# Notes on Mitsui’s Prime Number Theorem with Siegel zeros

Wataru Kai  Mathematical Institute, Tohoku University, Aoba 6-3, Sendai
980-8578, Japan<EMAIL_ADDRESS>
###### Abstract.

In these notes, we refine Mitsui’s Prime Number Theorem from 1957, which for a
number field $K$ predicts how many prime elements there are in bounded convex
sets in $K\otimes_{\mathbf{Q}}\mathbf{R}$, by incorporating potential Siegel
zeros of Hecke $L$-functions. This allows the norm of the modulus
$\mathbf{N}(\mathfrak{q})$ to grow at a pseudopolynomial rate $e^{\sqrt{\log
X}/O(1)}$ with respect to the size $X$ of the convex set as opposed to powers
of $\log X$. The extra flexibility and precision will be essential in our
future application to the study of linear patterns of prime elements. We also
hope that our updated exposition will make Mitsui’s work accessible to a wider
mathematical audience.

###### Key words and phrases:

prime elements, number fields, prime number theorem

###### 1991 Mathematics Subject Classification:

11R45, 11N32

###### Contents

  1. 1 Introduction
  2. 2 Recollection of Prime Ideal Theorem
  3. 3 The statement and first steps of the proof
  4. 4 Main theorem for thin cones—Fourier analysis
  5. 5 End of proof
  6. A Conventions on $K\otimes_{\mathbf{Q}}\mathbf{R}$
  7. B Fourier analysis on a torus
  8. C Idèle class group as a torus
  9. D Euclid’s algorithm

##  1\. Introduction

###  1.1. The Siegel–Walfisz prime number theorem

Prime Number Theorem states in one of the several equivalent forms:

(1.1) $\displaystyle\sum_{0<p<X}\log(p)=X+o(X)\quad\text{ as }X\to+\infty.$

Here the sum $\sum_{0<p<X}$ is the sum over prime numbers $p$ satisfying
$0<p<X$.

A little more generally, the Siegel–Walfisz prime number theorem says for any
modulus $1<q<\log^{A}X$ ($A\geq 1$ is a prescribed constant) and
$a\in(\mathbf{Z}/q\mathbf{Z})^{\times}$, we have the estimate

(1.2) $\displaystyle\sum_{\begin{subarray}{c}0<p<X\\\ p\equiv a\
\mathrm{mod}\,q\end{subarray}}\log(p)=\frac{1}{\varphi(q)}X+O_{A}(Xe^{-\sqrt{\log
X}/O_{A}(1)}).$

Formulated this way, the restriction $q<\log^{A}X$ cannot be improved by
today’s human technology because the contemporary analytic number theory has
not been able to exclude the existence of notorious Siegel zeros of Dirichlet
$L$-functions. Siegel zeros are potential real zeros of Dirichlet
$L$-functions slightly smaller than $1$, which contradicts the Grand Riemann
Hypothesis. This concept will be reviewed in §2.

Indeed, a Siegel zero $0<\beta<1$ modulo $q$ would give rise to a second main
term $X^{\beta}/\beta\varphi(q)$ to (1.2). While this can be absorbed in the
error term if we stay in the realm $q<\log^{A}X$, we have to take it seriously
if we want to go beyond this bound. Indeed, by incorporating the term
$X^{\beta}/\beta\varphi(q)$ into the picture, the modulus $q$ is allowed to be
at least as large as $e^{\sqrt{\log X}/O(1)}$. Namely, for all $1\leq
q<e^{\sqrt{\log X}/O(1)}$ we have

(1.3) $\displaystyle\sum_{\begin{subarray}{c}0<p<X\\\ p\equiv a\
\mathrm{mod}\,q\end{subarray}}\log(p)=\frac{1}{\varphi(q)}\left(X-\psi_{\mathrm{Siegel}}(a)\frac{X^{\beta}}{\beta}\right)+O(Xe^{-\sqrt{\log
X}/O(1)})$

where $\psi_{\mathrm{Siegel}}\colon(\mathbf{Z}/q\mathbf{Z})^{\times}\to\\{\pm
1\\}$ is the potential Siegel character mod $q$. See e.g. [6, Theorem 5.27 on
p. 122].

###  1.2. Mitsui’s theorem and our main result

The number field analogue of the Siegel–Walfisz theorem has already been
documented. It is known as Mitsui’s prime number theorem, which can be
formulated as follows:

######  Theorem 1.2.1 (Mitsui [11, Corollary and Theorem on p. 35], 1956).

Let $K$ be a number field of degree $n=[K:\mathbf{Q}]=\dim_{\mathbf{Q}}K$. Let
$A>1$ be a fixed positive number. Let $X>1$ and consider a convex open set
$C\subset K\otimes_{\mathbf{Q}}\mathbf{R}$ contained in the region
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$ of size $X$ (§A.4). Let
$\mathfrak{q}\subset\mathcal{O}_{K}$ be a non-zero ideal such that

(1.4) $\displaystyle\mathbf{N}(\mathfrak{q})<\log^{A}X$

and choose any $\alpha\in(\mathcal{O}_{K}/\mathfrak{q})^{\times}$. Then we
have

(1.5) $\displaystyle\sum_{\begin{subarray}{c}\pi\in C,\\\ \text{prime element,
}\\\ \pi\equiv\alpha\text{\ in
}\mathcal{O}_{K}/\mathfrak{q}\end{subarray}}\log|\mathbf{N}(\pi)|=\frac{w_{K}}{\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\operatorname{Vol}(C)+O_{K,A}(X^{n}e^{-\sqrt{\log
X}/O_{K,A}(1)}).$

Here, there is a canonical $\mathbf{R}$-linear isomorphism
$K\otimes_{\mathbf{Q}}\mathbf{R}\cong\mathbf{R}^{n}$ and
$\operatorname{Vol}(C)$ refers to the volume in the Lebesgue measure $\mu$ on
$\mathbf{R}^{n}$.

The main purpose of this text is to improve Mitsui’s theorem by incorporating
the secondary main term coming from potential Siegel zeros $\beta$ of
$L$-functions $L(\psi,s)$ associated with mod $\mathfrak{q}$ characters
$\psi$. To be specific, let $\psi_{\mathrm{Siegel}}$ be the potential Siegel
character mod $\mathfrak{q}$ and $\beta$ the Siegel zero (see §2.2 for these
notions). In our main Theorem 3.1.1, the right hand side of (1.5) will be
refined to the expression

(1.6)
$\frac{w_{K}}{\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int_{C}1-\psi_{\mathrm{Siegel}}(-;\alpha)N(-)^{\beta-1}d\mu\\\
+O_{K}(X^{n}e^{-\sqrt{\log X}/O_{K}(1)}).$

This modification allows us to relax the restrction (1.4) to a
pseudopolynomial growth

(1.7) $\displaystyle\mathbf{N}(\mathfrak{q})<e^{\sqrt{\log X}/O_{K}(1)}.$

This quantitative improvement will be essential in our application in the
forthcoming work,111Now available at https://arxiv.org/abs/2306.16983 where
based on the author’s former joint work [7] we extend results of
Green–Tao–Ziegler [3, 4] and Tao–Teräväinen [14] on linear patterns of prime
numbers to general number fields. To illustrate what it is about, consider
polynomials of degree $1$:

(1.8) $\displaystyle ax+by+c\in\mathcal{O}_{K}[x,y],$

where the coefficients $(a,b,c)$ vary in a given finite subset
$T\subset\mathcal{O}_{K}^{3}$. Under some conditions on $T$ including the
hypothesis that the vectors $(a,b)\in K^{2}$ are pairwise linearly
independent, we will be able to conclude that for any large enough convex body
$C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{2}$, there are points $(x,y)\in
C\cap\mathcal{O}_{K}^{2}$ such that the values of (1.8) are prime elements of
$\mathcal{O}_{K}$ simultaneously for all $(a,b,c)\in T$.

This will furthermore be applied to a Hasse principle type problem on
$K$-rational points on certain specific type of varieties as was done in [5]
by Harpaz–Skorobogatov–Wittenberg over $K=\mathbf{Q}$.

###  1.3. Remarks on the formulation

In Mitsui’s original account [11, Theorem on p. 35], the theorem is formulated
for sets $C$ of the following form

(1.9) $C=(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X_{1},\dots,X_{r_{1}+r_{2}}}\\\
:=\\{x\in K\otimes_{\mathbf{Q}}\mathbf{R}\mid|\sigma_{i}(x)|<X_{i}\
1\leq\forall i\leq r_{1}+r_{2}\\},$

where $X_{i}$ are positive numbers satisfying $X_{i}\leq X_{j}^{a}$ for all
$i,j$ for a prescribed constant $a\geq 1$. Also, the error term in his
formulation is

$O_{K,A,a}(Ne^{-\sqrt{\log N}/O_{K,A,a}(1)}),$

where $N$ is the supremum of norms in the region
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X_{1},\dots,X_{r_{1}+r_{2}}}$. His
arguments can easily be adapted to give Theorem 1.2.1. It is not clear if the
theorem stays true for general convex bodies $C$ in
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X_{1},\dots,X_{r_{1}+r_{2}}}$.

We formulated our Theorem 3.1.1 for convex bodies $C$ in
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$ in parallel with Theorem 1.2.1
because our intended application requires this case. It can also be shown for
$C=(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X_{1},\dots,X_{r_{1}+r_{2}}}$ with
essentially the same proof but we do not present the argument specifically
needed for this case.222Here is a hint for the interested reader. Note that
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X_{1},\dots,X_{r_{1}+r_{2}}}$ is an
annulus sector in the sense of Definition 5.1.2. Its logarithmic image in
$\mathbf{R}^{r_{1}+r_{2}}$ is a cuboid in which the ratios of the side lengths
are bounded by $a$. Then one runs the process of §5.

In addition to prime elements, his theorem in [11] contains statements about
how many prime ideal numbers there are—a concept which has something to do
with the fact that the number ring $\mathcal{O}_{K}$ is not necessarily a
unique factorization domain (UFD)—in the convex body $C$. Our result will also
be shown in this generality, replacing prime ideal numbers by prime elements
in ideals (§1.5.2). This generality will be useful for example when one wants
to find linear patterns of points in $\mathbf{Z}^{2}$ at which a given
quadratic form $ax^{2}+bxy+cy^{2}$ takes primes values; see [7, §10].

###  1.4. Outline of the method

The method is largely the same as Mitsui’s in [11]. On top of the
consideration of potential Siegel zeros, there are two expositional
improvements which we hope are making this text more accessible to the modern
reader. One is that we got rid of the Byzantine notion of ideal numbers (see
e.g. [13, p. 486] for this notion) and used exclusively the usual notion of
ideals to formulate and prove the results. The other is that Mitsui’s highly
unpenetrable seemingly ad hoc computations and estimates has mostly been
replaced by general discussions of idèles and Fourier analysis.333Actually
these two aspects have been preventing the author from being able to read
through his paper to this day.

Now we outline Mitsui’s [11] method. The case of quadratic fields is also
sketched in references like [6, p. 130] [9, p. 318].444 By the way it’s a
pitty that they don’t mention Mitsui’s work even though he pretty much
predates these books. Let us assume that there is no Siegel zeros because we
want to focus on the main ideas here.

First, from a known zero-free region of Hecke $L$-functions, we have the
following estimate of the weighted sum of prime ideals (Theorem 2.3.2). Below,
$\psi$ is an arbitrary mod $\mathfrak{q}$ Hecke character555 Let us not forget
that we are assuming there is no Siegel zero. Also, in the main body of the
text $\psi(\mathfrak{p})$ is written $\psi([\mathfrak{p}])$ for good reason.

(1.10) $\displaystyle\sum_{\begin{subarray}{c}\mathfrak{p}\\\ \text{prime to
}\mathfrak{q}\\\ \mathbf{N}(\mathfrak{p})<N\end{subarray}}\psi(\mathfrak{p})\
\log\mathbf{N}(\mathfrak{p})=\begin{cases}N+O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)})&\text{ if }\psi\equiv 1,\\\\[8.53581pt] O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)})&\text{ otherwise}.\end{cases}$

For simplicity let us further assume $\mathcal{O}_{K}$ is a UFD. If we choose
an $\mathcal{O}_{K}^{\times}$-fundamental domain
$\mathcal{D}\subset\mathcal{O}_{K}\setminus\\{0\\}$, prime ideals (now all
principal) correspond to unique prime elements in $\mathcal{D}$. Thus the left
hand side of (1.10) becomes a sum over prime elements $\pi\in\mathcal{D}$ with
$\mathbf{N}(\pi)<N$.

