [60, Theorem
I.1]) or Propositions 3.2.1 and 3.2.2. ∎

It will be more convenient to work with the function

$\Phi(s)=\sum_{[K:F]=2}\frac{1}{\operatorname{Nm}({\rm
Disc}(K/F))^{2s}}\left(1+\Phi_{K,2}(C_{2},s)\right)$ (13)

rather than the function $\tilde{\Phi}(s)$ defined in (7). From [18, Theorem
1.1], we have

$1+\Phi_{K,2}(C_{2},s)=\frac{2^{-r_{2}(K)}}{\zeta_{K}(2s)}\sum_{\mathfrak{c}\mid
2{\mathcal{O}}_{K}}\operatorname{Nm}(2/\mathfrak{c})^{1-2s}\sum_{\chi}L_{K}(s,\chi),$
(14)

where $\mathfrak{c}$ runs over integral ideals and $\chi$ runs over all the
quadratic characters of the ray class group ${\rm Cl}_{\mathfrak{c}^{2}}(K)$.
Note that we would not expect a summation formula in terms of $L$-functions
like (14) to exist if we did not add 1 to $\Phi_{K,2}(C_{2},s)$, and so
computing the residue of $\tilde{\Phi}(s)$ would be much more difficult to do
directly as we would not be able to utilize the study of $L_{K}(s,\chi)$.

By equation (9), $\Phi(s)$ can be expressed as

$\Phi(s)=2\Phi_{F,4}(D_{4},s)+\Phi_{F,2}(C_{2},2s)+\Phi_{F,4}(C_{4},s)+3\Phi_{F,4}(V_{4},s).$
(15)

From equation (14) we also have

$\Phi_{F,2}(C_{2},s)=-1+\frac{2^{-r_{2}(F)}}{\zeta_{F}(2s)}\sum_{\mathfrak{c}\mid
2{\mathcal{O}}_{F}}\frac{\operatorname{Nm}(2/\mathfrak{c})}{\operatorname{Nm}(2/\mathfrak{c})^{2s}}\sum_{\chi}L_{F}(s,\chi),$
(16)

where $\mathfrak{c}$ runs over all integral ideals of $F$ dividing $2$ and
$\chi$ runs over all quadratic characters of the ray class group
$\mathcal{C}\ell_{\mathfrak{c}^{2}}(F)$ modulo $\mathfrak{c}^{2}$.

We now focus on $\Phi(s)$. It follows from (14) and Lemma 2.1.1 that
$1+\Phi_{K,2}(C_{2},s)$ is meromorphic in the complex plane and has a simple
pole at $s=1$. If we denote by $R$ the residue of $\Phi(s)$ at $s=1$, by using
the convexity bound

$L_{K}(s,\chi)\ll\operatorname{Nm}({\rm Disc}(K/F))^{\frac{1}{2}},$

combined with equations (13) and (14), it follows that $\Phi(s)-R$ is
absolutely convergent for $\Re(s)\geq 3/4+\epsilon.$ From equations (15) and
(16), in conjunction with Propositions 3.2.1 and 3.2.2 we also get the
following.

######  Proposition 3.2.4.

The function $\Phi_{F,4}(D_{4},s)$ has meromorphic continuation to the whole
complex plane and has a simple pole at $s=1$. Furthermore,
$\frac{1}{2}\Phi(s)$ and $\Phi_{F,4}(D_{4},s)$ have the same residue at the
simple pole at $s=1$. If $\Phi(1)$ denotes the residue of $\Phi(s)$ at the
pole $s=1$, then $\Phi_{F,4}(D_{4},s)-\frac{\Phi(1)}{2}$ is absolutely
convergent for $\Re(s)\geq 3/4+\epsilon$.

##  4 Proof of the main theorem

We will now prove Theorem 1, originally stated in Remark 4 of [18, Corollary
6.1]. We denote $c_{F,4}(D_{4})$ the constant in the main term, i.e.

$c_{F,4}(D_{4})=\sum_{[K:F]=2}\frac{1}{2^{r_{2}(K)+1}\operatorname{Nm}({\rm
Disc}(K/F))^{2}}\frac{\zeta_{K}(1)}{\zeta_{K}(2)}.$

Recall that $\zeta_{K}(1)$ denotes the value of the residue of the Dedekind
zeta function of $K$ at $s=1.$ With this notation, we need to prove that

$N_{F,4}(D_{4},X)=c_{F,4}(D_{4})X+O(X^{1-\frac{1}{2d}+\epsilon})$

where $d=[F:\mathbb{Q}]$. By Perron’s formula, we have that

$N_{F,4}(D_{4},X)=\frac{1}{2\pi
i}\int_{\Re(s)=2}\Phi_{F,4}(D_{4},s)\frac{X^{s}}{s}\,ds.$ (17)

When we shift the contour to the left to the line $\Re(s)=1+\epsilon$, we do
not encounter any poles. So we can rewrite

$N_{F,4}(D_{4},X)=\frac{1}{2\pi
i}\int_{\Re(s)=1+\epsilon}\Phi_{F,4}(D_{4},s)\frac{X^{s}}{s}\,ds.$

Using standard contour integration, we get that

$N_{F,4}(D_{4},X)=\frac{1}{2\pi i}\int_{1+\epsilon-
iT}^{1+\epsilon+iT}\Phi_{F,4}(D_{4},s)\frac{X^{s}}{s}\,ds+O\left(\frac{X^{1+\epsilon}}{T}\right),$
(18)

and we will choose $T$ suitably. We shift the line of integration to
$\Re(s)=3/4+\epsilon$ and encounter the simple pole at $s=1$. The residue of
the pole at $s=1$ will contribute the main term which will be of size $X$, and
the error term will come from bounding the contribution of the integrals on
the other three sides of the contour of integration.