$\mathbf{Q}$$\mathbf{Q}\sqrt{2}$$\mathcal{D}$ Figure 1. A fundamental domain
for $K=\mathbf{Q}(\sqrt{2})$ truncated at some norm. The scale is adjusted so
that the point $x+y\sqrt{2}\in\mathbf{Q}(\sqrt{2})$ looks like
$(x,y\sqrt{2})\in\mathbf{R}^{2}$. The norm is
$\mathbf{N}(x+y\sqrt{2})=\left|x^{2}-(y\sqrt{2})^{2}\right|$. The hyperbolas
pass through points of the same norm. In this example we know
$(K\otimes_{\mathbf{Q}}\mathbf{R})/\mathcal{O}_{K}^{\times}\cong\mathbf{R}^{\times}_{>0}\times(\mathbf{R}/\mathbf{Z})$.

Though not strictly needed, there is a standard, geometric way of choosing
$\mathcal{D}$. For example when $K=\mathbf{Q}(\sqrt{2})$, an instance of
$\mathcal{D}$ is shown in Figure 1.

Mod $\mathfrak{q}$ Hecke characters are characters of the mod $\mathfrak{q}$
idèle class group $C(\mathfrak{q})$, which is a commutative Lie group. By
modding out a subgroup $\cong\mathbf{R}^{\times}_{>0}\cong\mathbf{R}$, we
obtain a compact commutative Lie group
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$. It is a finite cover of another
group

(1.11) $\displaystyle
C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}\twoheadrightarrow(K\otimes_{\mathbf{Q}}\mathbf{R})/(\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}),$

where the kernel is $(\mathcal{O}_{K}/\mathfrak{q})^{\times}$. Fourier
analysis (§B) can be applied to this torus. This allows us to count prime
elements in a specified mod $\mathfrak{q}$ class and inside thin cones as in
Figure 2. Such cones are written as $\mathbf{R}^{\times}_{>0}P$ in the main
body of the text. Thus we will obtain:666 Let us repeat that we are assuming
that there is no Siegel zero.

(1.12) $\displaystyle\sum_{\begin{subarray}{c}\pi\text{ prime element}\\\
\pi\equiv\alpha\ \mathrm{mod}\,\mathfrak{q}\\\
\pi\in\mathbf{R}^{\times}_{>0}P\\\
|\mathbf{N}(\pi)|<N\end{subarray}}\log|\mathbf{N}(\pi)|=\frac{\text{const.}}{\varphi_{K}(\mathfrak{q})}\
N+O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}),$

where the constant depends only on $K$ and $\mathbf{R}^{\times}_{>0}P$, and is
proportional to the volume of the set $\mathbf{R}^{\times}_{>0}P\cap\\{x\in
K\otimes_{\mathbf{Q}}\mathbf{R}\mid|\mathbf{N}(x)|<1\\}$. See Corollary 4.2.1.

$\mathbf{Q}$$\mathbf{Q}\sqrt{2}$$\mathcal{D}$$\mathbf{R}^{\times}_{>0}P$
Figure 2. Fourier analysis allows us to focus on prime elements inside thin
cones like this (colored dark gray).

In §5, we will use (1.12) to count prime elements in a given convex set $C$ by
approximating it with segments of thin cones (Figure 3). This approximation
can be done within the claimed error.

Management of the error, notably its independence from $\mathfrak{q}$,
requires somewhat careful and boring analysis, which is done in Appendices
A–D.

$C$$\mathbf{Q}$$\mathbf{Q}\sqrt{2}$ Figure 3. Approximation of a convex body
$C$ by segments of thin cones

###  1.5. General notation and terminology

####  1.5.1. Multiplicative groups of complex numbers

Here are some subgroups of $\mathbf{C}^{\times}$ that we frequently use in
this paper.

  * •

We write $\mathbf{R}^{\times}_{>0}$ for the multiplicative group of positive
real numbers. We know $\mathbf{R}^{\times}_{>0}\cong\mathbf{R}$ via the
logarithm.

  * •

We write $S^{1}\subset\mathbf{C}^{\times}$ for the subgroup of complex numbers
of absolute value $1$. It is canonically isomorphic to $\mathbf{R}/\mathbf{Z}$
by the map $\theta\ (\mathrm{mod}\,\mathbf{Z})\mapsto e^{2\pi i\theta}$.

  * •

We know

(1.13)
$\displaystyle\mathbf{R}^{\times}\cong\mathbf{R}^{\times}_{>0}\times\\{\pm
1\\}\cong\mathbf{R}\times\\{\pm 1\\}\text{ and }$ (1.14)
$\displaystyle\mathbf{C}^{\times}\cong\mathbf{R}^{\times}_{>0}\times
S^{1}\cong\mathbf{R}\times(\mathbf{R}/\mathbf{Z}),$

where we choose the projection
$\mathbf{C}^{\times}\to\mathbf{R}^{\times}_{>0}$ to be $z\mapsto|z|^{2}$.

####  1.5.2. Number rings

We write $\operatorname{Ideals}_{K}$ for the multiplicative monoid of non-zero
ideals of $\mathcal{O}_{K}$. We write $h_{K}:=\\#\operatorname{Cl}(K)$ for the
class number. For each class $\lambda\in\operatorname{Cl}(K)$, we fix a
representing fractional ideal $\mathfrak{a}_{\lambda}\subset K$. For the
trivial class $0\in\operatorname{Cl}(K)$, let us choose
$\mathfrak{a}_{0}:=\mathcal{O}_{K}$. Let
$\mathfrak{a}_{\lambda}^{-1}=\\{\alpha\in
K\mid\alpha\mathfrak{a}_{\lambda}\subset\mathcal{O}_{K}\\}$ be the inverse
fractional ideal. We have a bijection

(1.15) $\displaystyle\Bigl{\\{}(0)\neq\mathfrak{a}\subset\mathcal{O}_{K}\text{
ideals }\Bigm{|}[\mathfrak{a}]=\lambda\text{ in
}\operatorname{Cl}(K)\Bigr{\\}}$ $\displaystyle\ \cong\
(\mathfrak{a}_{\lambda}^{-1}\setminus\\{0\\})/\mathcal{O}_{K}^{\times}$ (1.16)
$\displaystyle\mathfrak{a}=a\mathfrak{a}_{\lambda}$ $\displaystyle\
\mathrel{\reflectbox{$\mapsto$}}\ a.$

It follows that if we are given an $\mathcal{O}_{K}^{\times}$-fundamental
domain $\mathcal{D}$ for $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$, there
is a bijection

(1.17)
$\displaystyle\operatorname{Ideals}_{K}\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\mathfrak{a}_{\lambda}^{-1}\cap\mathcal{D}.$

Let us call an element $\pi\in\mathfrak{a}$ of a non-zero fractional ideal a
prime element of $\mathfrak{a}$ if $\pi\mathfrak{a}^{-1}$ is a non-zero prime
ideal (where $\mathfrak{a}^{-1}$ is the inverse fractional ideal of
$\mathfrak{a}$).777 An algebro-geometric justification for this piece of
terminology is that for any integral scheme $X$ and a line bundle
$\mathcal{L}$ on it, we could call a non-zero section
$s\colon\mathcal{O}_{X}\to\mathcal{L}$ prime if its zero locus $(s)\subset X$
is an integral scheme. Denote by $\mathcal{P}(\mathfrak{a})$ the set of prime
elements of $\mathfrak{a}$. In this paper, sums and products
$\bigoplus_{\mathfrak{p}},\sum_{\mathfrak{p}},\prod_{\mathfrak{p}}$ indexed by
the letter $\mathfrak{p}$ will always mean those indexed by non-zero prime
ideals, often subject to some additional conditions. Likewise, sums and
products $\bigoplus_{\pi},\sum_{\pi},\prod_{\pi}$ indexed by $\pi$ will mean
those indexed by prime elements.

Euler’s totient function $\varphi_{K}$ for $K$ is defined as the function
$\mathfrak{a}\mapsto\\#(\mathcal{O}_{K}/\mathfrak{a})^{\times}$,
$\operatorname{Ideals}_{K}\to\mathbf{N}$.

####  1.5.3. Landau’s $O(-)$ symbol

The symbol $O_{a,b,c}(1)$ denotes a complex value whose absolute value is
bounded from above by a constant $C_{a,b,c}>0$ depending only on the
parameters $a,b,c$. The specific value can be different from occasion to
occasion. In particular, an $O(1)$ means a complex value whose absolute value
is bounded by an absolute constant independent of any parameter even if some
parameters are at play.

When $f(t)$ is a positively valued function of $t$, the expression
$O_{a,b,c}(f(t))$ means $O_{a,b,c}(1)\cdot f(t)$. Often $t$ will be a positive
real parameter, and the bound may be valid only when $t$ is large enough. We
allow this threshhold to depend on the parameters $a,b,c$. We shall indicate
it if the threshhold depends on more (or less) parameters.

We make an exception of allowing ourselves to use $O_{K}(-)$ even when the
bounds depend on additional choices related to $K$, as long as those choices
are independent of the problem at hand and can be made once the number field
$K$ is given. Such choices include an isomoprhism
$\mathcal{O}_{K}^{\times}\cong(\mathbf{Z}/w\mathbf{Z})\times\mathbf{Z}^{r}$
and representatives $\mathfrak{a}_{\lambda}$
($\lambda\in\operatorname{Cl}(K)$) for the ideal class group.

The expression $f(t)\ll_{a,b,c}g(t)$ means $f(t)=O_{a,b,c}(g(t))$. The
expression $f(t)\asymp_{a,b,c}g(t)$ means that $f(t)\ll_{a,b,c}g(t)$ and
$g(t)\ll_{a,b,c}f(t)$ both hold.

It goes without saying that a statement of the form

  * •

If $A=B/O_{a,b,c}(1)$, then $X=O_{d,e,f}(Y)$, or

  * •

If $A\ll_{a,b,c}B$, then $X\ll_{d,e,f}Y$

should be read as the statement that there are large enough positive constants
$C_{a,b,c}>1$ and $C_{d,e,f}>1$ depending only on the parameters indicated in
subscript such that the following holds:

> If $|A|<B/C_{a,b,c}$, then $|X|<C_{d,e,f}Y$.

Observe that the larger the constants $C_{a,b,c},C_{d,e,f}$ are, the more
restrictive the hypothesis is and the weaker the conclusion. So the statement
gets more likely to hold.

####  1.5.4. Usage of some letters

The letter $X$ is often used to indicate the length which characterizes the
region in $K\otimes_{\mathbf{Q}}\mathbf{R}$ in which we search for prime
elements.

The letter $N$ is used to bound the norm of elements. It is often set to be
$N:=X^{n}$, where $n=[K:\mathbf{Q}]$.

The letters $M,Y>1$ will be used to denote generic positive parameters which
grow at a pseudopolynomial rate $e^{\sqrt{\log N}/O_{K}(1)}$. The specific
value of $O_{K}(1)$ may vary from occasion to occasion.

### Acknowledgments

I thank Federico Binda, Hiroyasu Miyazaki and Rin Sugiyama for fruitful on-
line and in-person discussions as well as feedback to earlier versions of this
paper. I was supported by Japan Society for the Promotion of Science (JSPS)
through JSPS Grant-in-Aid for Young Scientists (numbers JP18K13382 and
JP22K13886). This work was done while I was staying at the University of Milan
as a JSPS Overseas Research Fellow. I thank my colleagues there for the
pleasant working environment.

##  2\. Recollection of Prime Ideal Theorem

Let us recall the notion of Hecke characters, the associated $L$-functions,
their known zero-free regions and the Prime Ideal Theorem 2.3.2 that results.

###  2.1. Hecke characters

We denote the group of idèles by $\mathbf{I}_{K}$. A Hecke character is a
continuous homomorphism

(2.1) $\displaystyle\psi\colon\mathbf{I}_{K}/K^{\times}\to S^{1}.$

In the literature it is also called a Größencharakter. Every such $\psi$
factors through a quotient of the following form for some non-zero ideal
$\mathfrak{q}=\prod_{\mathfrak{p}\text{ finitely
many}}\mathfrak{p}^{n_{\mathfrak{p}}}$:

(2.2) $\displaystyle\psi\colon\mathbf{I}_{K}/K^{\times}$
$\displaystyle\twoheadrightarrow\left((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times\bigoplus_{\mathfrak{p}|\mathfrak{q}}K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\times\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}\right)/K^{\times}$
(2.3) $\displaystyle\longrightarrow S^{1}.$

In this case we say $\psi$ is a mod $\mathfrak{q}$ Hecke character. It is
called a primitive mod $\mathfrak{q}$ Hecke character if it is not a mod
$\mathfrak{q}^{\prime}$ Hecke character for any other divisor
$\mathfrak{q}^{\prime}|\mathfrak{q}$, $\mathfrak{q}^{\prime}\neq\mathfrak{q}$.
In this case $\mathfrak{q}$ is called the conductor of $\psi$. See [1, XV-§3],
[9, VII-§3 and XVI-§7] or [13, VI-§1 and VII-§6] for more background.

Let us write this quotient (the so-called idèle class group mod
$\mathfrak{q}$) in the middle as

(2.4) $\displaystyle
C(\mathfrak{q})=\widetilde{C}(\mathfrak{q})/K^{\times}:=\left((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times\bigoplus_{\mathfrak{p}|\mathfrak{q}}K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\times\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}\middle)\right/K^{\times}.$

###  2.2. The $L$-function

Let $\psi$ be a Hecke character with conductor $\mathfrak{q}$. For prime
ideals $\mathfrak{p}\nmid\mathfrak{q}$, let
$[\mathfrak{p}]\in\widetilde{C}(\mathfrak{q})$ be the element whose
$\mathfrak{p}$-component is $1$ and the other components are trivial.
Sometimes the symbol $[\mathfrak{p}]$ is used to mean its image to
$\bigoplus_{\mathfrak{p}^{\prime}\nmid\mathfrak{q}}\mathbf{Z}$ as well.
Multiplicatively we can define $[\mathfrak{a}]\in\widetilde{C}(\mathfrak{q})$
or $\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}$ for every ideal
$\mathfrak{a}$ prime to $\mathfrak{q}$.

The Hecke $L$-funtion associated to $\psi$ is defined by (recall
$\mathfrak{q}$ is the conductor)

(2.5) $\displaystyle L(s,\psi)$ $\displaystyle:=\sum_{\mathfrak{a}\text{ ideal
prime to
}\mathfrak{q}}\frac{\psi([\mathfrak{a}])}{\mathbf{N}(\mathfrak{a})^{s}}$ (2.6)
$\displaystyle=\prod_{\mathfrak{p}\nmid\mathfrak{q}}\frac{1}{1-\frac{\psi([\mathfrak{p}])}{\mathbf{N}(\mathfrak{p})^{s}}}$

for $\Re(s)>1$. $L(s,\psi)$ is known to be a meromorphic function on
$\mathbf{C}$ (see e.g. [1, Main Theorem 4.4.1 and p. 346] [9, Theorem 12 on p.
295 and p. 299] or [13, VII-§8 (8.6)]).

###  2.3. The zero-free region and prime ideal theorem

A classical zero-free region is known:

######  Theorem 2.3.1 (zero-free region).

There is a positive constant $0<c_{K}<1$ such that the following holds for
every non-zero ideal $\mathfrak{q}\subset\mathcal{O}_{K}$.

  1. (1)

For all but possibly one mod $\mathfrak{q}$ Hecke character $\psi$ (not
necessarily primitive), the $L$-function $L(s,\psi)$ does not have zeros or
poles in the region (where $s=\sigma+it$ as usual)

(2.7)
$\displaystyle\sigma>1-\frac{c_{K}}{\log(\mathbf{N}(\mathfrak{q})(|t|+4))}.$

  2. (2)

The potential exceptional $\psi$, if it exists, has to be real (i.e., takes
values in $\\{-1,+1\\}$). This $\psi$ is called the Siegel (or exceptional)
character mod $\mathfrak{q}$ and denoted by $\psi_{\mathrm{Siegel}}$.

  3. (3)

The $L$-function $L(\psi_{\mathrm{Siegel}},s)$ can have at most one zero
$\beta$ and the zero has to be real:

(2.8) $\displaystyle 1-\frac{c_{K}}{\log(4\mathbf{N}(\mathfrak{q}))}<\beta<1.$

The zero $\beta$ is called the Siegel (or exceptional) zero mod
$\mathfrak{q}$.

###### Proof.

See e.g. [6, Theorem 5.35] or [8, Lemma 2.3], and [12, Theorem 11.7] or [15,
Theorem 1.9]. ∎

It is known that an $L$-function with a zero-free region as in Theorem 2.3.1
is accompanied by a Prime Ideal Theorem where the prime ideals are counted
with weights. To us, the pseudopolynomial control of the error term in the
following theorem is important.

######  Theorem 2.3.2 (Prime Ideal Theorem).

Let $N>1$ be a positive number and $\mathfrak{q}\in\operatorname{Ideals}_{K}$
be an ideal with norm

(2.9) $\displaystyle\mathbf{N}(\mathfrak{q})<e^{\sqrt{\log N}/O_{K}(1)}.$

Let $\psi$ be a mod $\mathfrak{q}$ Hecke character. Then we have

(2.10) $\displaystyle\sum_{\begin{subarray}{c}\mathfrak{p}\\\
\mathfrak{p}\nmid\mathfrak{q},\
\mathbf{N}(\mathfrak{p})<N\end{subarray}}\psi([\mathfrak{p}])\log\mathbf{N}(\mathfrak{p})=N\cdot
1_{\psi\equiv
1}-\frac{N^{\beta}}{\beta}1_{\psi\mathrm{:Siegel}}+O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)}),$

where $\beta$ is the potential Siegel zero mod $\mathfrak{q}$.

The terms $1_{[\cdots]}$ are present only when the condition in the subscript
is true and equal the constant function $1$ in such cases.

###### Proof.

See e.g. [6, Theorem 5.13]. ∎

##  3\. The statement and first steps of the proof

Before stating our main result, we have to introduce a few pieces of notation.
Let $\mathfrak{a},\mathfrak{q}\subset\mathcal{O}_{K}$ be non-zero fractional
ideals. Write
$(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}\subset\mathfrak{a}/\mathfrak{q}\mathfrak{a}$
for the set of residue classes which can be a single generator of
$\mathfrak{a}/\mathfrak{q}\mathfrak{a}$ as an $\mathcal{O}_{K}$-module. By
Proposition C.2.4, we have a canonical set map
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}\to
C(\mathfrak{q})$. Thus for any function $\psi$ on $C(\mathfrak{q})$, we may
consider its pullback to
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$.
We write this restriction as $(x,\alpha)\mapsto\psi(x;\alpha)$. In particular
if we fix a residue class
$\alpha\in(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$, we obtain a
function

(3.1)
$\displaystyle\psi(-;\alpha)\colon(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\to\mathbf{C}.$

###  3.1. The statement

######  Theorem 3.1.1.

Let $K$ be a number field of degree $n=[K:\mathbf{Q}]=\dim_{\mathbf{Q}}K$. Fix
a non-zero fractional ideal $\mathfrak{a}\subset\mathcal{O}_{K}$.

Let $X>1$ be a real number. Set $N:=X^{n}$.

Let
$C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X\mathbf{N}(\mathfrak{a})^{1/n}}$
be a convex open set and $\mathfrak{q}$ be an ideal whose norm obeys a bound

(3.2) $\displaystyle\mathbf{N}(\mathfrak{q})<e^{\sqrt{\log N}/O_{K}(1)}$

with a large enough constant $O_{K}(1)>1$. Let
$\alpha\in(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$. Then we have

(3.3) $\sum_{\begin{subarray}{c}\pi\in C\cap\mathfrak{a},\\\ \pi=\alpha\text{
in
}\mathfrak{a}/\mathfrak{q}\mathfrak{a}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}^{-1})=\\\\[10.0pt]
\frac{1}{\mathbf{N}(\mathfrak{a})\varphi_{K}(\mathfrak{q})}\frac{w_{K}}{2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int_{C}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}^{-1})^{\beta-1}d\mu_{\mathrm{add}}\\\\[10.0pt]
+O_{K}\left(Ne^{-\sqrt{\log N}/O_{K}(1)}\right),$

where by convention we ignore the term
$\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}^{-1})^{\beta-1}$
if there is no Siegel zero mod $\mathfrak{q}$; in this case the integral is
$\operatorname{Vol}_{\mathrm{add}}(C)$.

Our convention for the measure
$\mu_{\mathrm{add}},\operatorname{Vol}_{\mathrm{add}}$ is in §A.5.

Though this is not needed in the sequel, the coefficient
${w_{K}}/(2^{r_{1}}\pi^{r_{2}}h_{K}R_{K})$ is equal to
${2^{r_{2}}}/(\operatorname*{res}_{s=1}(\zeta_{K}(s))\sqrt{|D_{K}|})$ by Class
Number Formula [9, Theorem 5 on p. 161], where $D_{K}$ is the discriminant of
$K$.

Requirements for the $O_{K}(1)$ constant in (3.2) will be specified in
§§4.1.2, 5.2.5.

###  3.2. First reduction

Here we show that in Theorem 3.1.1 we may assume $\mathfrak{a}$ equals the
pre-chosen representative $\mathfrak{a}_{\lambda}$ of its ideal class
$\lambda$ without loss of generality. This will also allow us to simplify the
range of the convex set $C$ to
$C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$, omitting
$\mathbf{N}(\mathfrak{a}_{\lambda})^{1/n}$, because
$\mathbf{N}(\mathfrak{a}_{\lambda})$ is an $O_{K}(1)$ when representatives
$\mathfrak{a}_{\lambda}$ are fixed beforehand. The reader who is not
interested in Theorem 3.1.1 for a general fractional ideal $\mathfrak{a}$ may
of course skip over this step to §3.3.

By the bijection (1.15)

(3.4)
$\displaystyle\operatorname{Ideals}_{K}\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}(\mathfrak{a}_{\lambda}^{-1}\setminus\\{0\\})/\mathcal{O}_{K}^{\times}$

there are an index $\lambda$ and an element
$a_{0}\in\mathfrak{a}_{\lambda}^{-1}$ such that
$\mathfrak{a}=a_{0}\mathfrak{a}_{\lambda}$. We have

(3.5)
$\displaystyle\mathbf{N}(\mathfrak{a})=\mathbf{N}(a_{0})\mathbf{N}(\mathfrak{a}_{\lambda}).$

We argue that Theorem 3.1.1 for $\mathfrak{a}$ follows from the case of
$\mathfrak{a}_{\lambda}$ if we multiply some parameters by $a_{0}^{-1}\in
K^{\times}\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$.

####  3.2.1. The left hand side

Via the correspondence of prime elements
$\mathcal{P}(\mathfrak{a})\xrightarrow{\cong}\mathcal{P}(\mathfrak{a}_{\lambda})$,
$\pi\mapsto a_{0}^{-1}\pi=:\pi^{\prime}$, the left hand sides of (3.3) for
$\mathfrak{a}$ and $\mathfrak{a}_{\lambda}$ are equal:

(3.6) $\displaystyle\sum_{\begin{subarray}{c}\pi\in C\cap\mathfrak{a},\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}/\mathfrak{q}\mathfrak{a}\end{subarray}}\log
N(\pi\mathfrak{a}^{-1})=\sum_{\begin{subarray}{c}\pi^{\prime}\in
a_{0}^{-1}C\cap\mathfrak{a}_{\lambda},\\\ \pi^{\prime}\equiv
a_{0}^{-1}\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log
N(\pi^{\prime}\mathfrak{a}_{\lambda}^{-1}).$

There is a constant $A=O_{K}(1)$ independent888 The independence of $A$ from
$\mathfrak{a}$ is not needed if one is fine with the dependence on
$\mathfrak{a}$ of the constants in the error term in the theorem. But this
independence plays a role in our future work.  of $\mathfrak{a}$ such that
$a_{0}$ can be taken from
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<A\mathbf{N}(\mathfrak{a}\mathfrak{a}_{\lambda}^{-1})^{1/n}}$,
see e.g. [7, Lemma 4.11] or [10, Lemma 4.2], the latter of which is stated for
principal ideals but whose proof is valid for the general case if
appropriately read. It follows that
$a_{0}^{-1}C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<A^{\prime}X\mathbf{N}(\mathfrak{a}_{\lambda})^{1/n}}$
for another $A^{\prime}={O_{K}(1)}$. Write $X^{\prime}:=A^{\prime}X$ and

(3.7) $\displaystyle N^{\prime}:=(X^{\prime})^{n}\asymp_{K}N.$

####  3.2.2. The right hand side

On the right hand sides, since the Jacobian of the multiplication map
$[a_{0}]\colon a_{0}^{-1}C\to C$ is $\mathbf{N}(a_{0})$ we have the
compatibility of measures
$[a_{0}]^{*}d\mu_{\mathrm{add}}=N(a_{0})d\mu_{\mathrm{add}}$.