Next we bound the integral on the two horizontal lines of the contour and on
the vertical line from $3/4+\epsilon-iT$ and $3/4+\epsilon+iT$. The integral
on the horizontal lines will be bounded by $X^{1+\epsilon}/T$ and for the
integral on the vertical line, we have

$\frac{1}{2\pi i}\int_{3/4+\epsilon-
iT}^{3/4+\epsilon+iT}\Phi_{F,4}(D_{4},s)\frac{X^{s}}{s}\,ds\ll
X^{3/4+\epsilon},$ (19)

where we have used Lemma 3.2.4.

Note that using the Lindelöf bound (6) to bound $\Phi(s)$ yields an upper
bound of size $X^{1/4+\epsilon}$ for the vertical integrals.

Finally, we evaluate the residue of the simple pole at $s=1$, which will give
us the constant $c_{F,4}(D_{4})$, following [18]. Using (15) and Proposition
3.2.4, we see that, since the difference between $\Phi_{F,4}(D_{4},s)$ and
$\frac{1}{2}\Phi(s)$ is holomorphic for $\Re(s)>1/2$, the two functions
$\Phi_{F,4}(D_{4},s)$ and $\frac{1}{2}\Phi(s)$ have the same residue at $s=1.$
From equations (5), (13), (14), and (15) it follows that the value of the
residue at $s=1$ is equal to

$\displaystyle\frac{1}{2}\sum_{[K:F]=2}\frac{2^{-r_{2}(K)}\zeta_{K}(1)}{\operatorname{Nm}({\rm
Disc}(K/F))^{2}\zeta_{K}(2)}\sum_{\mathfrak{c}|2{\mathcal{O}}_{K}}\operatorname{Nm}(2/\mathfrak{c})^{-1}\prod_{\mathfrak{p}|\mathfrak{c}}(1-\operatorname{Nm}(\mathfrak{p})^{-1}),$
(20)

where ${\mathfrak{c}}$ runs over integral ideals of ${\mathcal{O}}_{K}$ and
${\mathfrak{p}}$ runs over prime ideals of ${\mathcal{O}}_{K}$. Combining
(18), (19), (20) and Lemma 4.0.1 and choosing $T=X^{1/4}$, the conclusion
follows.

Note that if we assume the Lindelöf hypothesis, then using the observation
following (19), it follows that we can choose $T=X^{3/4}$ and then the
improved bound in the second part of Theorem 1 follows.

######  Lemma 4.0.1.

For a number field $E$,

$\sum_{\mathfrak{c}|2{\mathcal{O}}_{E}}\operatorname{Nm}(2/\mathfrak{c})^{-1}\prod_{\mathfrak{p}|\mathfrak{c}}(1-\operatorname{Nm}(\mathfrak{p})^{-1})=1,$

where ${\mathfrak{c}}$ runs over integral ideals of ${\mathcal{O}}_{E}$ and
${\mathfrak{p}}$ runs overs prime ideals of ${\mathcal{O}}_{E}$.

###### Proof.

Following §3.3 of [18], the above formula can be proven by writing

$\displaystyle\sum_{\mathfrak{c}|2}$
$\displaystyle\operatorname{Nm}(2/\mathfrak{c})^{-1}\prod_{\mathfrak{p}|\mathfrak{c}}(1-\operatorname{Nm}(\mathfrak{p})^{-1})=\frac{1}{\operatorname{Nm}(2)}\sum_{\mathfrak{c}|2}\operatorname{Nm}(\mathfrak{c})\prod_{\mathfrak{p}|\mathfrak{c}}(1-\operatorname{Nm}(\mathfrak{p})^{-1})$
$\displaystyle=\frac{1}{\operatorname{Nm}(2)}\prod_{\mathfrak{p}|2}\left(1+\sum_{j=1}^{v_{\mathfrak{p}}(2)}\operatorname{Nm}(\mathfrak{p}^{j})(1-\operatorname{Nm}(\mathfrak{p})^{-1})\right),$

where $v_{\mathfrak{p}}(2)$ is the valuation of $2$ at $\mathfrak{p}$. The
previous formula simplifies to

$\displaystyle\sum_{\mathfrak{c}|2}$
$\displaystyle\operatorname{Nm}(2/\mathfrak{c})^{-1}\prod_{\mathfrak{p}|\mathfrak{c}}(1-\operatorname{Nm}(\mathfrak{p})^{-1})=\frac{1}{\operatorname{Nm}(2)}\prod_{\mathfrak{p}|2}\operatorname{Nm}(\mathfrak{p})^{v_{\mathfrak{p}}(2)}=1.$

∎

##  5 Acknowledgements

We would like to thank Brandon Alberts, Henri Cohen, Erik Holmes, Jordan
Ellenberg, Dick Gross, Kiran S. Kedlaya, Robert Lemke Oliver, Arul Shankar,
Jiuya Wang, Melanie Matchett Wood and the referee for helpful discussions. We
would also like to thank Wei Ho and Renate Scheidler, the WIN5 organizers, for
providing the impetus for this project.

## References

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  * [2] Brandon Alberts  “The weak form of Malle’s conjecture and solvable groups”  In _Res. Number Theory_ 6.1, 2020, pp. Paper No. 1023  DOI: 10.1007/s40993-019-0185-7
  * [3] Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang and Melanie Matchett Wood Preprint (2022)
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