We can compute the pullback of the norm function as
$[a_{0}]^{*}N((-)\mathfrak{a}^{-1})=N((-)\mathfrak{a}_{\lambda}^{-1})$.

We have
$a_{0}^{-1}\alpha\in(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}$.
By the commutative diagram

(3.14)

we have $[a_{0}]^{*}\psi(-;\alpha)=\psi(-;a_{0}^{-1}\alpha)$.

It follows that

(3.15)
$\displaystyle\int_{C}1-\psi_{\mathrm{Siegel}}(-;\alpha)N((-)\mathfrak{a}^{-1})^{\beta-1}d\mu_{\mathrm{add}}$
(3.16) $\displaystyle=\mathbf{N}(a_{0})$
$\displaystyle\int_{a_{0}^{-1}C}1-\psi_{\mathrm{Siegel}}(-;a_{0}^{-1}\alpha)N((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu_{\mathrm{add}}.$

####  3.2.3. End of reduction to $\mathfrak{a}_{\lambda}$

Now if we assume the validity of Theorem 3.1.1 for $\mathfrak{a}_{\lambda}$
and the convex body
$a_{0}^{-1}C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X^{\prime}\mathbf{N}(\mathfrak{a}_{\lambda})^{1/n}}$,
we have

(3.17) $\sum_{\begin{subarray}{c}\pi^{\prime}\in
a_{0}^{-1}C\cap\mathfrak{a}_{\lambda},\\\ \pi^{\prime}\equiv
a_{0}^{-1}\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log
N(\pi^{\prime}\mathfrak{a}_{\lambda}^{-1})=\\\
\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int_{a_{0}^{-1}C}1-\psi_{\mathrm{Siegel}}(-;a_{0}^{-1}\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu_{\mathrm{add}}\\\\[10.0pt]
+O_{K}(N^{\prime}e^{-\sqrt{\log N^{\prime}}/O_{K}(1)}).$

By (3.5) (3.6) (3.7) (3.15), this is equivalent to Theorem 3.1.1 for
$\mathfrak{a}$ and
$C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X\mathbf{N}(\mathfrak{a})^{1/n}}$.

This reduces the problem to the case of $\mathfrak{a}=\mathfrak{a}_{\lambda}$
for some $\lambda\in\operatorname{Cl}(K)$.

###  3.3. From prime ideals to prime elements

We already know how many prime ideals there are within bounded norms (Theorem
2.3.2). Here is the first step to deduce information on the number of prime
elements in bounded regions.

Let the situation be as in Theorem 3.1.1 and let $\psi$ be a mod
$\mathfrak{q}$ Hecke character.

####  3.3.1. Prime ideals and prime elements

Let $\mathcal{D}\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ be an
arbitrary $\mathcal{O}_{K}^{\times}$-fundamental domain.

We temporarily use the bijection (1.17) in the form

(3.18)
$\displaystyle\operatorname{Ideals}_{K}\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\mathfrak{a}_{\lambda}\cap\mathcal{D}.$

Namely we use the complete set of representatives
$\\{\mathfrak{a}_{\lambda}^{-1}\\}_{\lambda}$ as opposed to
$\\{\mathfrak{a}_{\lambda}\\}_{\lambda}$. It follows that for every
$\mathfrak{p}$, there are a unique $\lambda\in\operatorname{Cl}(K)$ and a
$\pi\in\mathfrak{a}_{\lambda}\cap\mathcal{D}$ such that
$\mathfrak{p}=\pi\mathfrak{a}_{\lambda}^{-1}$. In this situation $\pi$ is a
prime element of $\mathfrak{a}_{\lambda}$ by the very definition (§1.5.2) of
this notion. This establishes the bijection

(3.19) $\displaystyle\\{\mathfrak{p}\subset\mathcal{O}_{K}\mid\text{ non-zero
prime ideals
}\\}\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\mathcal{P}(\mathfrak{a}_{\lambda})\cap\mathcal{D}.$

####  3.3.2. To subtract a diagonal image

Now let $\mathfrak{p}$ be a non-zero prime ideal not dividing $\mathfrak{q}$
and $\pi\in\mathfrak{a}_{\lambda}\cap\mathcal{D}$ the corresponding prime
element. Let us write
$\operatorname{\mathbf{diag}}(\pi)\in\widetilde{C}(\mathfrak{q})$ for the
canonical diagonal image

(3.20)
$\displaystyle\operatorname{\mathbf{diag}}(\pi)\in\widetilde{C}(\mathfrak{q})=(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times\bigoplus_{\mathfrak{p}_{1}|\mathfrak{q}}K^{\times}/(1+\mathfrak{p}_{1}^{n_{\mathfrak{p}_{1}}}\mathcal{O}_{K,\mathfrak{p}_{1}})\times\bigoplus_{\mathfrak{p}_{2}\nmid\mathfrak{q}}\mathbf{Z}.$

We know $v_{\mathfrak{p}_{2}}(\pi)=0$ for $\mathfrak{p}_{2}\nmid\mathfrak{q}$
different from $\mathfrak{p}$. This implies that
$\operatorname{\mathbf{diag}}(\pi)=(\pi;\operatorname{diag}(\pi);[\mathfrak{p}])$,
where
$\operatorname{diag}(\pi)\in\bigoplus_{\mathfrak{p}_{1}\mid\mathfrak{q}}K^{\times}/(1+\mathfrak{p}_{1}^{n_{\mathfrak{p}_{1}}}\mathcal{O}_{K,\mathfrak{p}_{1}})$
is also the diagonal image, which we will simply write as $\pi$ in the sequel.
Since the Hecke character $\psi$ kills the image of $K^{\times}$, we know

(3.21) $\displaystyle\psi([\mathfrak{p}])$
$\displaystyle=\psi([\mathfrak{p}]\operatorname{\mathbf{diag}}(\pi)^{-1})$
(3.22) $\displaystyle=\psi(\pi^{-1};\pi^{-1};0)$ (3.23)
$\displaystyle=\overline{\psi}(\pi;\pi;0).$

By changing the notation $\psi\leftrightarrow\overline{\psi}$, from Theorem
2.3.2 we deduce:

######  Corollary 3.3.3.

Under the notation of Theorem 2.3.2 and this §3.3, we have

(3.24)
$\displaystyle\sum_{\lambda\in\operatorname{Cl}(K)}\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap\mathcal{D},\\\
\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})<N\end{subarray}}\psi(\pi;\pi;0)\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})$
(3.25) $\displaystyle=$ $\displaystyle N\cdot 1_{\psi\equiv
1}-\frac{N^{\beta}}{\beta}1_{\psi\mathrm{:Siegel}}+O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)}).$

##  4\. Main theorem for thin cones—Fourier analysis

Here we will combine Fourier analysis recalled in §§B–C and Corollary 3.3.3 to
prove Theorem 3.1.1 for a special type of subsets (Corollary 4.2.3).

Let the situation be as in Theorem 3.1.1.

By Proposition C.2.4 we have a chart

(4.1) $\displaystyle C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$
$\displaystyle\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\frac{((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0})\times(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}}{\mathcal{O}_{K}^{\times}}.$

Let $\lambda\in\operatorname{Cl}(K)$ be an ideal class and let
$\alpha\in(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}$.

###  4.1. Thin cones

Let
$P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$
be a connected open set such that the map
$P\to(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/(\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0})$
is injective. There is an $\mathcal{O}_{K}^{\times}$-fundamental domain
$\mathcal{D}\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ containing
$\mathbf{R}^{\times}_{>0}P$. Fix any such $\mathcal{D}$.

It follows that $P\times\\{\alpha\\}\to
C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ is also injective. Write
$P\times\\{\alpha\\}\subset C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ for the
image as well. Write $(\mathbf{R}^{\times}_{>0}P)\times\\{\alpha\\}$ for the
corresponding subset of either
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}$
or $C(\mathfrak{q})$.

####  4.1.1. Applying Fourier analysis

Suppose further $P$ is convex and small enough, depending on
$\varphi_{K}(\mathfrak{q})$, to satisfy the hypothesis of Proposition C.2.2.
The term thin cones refers to subsets of the form $\mathbf{R}^{\times}_{>0}P$
for such a $P$. We then have a decomposition of the indicator function
$1_{P\times\\{\alpha\\}}$ on $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ as in
Proposition C.2.2:

(4.2) $\displaystyle
1_{P\times\\{\alpha\\}}=\sum_{\psi\in\Psi\subset(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}c_{\psi}\psi+G+H.$

Now we take the sum $\sum_{\psi\in\Psi}c_{\psi}$ of (3.24). Combined with
(4.2) we obtain the following

(4.3)
$\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap\mathcal{D},\\\
\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})<N,\\\
(\pi;\pi;0)\in(\mathbf{R}^{\times}_{>0}P)\times\\{\alpha\\}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})+G((\pi;\pi;0))+H((\pi;\pi;0))\\\
=\frac{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P\times\\{\alpha\\})}{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}\Bigl{(}N(1+O_{n}(1/M))\\\
-\frac{N^{\beta}}{\beta}(\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})+O_{n}(1/M))\Bigr{)}\\\
+O_{K}(\varphi_{K}(\mathfrak{q})Y^{n-1})O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}).$

Here, note that as $\psi_{\mathrm{Siegel}}$ takes values in the discrete set
$\\{\pm 1\\}$ and $P\times\\{\alpha\\}$ is connected, the value
$\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})\in\\{\pm 1\\}$ is well defined.

The parameters $Y,M>1$ are from Proposition C.2.2. Let us specify their
values.

####  4.1.2. Estimate of errors

First, we want that the term

(4.4) $\displaystyle
O_{K}(\varphi_{K}(\mathfrak{q})Y^{n-1})O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$

in (4.3) remain in the form $O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$. For this,
we declare that the $O_{K}(1)$ constant in (3.2) in the statement of Theorem
3.1.1 be large enough (note that
$\varphi_{K}(\mathfrak{q})<\mathbf{N}(\mathfrak{q})$) and that the parameter
$Y$ be an $e^{\sqrt{\log N}/O_{K}(1)}$ with $O_{K}(1)$ large enough. (The
threshholds for these choices of course depends on the $O_{K}(1)$ constant in
the exponent in the displayed formula.)

Having fixed $Y$, we take $M$ to be an $e^{\sqrt{\log N}/O_{K}(1)}$ with
$O_{K}(1)$ even larger so that $H((\pi;\pi;0))=O_{n}(M\frac{\log Y}{Y})$ will
be an $O_{K}(e^{-\sqrt{\log N}/O_{K}(1)})$; it follows then that in (4.3)

(4.5) $\displaystyle\sum_{\pi}H((\pi;\pi;0))=O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)})$

because the number of $\pi$’s is at most
$\\#\\{x\in\mathfrak{a}_{\lambda}\cap\mathcal{D}\mid\mathbf{N}(x\mathfrak{a}_{\lambda}^{-1})<N\\}=O_{K}(N)$.

The above choices allow us to conclude also

(4.6) $\displaystyle\sum_{\pi}G((\pi;\pi;0))=O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)}).$

As a result, for every $\alpha,P$ as in the beginning of §4.1.1 we get:

(4.7)
$\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap\mathbf{R}^{\times}_{>0}P,\\\
\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})<N,\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=\\\
\frac{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P\times\\{\alpha\\})}{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}\left(N-\frac{N^{\beta}}{\beta}\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})\right)+O_{K}(Ne^{-\sqrt{\log
N}/O_{K}(1)}).$

###  4.2. Slight change of notation

Let us rewrite the condition $\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})<N$ in
the left hand side of (4.7) as
$\mathbf{N}(\pi)<\mathbf{N}(\mathfrak{a}_{\lambda})N=:N_{1}$ and write this
$N_{1}$ as $N$ afresh. Also, we may consider this formula for two values
$N^{\prime}<N$ and take the difference. This gives:

######  Corollary 4.2.1.

Let $N^{\prime}<N$. Let $\mathfrak{q}$ be a non-zero ideal whose norm
satisfies

(4.8) $\displaystyle\mathbf{N}(\mathfrak{q})<e^{\sqrt{\log
N^{\prime}}/O_{K}(1)}.$

Let $\lambda\in\operatorname{Cl}(K)$,
$\alpha\in(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}$
be arbitrary, and
$P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$
be a convex connected open set small enough (depending on
$\varphi_{K}(\mathfrak{q})$) to satisfy the hypothesis of Proposition C.2.2.

Then we have

(4.9)
$\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap\mathbf{R}^{\times}_{>0}P,\\\
N^{\prime}<\mathbf{N}(\pi)<N,\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=\\\
\frac{1}{\mathbf{N}(\mathfrak{a}_{\lambda})}\frac{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P\times\\{\alpha\\})}{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}\left((N-N^{\prime})-\frac{\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})}{\beta\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}(N^{\beta}-N^{\prime\beta})\right)\\\
+O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}).$

####  4.2.2. Description as an integral

It will be convenient to note that we can describe the previous quantity as an
integral

(4.10)
$\displaystyle(N-N^{\prime})-\frac{\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})}{\beta\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}(N^{\beta}-N^{\prime\beta})=\int_{N^{\prime}}^{N}1-\frac{\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})}{\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}x^{\beta-1}dx.$

Recall the basis $\bm{u}_{1},\dots,\bm{u}_{r_{1}+r_{2}}$ and the associated
coordinates $\xi_{1},\dots,\xi_{r_{1}+r_{2}}$ of $\mathbf{R}^{r_{1}+r_{2}}$
from §A.5. The norm map
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\to\mathbf{R}$ looks like the
composite

(4.11)
$\displaystyle(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\cong\mathbf{R}^{r_{1}+r_{2}}\times(S^{1})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}\xrightarrow{\text{projection}}\mathbf{R}^{r_{1}+r_{2}}$
$\displaystyle\to\mathbf{R}$ (4.12)
$\displaystyle(\xi_{1},\dots,\xi_{r_{1}+r_{2}})$ $\displaystyle\mapsto
e^{\xi_{r_{1}+r_{2}}}.$

For the reader’s eyeballs’ comfort, write $\xi:=\xi_{r_{1}+r_{2}}$. By the
substitution $x=e^{\xi}$ we can compute the previous integral (multiplied by
the constant factor $\operatorname{Vol}_{\mathrm{mult}}(P)$) as

(4.13)
$\operatorname{Vol}_{\mathrm{mult}}(P)\int_{N^{\prime}}^{N}1-\frac{\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})}{\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}x^{\beta-1}dx\\\
=\int\limits_{[\log N^{\prime},\log N]\times
P}e^{\xi}-\frac{\psi_{\mathrm{Siegel}}(-;\alpha)}{\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}e^{\beta\xi}\
|d\xi d\xi_{1}\dots\xi_{r_{1}+r_{2}-1}|.$

See (A.16) for the definition of the subset $[\log N^{\prime},\log N]\times P$
of $\mathbf{R}\times H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}$.

By (A.20) noting $\mathbf{N}(-)=e^{\xi}$, this can be written as an integral
in $d\mu_{\mathrm{add}}$ (let us also switch to the additive notation
$[N^{\prime},N]\cdot P\subset K\otimes_{\mathbf{Q}}\mathbf{R}$ from §A.3):

(4.14)
$\displaystyle=2^{r_{2}}\sqrt{r_{1}+r_{2}}\int\limits_{[N^{\prime},N]\cdot
P}1-\frac{\psi_{\mathrm{Siegel}}(-;\alpha)}{\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}\mathbf{N}(-)^{\beta-1}d\mu_{\mathrm{add}}.$

Now we can rewrite the main terms in (4.9). By the choice of
$\mu_{\mathrm{id\grave{e}le}}$ we have
$\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P\times\\{\alpha\\})=\operatorname{Vol}_{\mathrm{mult}}(P)$.
We also know the value (C.24) of
$\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})$.
We conclude that the main terms of (4.9) are rewritten as

(4.15)
$\displaystyle\frac{1}{\mathbf{N}(\mathfrak{a}_{\lambda})}\frac{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P)}{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}\left((N-N^{\prime})-\frac{\psi_{\mathrm{Siegel}}(P\times\\{\alpha\\})}{\beta\mathbf{N}(\mathfrak{a}_{\lambda})^{\beta-1}}(N^{\beta}-N^{\prime\beta})\right)$
(4.16) $\displaystyle=$
$\displaystyle\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int\limits_{[N^{\prime},N]\cdot
P}1-\psi_{\mathrm{Siegel}}(-;\alpha)\left(\frac{\mathbf{N}(-)}{\mathbf{N}(\mathfrak{a}_{\lambda})}\right)^{\beta-1}d\mu_{\mathrm{add}}.$

Namely:

######  Corollary 4.2.3.

Let $1<N^{\prime}<N$ be positive numbers and $\mathfrak{q}$ be an ideal whose
norm obeys the bound:

(4.17) $\displaystyle\mathbf{N}(\mathfrak{q})<e^{\sqrt{\log
N^{\prime}}/O_{K}(1)}.$

Then Theorem 3.1.1 is true for subsets of the form $C=[N^{\prime},N]\cdot P$
where
$P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$
is an open set which maps injectively into
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}\mathcal{O}_{K}^{\times}$
and such that the image is a small convex set of size
$\ll_{n}\varphi(\mathfrak{q})^{-(n-2)}$ in the sense of Definition B.1.1.

Let us repeat that the definition of the set $[N^{\prime},N]\cdot P$ can be
found at (A.16).

##  5\. End of proof

Lastly, we will “integrate” the estimate (4.9) (4.15) in various small convex
subsets
$P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$
to prove Theorem 3.1.1. Some care must be taken to make sure that the bound of
the error will remain uniform in the ideal parameter $\mathfrak{q}$.

Let $Y>1$ be again an auxiliary paramater which will be set to be an
$e^{\sqrt{\log N}/O_{K}(1)}$.

###  5.1. Reduction to the case of annulus sectors

Let the notation be as in Theorem 3.1.1: in particlular let $X>1$ be a
positive number. Write $N:=X^{n}$. Let
$C\subset(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$ be a convex open set.

Let $\mathfrak{q}\subset\mathcal{O}_{K}$ be an ideal whose norm obeys the
upper bound (3.2) and let
$\alpha\in(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}$
for some chosen ideal class $\lambda\in\operatorname{Cl}(K)$.

####  5.1.1. Elements close to the axes

The number of elements of $\mathfrak{a}_{\lambda}$ in the region

(5.1) $\displaystyle\\{x\in(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}\mid\ $ some
of the coordinates (5.2) $\displaystyle x=(x_{1},\dots,x_{r_{1}+r_{2}})\in
K\otimes_{\mathbf{Q}}\mathbf{R}\cong\mathbf{R}^{r_{1}}\times\mathbf{C}^{r_{2}}$
(5.3) $\displaystyle\text{ has size }|x_{i}|<1\\}$

is $O_{K}(X^{n-1})=O_{K}(N/X)$. Hence the contribution of these points to the
sum
$\sum_{\pi}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})\leq\sum_{\pi}\log N$
is $O_{K}(N\frac{\log X}{X})$. Also, the contribution of this region to the
integral
$\int_{C}(1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})d\mu$
is $O_{n}(N/X)$.

Therefore the contribution from points in (5.1) can be ignored.

######  Definition 5.1.2.

An annulus sector of size $X/Y$ is a connected component of a subset of
$K\otimes_{\mathbf{Q}}\mathbf{R}$ of the following Baumkuchen-like shape,
determined by the choice of parameters $a_{1},\dots,a_{r_{1}+r_{2}}$,
$\theta_{1},\dots,\theta_{r_{2}}\in\mathbf{R}$:

(5.4) $\displaystyle B=$ $\displaystyle
B(a_{1},\dots,a_{r_{1}+r_{2}};\theta_{1},\dots,\theta_{r_{2}})$ (5.5)
$\displaystyle=$ $\displaystyle\\{x\in K\otimes_{\mathbf{Q}}\mathbf{R}\mid\
0<|x|_{\sigma_{i}}-a_{i}<X/Y\ \text{ for }\forall i,$ (5.6) $\displaystyle
0<\arg(|x|_{\sigma_{i}})-\theta_{i}<1/Y\pmod{2\pi\mathbf{Z}}\ \text{ for
}r_{1}+1\leq\forall i\leq r_{1}+r_{2}\\}.$

We want to approximate the convex body $C$ by the union of disjoint annulus
sectors of size $\leq X/Y$. First we cover
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$ with $O_{n}(Y^{n})$ many disjoint
annulus sectors $B_{j}$ of size $\leq X/Y$. Then let

(5.7) $\displaystyle B_{j}\quad(j\in J)$

be those annulus sectors contained in $C$ and not intersecting with the region
(5.1).

######  Lemma 5.1.3.

We have $\operatorname{Vol}(C\setminus\bigsqcup_{j\in J}B_{j})\ll_{n}N/Y$.
Consequently, there are $\ll_{K}N/Y$ points in $C\setminus\bigsqcup_{j\in
J}B_{j}$.

###### Proof.

Let $B_{j}$ ($j\in J^{\prime}$) be those annulus sectors intersecting with
$\partial C$ or the region (5.1). Since an annulus sector of size $\leq X/Y$
has diameter $O_{n}(X/Y)$, the volume covered by $B_{j}$’s ($j\in J^{\prime}$)
is $O_{n}(\frac{X}{Y}\cdot\bigl{(}\operatorname{Area}(\partial
C)+O_{n}(X^{n-1})\bigr{)})$. Because $\operatorname{Area}(\partial
C)=O_{n}(X^{n-1})$ (see e.g. [3, Lemma A.1]), this quantity is an
$O_{n}(X^{n}/Y)=O_{n}(N/Y)$. ∎

####  5.1.4. Annulus sectors inside $C$

By Lemma 5.1.3, we have

(5.8) $\displaystyle\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap
C,\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=\left(\sum_{j\in
J}\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap B_{j},\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})\right)+O_{n}((N\log
N)/Y).$

Assuming that the conclusion of Theorem 3.1.1 holds for $B_{j}$, $j\in J$
(with uniform $O_{K}(1)$ constants of course), we can write

(5.9) $\displaystyle\sum_{j\in J}$
$\displaystyle\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap
B_{j},\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=$
(5.10)
$\displaystyle\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int\limits_{\bigsqcup_{j\in
J}B_{j}}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu$
(5.11) $\displaystyle+\\#J\cdot O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}).$

We know $\\#J=O_{n}(Y^{n})$ bacause the total number of annulus sectors obeys
this bound. It follows that the last error term in (5.9) is an
$O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$ if the constant in $Y=e^{\sqrt{\log
N}/O_{K}(1)}$ is taken large enough.

The difference between $\int_{\bigsqcup_{j\in J}B_{j}}$ and $\int_{C}$ is,
again by Lemma 5.1.3, bounded by $O(1)\cdot\\#(C\setminus\bigsqcup_{j\in
J}B_{j})=O_{n}(N/Y)$.

Thus we are reduced to proving Theorem 3.1.1 for $C=B_{j}$ an annulus sector
of size $\leq X/Y$ disjoint from the region (5.1).

###  5.2. Proof for annulus sectors

Now we claim and prove that Theorem 3.1.1 is true for annulus sectors
$B\subset K\otimes_{\mathbf{Q}}\mathbf{R}_{<X}$ of size $<X/10$ say, disjoint
from (5.1).

Let $Y_{2}>1$ be a new auxiliary integer parameter999 Here we could have used
the letter $Y$ afresh but we prefer to use a different symbol bacause in the
logical order, the $Y$ in §5.1 cannot be specified until we specify $Y_{2}$.
of the form $e^{\sqrt{\log N}/O_{K}(1)}$.

####  5.2.1. Annulus sectors in the multiplicative coordinates

Via the bijection
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\cong\mathbf{R}^{r_{1}+r_{2}}\times(\mathbf{R}/\mathbf{Z})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$, our annulus sectors look like cubes contained in $[0,\log
X]^{r_{1}+r_{2}}\times(\mathbf{R}/\mathbf{Z})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$ whose facets are parallel to coordinate hyperplanes. The left-
hand bound “$0$” in $[0,\log X]^{r_{1}+r_{2}}$ is due to the assumption that
our set is disjoint from (5.1).

####  5.2.2. Cubes in $H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}$

We consider the image of $[0,\log X]^{r_{1}+r_{2}}$ by the projection
$\mathbf{R}^{r_{1}+r_{2}}=\mathbf{R}\times H\to H$. We cover it with cubes of
size $1/Y_{2}$ with respect to a fixed $\mathbf{R}$-basis of $H$. This
requires $O_{n}((Y_{2}\log X)^{r_{1}+r_{2}-1})$ cubes.

We also cover the torus $(\mathbf{R}/\mathbf{Z})^{r_{2}}$ by $Y_{2}^{r_{2}}$
cubes of size $1/Y_{2}$.

######  Definition 5.2.3.

By a cube in
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}=H\times(\mathbf{R}/\mathbf{Z})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$ let us mean a subset $P$ obtained as the product of a cube in
$H$ and a cube in $(\mathbf{R}/\mathbf{Z})^{r_{2}}$ of sizes $1/Y_{2}$ as in
§5.2.2, placed in one of the connected components $\\{\pm 1\\}^{r_{1}}$. There
are $O_{n}(Y_{2}^{n-1}\log^{r_{1}+r_{2}-1}X)$ of them.

####  5.2.4. Sandwiching by special type of subsets

Let $P$ be a cube in
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$ in the
sense of Definition 5.2.3 and consider the intersection
$B\cap\mathbf{R}^{\times}_{>0}P$ in
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$.

For $1<a<b$, recall from §A.3 that we write $[a,b]\cdot
P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ for the set of points
where the projection to $H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}$
belong to $P$ and the value of $\mathbf{N}(-)$ is in $[a,b]$.

By §5.1.1, up to negligible error there are two real numbers
$\log(N/X)<a<b<\log N$ satisfying

(5.12) $\displaystyle[a,b]\times P\subset B\cap(\mathbf{R}\times
P)\subset[a-\varepsilon,b+\varepsilon]\times P\quad\text{ in }\mathbf{R}\times
H\times(S^{1})\times\\{\pm 1\\}^{r_{1}},$

where $\varepsilon=O_{n}(1/Y_{2})$ by §5.2.1. In the additive notation, this
means

(5.13) $\displaystyle[e^{a},e^{b}]\cdot P\subset
B\cap\mathbf{R}^{\times}_{>0}P\subset[e^{a}(1-\varepsilon),e^{b}(1+\varepsilon)]\cdot
P\quad\text{ in
}K\otimes_{\mathbf{Q}}\mathbf{R}=\mathbf{R}^{r_{1}}\times\mathbf{C}^{r_{2}}.$

It follows that

(5.14)
$\displaystyle\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap([e^{a},e^{b}]\cdot
P)\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})$
$\displaystyle\leq\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap(B\cap\mathbf{R}^{\times}_{>0}P)\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})$
(5.15)
$\displaystyle\leq\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap([e^{a}(1-\varepsilon),e^{b}(1+\varepsilon)]\cdot
P)\\\ \pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1}).$

####  5.2.5. Applying the main theorem for the special case

We want to apply Corollary 4.2.3 to the sets $[e^{a},e^{b}]\cdot P$ and
$[e^{a}(1-\varepsilon),e^{b}(1+\varepsilon)]\cdot P$ to rewrite (5.14). We
have to verify the hypotheses on $\mathbf{N}(\mathfrak{q})$ and the size of
$P$:

  * •

We know
$e^{\sqrt{\log(e^{a}(1-\varepsilon))}}>e^{\sqrt{\log(N/X)-\varepsilon}}>e^{\sqrt{\log
N}/2}>\mathbf{N}(\mathfrak{q})$ by (3.2);

  * •

By the construction of $P$, the image of $P\times\\{\alpha\\}$ to
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$ is a
cube of size $\ll_{K}1/Y_{2}$. By declaring that we shall take the $O_{K}(1)$
in the bound (3.2) of $\mathbf{N}(\mathfrak{q})$ large enough depending on
that in $Y_{2}$, we can ensure that the image of $P\times\\{\alpha\\}$ is
small enough (and automatically convex) to satisfy
$1/Y_{2}\ll_{K}\mathbf{N}(\mathfrak{q})^{-(n-2)}<\varphi_{K}(\mathfrak{q})^{-(n-2)}$
as required in Proposition C.2.2.

Under these circumstances we can apply Corollary 4.2.3 and obtain an estimate
of our sum from below:

(5.16)
$\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap(B\cap\mathbf{R}^{\times}_{>0}P)\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})\quad>\\\
\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int\limits_{[e^{a},e^{b}]\cdot
P}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu_{\mathrm{add}}\\\
+O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}),$

and from above by the same expression except that the domain of integration
$[e^{a},e^{b}]\cdot P$ is replaced by
$[e^{a}(1-\varepsilon),e^{b}(1+\varepsilon)]\cdot P$.

####  5.2.6. Difference of upper and lower bounds

The difference of the main terms of the upper and lower bounds in (5.16) is
bounded by

(5.17)
$\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\cdot\\\
\int\limits_{([e^{a}(1-\varepsilon),e^{b}(1+\varepsilon)]\cdot
P)\setminus([e^{a},e^{b}]\cdot
P)}\Bigl{|}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}^{-1})^{\beta-1}\Bigr{|}d\mu_{\mathrm{add}}$

and is an $O_{K}(N/(Y_{2})^{n})$ because: the factor in front of the integral
is an $O_{K}(1)$, the integrand always has absolute value $\leq 2$ and the
domain of integration has volume $<2N\varepsilon\operatorname{Area}(P)\ll
N/(Y_{2})^{n}$. By (5.16) and the triangle inequality we conclude

(5.18)
$\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap(B\cap\mathbf{R}^{\times}_{>0}P)\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=\\\
\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int\limits_{B\cap\mathbf{R}^{\times}_{>0}P}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu_{\mathrm{add}}\\\\[10.0pt]
+O_{K}(N/(Y_{2})^{n})+O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)}).$

####  5.2.7. To take the sum over the cubes

Take the sum $\sum_{P}$ of (5.18) over the cubes $P$ in the sense of
Definition 5.2.3. Since there are $O_{K}((Y_{2})^{n-1}\log^{r_{1}+r_{2}-1}X)$
of them, we obtain

(5.19) $\sum_{\begin{subarray}{c}\pi\in\mathfrak{a}_{\lambda}\cap B\\\
\pi\equiv\alpha\text{ in
}\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda}\end{subarray}}\log\mathbf{N}(\pi\mathfrak{a}_{\lambda}^{-1})=\\\
\frac{w_{K}}{\mathbf{N}(\mathfrak{a}_{\lambda})\varphi_{K}(\mathfrak{q})2^{r_{1}}\pi^{r_{2}}h_{K}R_{K}}\int_{B}1-\psi_{\mathrm{Siegel}}(-;\alpha)\mathbf{N}((-)\mathfrak{a}_{\lambda}^{-1})^{\beta-1}d\mu_{\mathrm{add}}\\\\[10.0pt]
+O_{K}\left(N\frac{\log^{r_{1}+r_{2}-1}X}{Y_{2}}\right)\\\
+O_{K}((Y_{2})^{n-1}\log^{r_{1}+r_{2}-1}X\cdot Ne^{-\sqrt{\log N}/O_{K}(1)}).$

The first error term in (5.19) is an $O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$
with the $O_{K}(1)$ in the exponent being more or less the same as that in
$Y_{2}=e^{\sqrt{\log N}/O_{K}(1)}$.

By taking the $O_{K}(1)$ constant in $Y_{2}$ large enough compared to the
constant in $e^{-\sqrt{\log N}/O_{K}(1)}$ in the second error term, we can
make the second error term also an $O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$.

If follows that if we set the $O_{K}(1)$ constant in $Y_{2}$ large enough, the
two error terms in (5.19) become a new $O_{K}(Ne^{-\sqrt{\log N}/O_{K}(1)})$.
Having fixed $Y_{2}$, we can now finalize the $O_{K}(1)$ constant in the
restriction (3.2) on $\mathbf{N}(\mathfrak{q})$ following the requirements
from §§4.1.2, 5.2.5. This proves Theorem 3.1.1 for annulus sectors $B$ and
accomplishes the goal of §5.2.

Then we run the process of §5.1 to fix the $O_{K}(1)$ constant in
$Y=e^{\sqrt{\log N}/O_{K}(1)}$ and deduce Theorem 3.1.1 in general. This
completes the proof of Theorem 3.1.1.

##  Appendix A Conventions on $K\otimes_{\mathbf{Q}}\mathbf{R}$

We have to fix some conventions in order to be able to do analysis on
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ and its quotients
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}$ and
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}$.
Following the usual convention, we number the embeddings as

(A.1)
$\displaystyle\sigma_{1},\dots,\sigma_{r_{1}};\quad\sigma_{r_{1}+1},\dots,\sigma_{r_{1}+r_{2}};\quad\sigma_{r_{1}+r_{2}+1},\dots,\sigma_{r_{1}+2r_{2}}$

where $\sigma_{1},\dots,\sigma_{r_{1}}$ are the different real embeddings and
$\sigma_{r_{1}+1},\dots,\sigma_{r_{1}+r_{2}}$ represent the different
conjugate pairs of imaginary embeddings.

###  A.1. Norm functions

For a non-zero ideal $\mathfrak{a}\subset\mathcal{O}_{K}$, we write
$\mathbf{N}(\mathfrak{a}):=\\#(\mathcal{O}_{K}/\mathfrak{a})$ for its norm.
For a non-zero element $\alpha\in\mathcal{O}_{K}$, by slight abuse of notation
we write
$\mathbf{N}(\alpha):=\mathbf{N}(\alpha\mathcal{O}_{K})=|N_{K/\mathbf{Q}}(\alpha)|$.
The norm functions $\mathbf{N}(-)$ and $N_{K/\mathbf{Q}}$ extend to
$K\otimes_{\mathbf{Q}}\mathbf{R}$ naturally,

(A.6)

where we have denoted the extensions by the same symbols.

Let us emphasize that $\mathbf{N}(-)$ always takes positive values; not to be
confused with the ring-theoretic norm $N_{K/\mathbf{Q}}(-)$, which does take
negative values unless $K$ is totally imaginary.

###  A.2. The group structure of $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$

Let $\sigma_{i}\otimes_{\mathbf{Q}}\mathbf{R}\colon
K\otimes_{\mathbf{Q}}\mathbf{R}\to\mathbf{R}$ ($1\leq i\leq r_{1}$) and
$\sigma_{i}\otimes_{\mathbf{Q}}\mathbf{R}\colon
K\otimes_{\mathbf{Q}}\mathbf{R}\to\mathbf{C}$ ($r_{1}+1\leq i\leq
r_{1}+r_{2}$) be the unique $\mathbf{R}$-linear extentions. The map they
jointly define

(A.7) $\displaystyle
K\otimes_{\mathbf{Q}}\mathbf{R}\xrightarrow{\cong}\mathbf{R}^{r_{1}}\times\mathbf{C}^{r_{2}}$

is known to be an isomorphism of $\mathbf{R}$-algebras.

It follows that

(A.8) $\displaystyle(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\ \cong\
(\mathbf{R}^{\times})^{r_{1}}\times(\mathbf{C}^{\times})^{r_{2}}\ \cong\
\mathbf{R}^{r_{1}+r_{2}}\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}},$

where the second isomorphism follows the conventions in §1.5.1; in particular
the map $\mathbf{C}^{\times}\to\mathbf{R}$ is defined by
$z\mapsto\log(|z|^{2})$.

###  A.3. Decomposition of $\mathbf{R}^{r_{1}+r_{2}}$

Write

(A.9)
$\displaystyle\bm{u}_{r_{1}+r_{2}}:=\frac{1}{n}(\underset{r_{1}}{\underbrace{1,\dots,1}},\underset{r_{2}}{\underbrace{2,\dots,2}})\in\mathbf{R}^{r_{1}+r_{2}}.$

In (A.8), the diagonal image
$\mathbf{R}^{\times}_{>0}\subset(\mathbf{R}^{\times})^{r_{1}}\times(\mathbf{C}^{\times})^{r_{2}}$
corresponds to the subspace

(A.10)
$\displaystyle\mathbf{R}\cdot\bm{u}_{r_{1}+r_{2}}\times\\{1\\}^{r_{2}}\times\\{+1\\}^{r_{1}}\quad\subset\quad\mathbf{R}^{r_{1}+r_{2}}\times(S^{1})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}.$

Let $H\subset\mathbf{R}^{r_{1}+r_{2}}$ be the following $\mathbf{R}$-linear
subspace of codimension $1$:

(A.11) $\displaystyle H:=\left\\{(x_{1},\dots,x_{r_{1}+r_{2}})\ \middle|\
\sum_{i=1}^{r_{1}+r_{2}}x_{i}=0\right\\}.$

This corresponds to elements
$x=(x_{1},\dots,x_{r_{1}};z_{1},\dots,z_{r_{2}})\in(\mathbf{R}^{\times})^{r_{1}}\times(\mathbf{C}^{\times})^{r_{2}}$
of norm $\pm 1$.

We have the direct sum decomposition

(A.12)
$\displaystyle\mathbf{R}^{r_{1}+r_{2}}=\mathbf{R}\cdot\bm{u}_{r_{1}+r_{2}}\oplus
H$

and hence the isomorphism

(A.13)
$\displaystyle(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}\cong
H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}.$

This also gives a decomposition

(A.14) $\displaystyle(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$
$\displaystyle=\mathbf{R}\times H\times(S^{1})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$ (A.15)
$\displaystyle=\mathbf{R}^{\times}_{>0}\times(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}.$

Given a subset
$P\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$,
we write $\mathbf{R}^{\times}_{>0}\cdot P$ for its inverse image in
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$. When we see it as a subset of
$\mathbf{R}\times H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}$, we prefer
to use the symbol $\mathbf{R}\times P$.

If furthermore $0<a<b$ are given, we write

(A.16) $\displaystyle[a,b]\cdot P:=\\{x\in\mathbf{R}^{\times}_{>0}\cdot P\mid
a\leq\mathbf{N}(x)\leq b\\}.$

When we see it as a subset of $\mathbf{R}\times
H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}$, we write it as $[\log a,\log
b]\times P$.

###  A.4. The domain $(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}$

For an embedding $\sigma\colon K\hookrightarrow\mathbf{C}$ and $x\in
K\otimes_{\mathbf{Q}}\mathbf{R}$, we set

(A.17)
$\displaystyle|x|_{\sigma}:=|(\sigma\otimes_{\mathbf{Q}}\mathbf{R})(x)|,$

where $|-|$ in the right hand side is the usual absolute value in
$\mathbf{C}$. For a positive number $X>1$, we define a subset
$(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}\subset
K\otimes_{\mathbf{Q}}\mathbf{R}$ by

(A.18) $\displaystyle(K\otimes_{\mathbf{Q}}\mathbf{R})_{<X}:=\\{x\in
K\otimes_{\mathbf{Q}}\mathbf{R}\mid|x|_{\sigma_{i}}<X\text{ for all }1\leq
i\leq r_{1}+r_{2}\\}.$

###  A.5. Measures

We will consider two measures on $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$.
One is the restriction of the Lebesgue measure by the inclusion
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\subset
K\otimes_{\mathbf{Q}}\mathbf{R}\cong\mathbf{R}^{r_{1}}\times\mathbf{C}^{r_{2}}\cong\mathbf{R}^{n}$,
where we took the isomorphism $\mathbf{C}\cong\mathbf{R}^{2}$;
$x+iy\leftrightarrow(x,y)$. We denote it by $\mu_{\mathrm{add}}$.

The other is obtained through
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\cong\mathbf{R}^{r_{1}+r_{2}}\times(S^{1})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$. We give $\mathbf{R}^{r_{1}+r_{2}}$ the Lebesgue measure and
$(S^{1})^{r_{2}}$ the Haar measure such that the volume of $(S^{1})^{r_{2}}$
is $(2\pi)^{r_{2}}$. (In other words, as far as volume is concerned, we regard
$S^{1}$ as $\mathbf{R}/2\pi\mathbf{Z}$, not $\mathbf{R}/\mathbf{Z}$.) Each
point of $\\{\pm 1\\}^{r_{1}}$ is given weight $1$. We denote the resulting
product measure on $(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ by
$\mu_{\mathrm{mult}}$.

By some calculus one finds

(A.19) $\displaystyle
d\mu_{\mathrm{mult}}=\frac{2^{r_{2}}}{\mathbf{N}(-)}d\mu_{\mathrm{add}}\quad\text{
on }(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times},$

where $\mathbf{N}(-)$ is the norm function in §A.1.

We endow $H\subset\mathbf{R}^{r_{1}+r_{2}}$ with the induced metric and hence
the measure $\mu_{H}$. This endows
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}=H\times(S^{1})^{r_{2}}\times\\{\pm
1\\}^{r_{1}}$ with the product measure. We denote it by $\mu_{\mathrm{mult}}$
as well.

Let $\bm{u}_{1},\dots,\bm{u}_{r_{1}+r_{2}-1}\in H$ be any orthonormal basis
and $(\xi_{1},\dots,\xi_{r_{1}+r_{2}})$ be the coordinate system corresponding
to the basis $\bm{u}_{1},\dots,\bm{u}_{r_{1}+r_{2}}$ of
$\mathbf{R}^{r_{1}+r_{2}}$. (See §A.3 for the vector $\bm{u}_{r_{1}+r_{2}}$.)
Note that the measure $\mu_{H}$ on $H$ is exactly the one associated with the
volume form $d\xi_{1}\dots d\xi_{r_{1}+r_{2}-1}$ on $H$. The inner product of
$\bm{u}_{r_{1}+r_{2}}$ and the unit normal vector
$(1,\dots,1)/\sqrt{r_{1}+r_{2}}$ of $H$ is $1/\sqrt{r_{1}+r_{2}}$. This
implies the following relation of measures:

(A.20) $\displaystyle|d\xi_{1}\dots
d\xi_{r_{1}+r_{2}}|=\sqrt{r_{1}+r_{2}}\,d\mu_{\mathrm{mult}}=\frac{2^{r_{2}}\sqrt{r_{1}+r_{2}}}{\mathbf{N}(-)}d\mu_{\mathrm{add}}.$

###  A.6. The regulator

Dirichlet’s Unit Theorem says that we have an isomoprhism as follows, which we
fix:

(A.21)
$\displaystyle\mathcal{O}_{K}^{\times}\cong(\mathbf{Z}/w_{K}\mathbf{Z})\times\mathbf{Z}^{r_{1}+r_{2}-1}.$

By its proof, we know that the inclusion map followed by the projection

(A.22)
$\displaystyle\mathcal{O}_{K}^{\times}\hookrightarrow(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\cong\mathbf{R}\times
H\times(S^{1})^{r_{2}}\times\\{\pm 1\\}^{r_{1}}\twoheadrightarrow H$

has image which is a full-rank lattice. The regulator $R_{K}>0$ of $K$ is the
real number satisfying
$\operatorname{Vol}_{\mathrm{mult}}(H/\mathbf{Z}^{r_{1}+r_{2}-1})=R_{K}\sqrt{r_{1}+r_{2}}$.

###  A.7. The structure of
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}$

We take this opportunity to compute the abelian group
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}$ further.
By §A.6 we know it decomposes as

(A.23)
$\displaystyle\mathbf{R}\times\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}=\mathbf{R}\times\frac{H\times\\{\pm
1\\}^{r_{1}}\times(S^{1})^{r_{2}}}{\mathcal{O}_{K}^{\times}}.$

We have the next commutative diagram with exact rows

(A.28)

This gives an exact sequence

(A.29) $\displaystyle 1\to\frac{\\{\pm
1\\}^{r_{1}}\times(S^{1})^{r_{2}}}{(\mathcal{O}_{K}^{\times})_{\mathrm{tors}}}\to\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}\to(\mathbf{R}/\mathbf{Z})^{r_{1}+r_{2}-1}\to
0.$

The left-hand term is isomorphic to $\\{\pm
1\\}^{r_{1}-1}\times(S^{1})^{r_{2}}$ if $r_{1}\neq 0$ and $(S^{1})^{r_{2}}$ if
$r_{1}=0$.

###  A.8. The neutral component of
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}$

From (A.29) it is now clear that
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/(\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0})$
is a torus (= commutative compact Lie group) of dimension $n-1$. As such, its
neutral component
$((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/(\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}))^{0}$
is isomorphic to $(S^{1})^{n-1}$. Choose and fix any such isomorphism

(A.30)
$\displaystyle\left(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}}\right)^{0}\cong(S^{1})^{n-1}.$

Of course there are natural ways of specifying such an isomorphism in the
spirit of (A.29) relying only on mild choices. But this is not needed in this
paper.

###  A.9. The volume of
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}$

Since $\mathcal{O}_{K}^{\times}$ is a discrete subgroup of
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathbf{R}^{\times}_{>0}$, and
translation by an element of $\mathcal{O}_{K}^{\times}$ preserves the measure
$\mu_{\mathrm{mult}}$ (see §A.5 for this measure), the quotient inherits the
measure. Let us write the resulting one also by $\mu_{\mathrm{mult}}$.

It will be convenient to know the volume of
$(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}$.
The left-hand term of (A.29) has volume $2^{r_{1}}(2\pi)^{r_{2}}/w_{K}$. The
right-hand term has volume $R_{K}\sqrt{r_{1}+r_{2}}$ (§A.6). By the definition
of $\mu_{\mathrm{mult}}$ as the product measure, it follows that

(A.31)
$\displaystyle\operatorname{Vol}_{\mathrm{mult}}((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}/\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0})=\frac{2^{r_{1}}(2\pi)^{r_{2}}R_{K}\sqrt{r_{1}+r_{2}}}{w_{K}}.$

##  Appendix B Fourier analysis on a torus

By a torus we mean a compact commutative Lie group. By so-called Lie’s three
theorems, such a group $\mathbf{T}$ is known to be isomorphic to a product

(B.1) $\displaystyle\mathbf{T}\cong G\times(S^{1})^{d}$

of a finite commutative group $G$ and a connected torus $(S^{1})^{d}$. Of
course the isomorphism is usually non-canonical.

###  B.1. Approximation of indicator functions of small convex subsets by
characters

Let $\mathbf{T}$ be a torus and

(B.2)
$\displaystyle\widehat{\mathbf{T}}:=\operatorname{Hom}_{\text{cont}}(\mathbf{T},S^{1})\cong\widehat{G}\times\mathbf{Z}^{d}$

be its Pontryagin dual. Every $\xi\in\widehat{\mathbf{T}}$ will be seen as a
$\mathbf{C}$-valued function on $\mathbf{T}$ by
$\mathbf{T}\xrightarrow{\xi}S^{1}\subset\mathbf{C}$. Fourier analysis on
$\mathbf{T}$ roughly says that functions on $\mathbf{T}$ can be written as
$\mathbf{C}$-linear combinations of elements of $\widehat{\mathbf{T}}$. The
precise statement we use in this paper is Proposition B.1.2 below.

######  Definition B.1.1.

Let $\mathbf{T}$ be a torus whose neutral component $\mathbf{T}^{0}$ is
equipped with an isomorphism $\mathbf{T}^{0}\cong(S^{1})^{d}$. A small convex
subset of $\mathbf{T}$ (resp. of size $\leq\delta$, $0<\delta<1/4$) is a
connected subset $P\subset\mathbf{T}$ such that for some $p\in P$ the
translate

(B.3) $\displaystyle
P-p\subset\mathbf{T}^{0}\cong(S^{1})^{d}\cong(\mathbf{R}/\mathbf{Z})^{d}$

is the projection of an open convex subset
$\widetilde{P}\subset[-\frac{1}{4},\frac{1}{4}]^{d}\subset\mathbf{R}^{d}$
(resp. $\widetilde{P}\subset[-\delta,\delta]^{d}$).

######  Proposition B.1.2.

Let $M,Y>1$ be positive parameters. Let $P$ be a small convex subset of a
torus of size $<1/5$ with respect to some identification
$\mathbf{T}^{0}\cong(S^{1})^{d}=\mathbf{R}^{d}/\mathbf{Z}^{d}$. Then the
indicator function $1_{P}$ can be approximated as:

(B.4) $\displaystyle 1_{P}$
$\displaystyle=\sum_{\xi\in\Xi}c_{\xi}\cdot\xi+G+H,$

where

  * •

$\Xi\subset\widehat{\mathbf{T}}$ is a subset of size
$O_{d}(\\#\pi_{0}(\mathbf{T})\cdot Y^{d})$,

  * •

the coefficients $c_{\xi}\in\mathbf{C}$ obey bounds $|c_{\xi}|\leq 1$,

  * •

the values of $G$ have absolute values $\leq 1$ and there is another small
convex set $P^{\prime}$ satisfying $\operatorname{Supp}(G)\subset
P^{\prime}\setminus P$ and $\operatorname{Vol}(P^{\prime}\setminus
P)=O_{n}(1/M)$,

  * •

$H$ is a function satisfying $\left\|H\right\|_{\infty}\leq
O_{d}(M\cdot\frac{\log Y}{Y})$.

Moreover, for $\xi\in\left(\pi_{0}(\mathbf{T})\right)\widehat{\
}\subset\widehat{\mathbf{T}}$ the coefficient $c_{\xi}$ can be explicitly
described: letting $\mu$ be any Haar measure on $\mathbf{T}$,

(B.5) $\displaystyle
c_{\xi}=\frac{\mu(P)}{\mu(\mathbf{T})}\overline{\xi(P)}+O_{d}(1/M),$

where the value $\xi(P)\in\mathbf{C}$ is well defined because
$\xi\in\left(\pi_{0}(\mathbf{T})\right)\widehat{\ }$ is constant on every
connected set.

###### Proof.

This follows from [3, Corollary A.3] and [2, Lemma A.9].

Note that although [2, Lemma A.9] is stated for the connected torus
$(\mathbf{R}/\mathbf{Z})^{d}$, the corresponding statement for a non-connected
commutative torus easily follows from the decompositions (B.1) (B.2). The
specific choice of these decompositions affect the resulting objects
$\Xi,G,H,\dots$ but do not affect the bounds of the error terms. ∎

In our application, the number $\\#\pi_{0}(\mathbf{T})$ of connected
components of $\mathbf{T}$ will be allowed to grow at a pseudopolynomial rate
$e^{\sqrt{\log N}/O_{K}(1)}$ at the fastest.

##  Appendix C Idèle class group as a torus

Let $\mathfrak{q}=\prod_{\mathfrak{p}}\mathfrak{p}^{n_{\mathfrak{p}}}$ be a
non-zero ideal and let $C(\mathfrak{q})$ be the mod $\mathfrak{q}$ idèle class
group from §2.1. We perform Fourier analysis from §B on the torus
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$. Its volume (C.24) explains the
coefficient in Mitsui’s Prime Number Theorem 1.2.1, 3.1.1.

###  C.1. The structure of $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$

In this subsection we review how the quotient
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ is a real torus and fix some
notation on the way.

####  C.1.1. To factor out $\operatorname{Cl}(K)$

Recall our notation

(C.1) $\displaystyle
C(\mathfrak{q})=\widetilde{C}(\mathfrak{q})/K^{\times}=\left((K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times\bigoplus_{\mathfrak{p}|\mathfrak{q}}K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\times\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}\right)/K^{\times}.$

Consider the $\mathfrak{p}$-adic valuation map $v_{\mathfrak{p}}\colon
K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\twoheadrightarrow\mathbf{Z}$
for each $\mathfrak{p}\mid\mathfrak{q}$. Its kernel is identified with
$\mathcal{O}_{K,\mathfrak{p}}^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})=(\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})^{\times}=(\mathcal{O}_{K}/\mathfrak{p}^{n_{\mathfrak{p}}})^{\times}$.
If we take the direct sum over $\mathfrak{p}\mid\mathfrak{q}$, by Chinese
Remainder Theorem we get $(\mathcal{O}_{K}/\mathfrak{q})^{\times}$.

It follows that we have the following commutative diagram with exact rows and
injective vertical maps, where $P_{K}$ denotes the group of principal
fractional ideals

(C.6)

By the snake lemma we obtain an exact sequence

(C.7) $\displaystyle
1\to\frac{\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathbf{R}^{\times}_{>0}}\times(\mathcal{O}_{K}/\mathfrak{q})^{\times}}{\mathcal{O}_{K}^{\times}}\to
C(\mathfrak{q})\to\operatorname{Cl}(K)\to 0.$

####  C.1.2. To factor out $(\mathcal{O}_{K}/\mathfrak{q})^{\times}$

We want to compute the first term of (C.7) further. Consider the commutative
diagram with exact rows:

(C.12)

As the vertical maps are injective, we conclude that
$(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathbf{R}^{\times}_{>0}}\times(\mathcal{O}_{K}/\mathfrak{q})^{\times})/\mathcal{O}_{K}^{\times}$
is an extension of
$\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathbf{R}^{\times}_{>0}\cdot\mathcal{O}_{K}^{\times}}$
by $(\mathcal{O}_{K}/\mathfrak{q})^{\times}$, the first of which has been
computed in §A.7.

####  C.1.3. The structure of
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$—conclusion

Our computation so far expresses $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ as
a twofold extension

(C.23)

In §A.7 we saw that
$\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}$
is a torus. It follows that $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ is also
a torus.

Its neutral component $(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})^{0}$
surjects to the neutral component of
$\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}$,
and the size of the kernel of this map is bounded from above by
$\\#(\mathcal{O}_{K}/\mathfrak{q})^{\times}=\varphi_{K}(\mathfrak{q})<\mathbf{N}(\mathfrak{q})$.

####  C.1.4. The volume of $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$

Via the diagram (C.23), the volume form of
$\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}$
induces a volume form on $C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$. Write the
resulting measure as $\mu_{\mathrm{id\grave{e}le}}$ and
$\operatorname{Vol}_{\mathrm{id\grave{e}le}}(-)$ the volume of subsets.101010
Note that this may be different from the measure adopted in the reader’s
favorite reference because every reference has its own convention on the
chioce of measures. Don’t worry, any two differ only by a constant factor
because all are Haar measures.  By the diagram and (A.31) we have

(C.24)
$\displaystyle\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})$
(C.25) $\displaystyle=$ $\displaystyle
h_{K}\varphi_{K}(\mathfrak{q})\operatorname{Vol}_{\mathrm{mult}}(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\mathbf{R}^{\times}_{>0}})$
(C.26) $\displaystyle=$
$\displaystyle\varphi_{K}(\mathfrak{q})2^{r_{1}}(2\pi)^{r_{2}}R_{K}h_{K}\sqrt{r_{1}+r_{2}}/w_{K}.$

###  C.2. Indicator function of convex sets

Now we combine Fourier analysis from §B and the description of
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$. The consequence, Proposition
C.2.2, is used when we compute weighted sum of prime elements in thin cones as
in Figure 2 in Itroduction.

####  C.2.1. A trivialization of the neutral component

Consider the covering map induced on the neutral components
$(\mathbf{C}(\mathfrak{q})/\mathbf{R}_{>0}^{\times})^{0}\twoheadrightarrow\left(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}\right)^{0}$.
In (A.30) we fixed an isomorphism between the target and
$(S^{1})^{n-1}=(\mathbf{R}/\mathbf{Z})^{n-1}$. This identifies the covering
map with a map of the form

(C.27)
$\displaystyle\mathbf{R}^{n-1}/\Lambda(\mathfrak{q})\twoheadrightarrow\mathbf{R}^{n-1}/\mathbf{Z}^{n-1}$

for some sublattice $\Lambda(\mathfrak{q})\subset\mathbf{Z}^{n-1}$ of index
dividing $\varphi_{K}(\mathfrak{q})$. By $\mathbf{Z}$-linear algebra (Euclid’s
algorithm, see Proposition D.1), one can find a basis

(C.28) $\displaystyle\bm{f}_{1},\dots,\bm{f}_{n-1}\in\Lambda(\mathfrak{q})$

whose coefficients as elements of $\mathbf{Z}^{n-1}$ have absolute values
$\leq\varphi_{K}(\mathfrak{q})$.

Now we are ready to state the following consequence of Fourier analysis.

######  Proposition C.2.2 (indicator function of a small convex set).

Let $M,Y>1$ be real numbers.

Let $P\subset C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ be a connected open
subset such that its translate to
$(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})^{0}$ projects injectively by the
map

(C.29)
$\displaystyle(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})^{0}\to\left(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}\right)^{0}\cong(S^{1})^{n-1}$

and the image is a small convex subset of size
$\ll_{n}\varphi_{K}(\mathfrak{q})^{-(n-2)}$ in the sense of Definition B.1.1.

Then one has a decomposition of the indicator function $1_{P}\colon
C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}\to\mathbf{C}$ of the form:

(C.30) $\displaystyle 1_{P}$
$\displaystyle=\sum_{\psi\in\Xi\subset(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})\widehat{\
}}c_{\psi}\psi+G+H,$

where

  * •

$\\#\Xi=O_{K}(\varphi_{K}(\mathfrak{q})Y^{n-1})$,

  * •

$|c_{\xi}|\leq 1$,

  * •

the values of $G$ have absolute value $\leq 1$, and there is another small
convex set $P^{\prime}$ such that $\operatorname{Supp}(G)\subset
P^{\prime}\setminus P$ and
$\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P^{\prime}\setminus
P)=O_{n}(1/M)$,

  * •

$\left\|H\right\|_{\infty}\leq O_{n}(M\frac{\log Y}{Y})$.

Moreover, for
$\psi\in\left(\pi_{0}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})\right)\widehat{\
}\subset(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})\widehat{\ }$ we have

(C.31) $\displaystyle
c_{\psi}=\frac{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(P)}{\operatorname{Vol}_{\mathrm{id\grave{e}le}}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})}\overline{\psi(P)}+O_{n}(1/M).$

###### Proof.

We continue to identify $(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})^{0}$ with
$\mathbf{R}^{n-1}/\Lambda(\mathfrak{q})$ as in (C.27). Take a basis of
$\Lambda(\mathfrak{q})$ as in (C.28). For this basis the matrix representing
the linear map
$\phi\colon\mathbf{Z}^{n-1}\cong\Lambda(\mathfrak{q})\hookrightarrow\mathbf{Z}^{n-1}$
has entries of absolute values $\leq\varphi_{K}(\mathfrak{q})$. It follows
that the $\mathbf{R}$-linear inverse map
$\phi_{\mathbf{R}}^{-1}\colon\mathbf{R}^{n-1}\to\Lambda(\mathfrak{q})\otimes_{\mathbf{Z}}\mathbf{R}\cong\mathbf{R}^{n-1}$
has coefficients $\ll_{n}\varphi_{K}(\mathfrak{q})^{n-2}$. It follows that if
the constant $0<c_{n}<1$ is chosen small enough to match the implied constant,
the inverse map $\phi_{\mathbf{R}}^{-1}$ sends the cube

(C.32)
$\displaystyle[-c_{n}\varphi_{K}(\mathfrak{q})^{-(n-2)},c_{n}\varphi_{K}(\mathfrak{q})^{-(n-2)}]^{n-1}$

into $(-1/5,1/5)^{n-1}$.

Under such a choice of $\Lambda(\mathfrak{q})\cong\mathbf{Z}^{n-1}$ and
$0<c_{n}<1$ we see that $P$ is a small convex set of size $<1/5$ in
$C(\mathfrak{q})/\mathbf{R}^{\times}_{>0}$ with respect to the corresponding
trivialization
$(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})^{0}\cong(S^{1})^{n-1}$.

Therefore Proposition B.1.2 applies and gives the desired result. Note that by
(C.23)

(C.33) $\displaystyle\\#\pi_{0}(C(\mathfrak{q})/\mathbf{R}^{\times}_{>0})\leq
h_{K}\cdot\varphi_{K}(\mathfrak{q})\cdot\\#\pi_{0}\left(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}\right)$

and
$\\#\pi_{0}\left(\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}}{\mathcal{O}_{K}^{\times}\cdot\mathbf{R}^{\times}_{>0}}\right)\leq
2^{r_{1}}=O_{n}(1)$. ∎

####  C.2.3. A chart

To produce connected open sets $P$ as in Proposition C.2.2, it is useful to
take the following kind of chart.

To state it, for non-zero ideals
$\mathfrak{a},\mathfrak{q}\subset\mathcal{O}_{K}$ let us denote by
$(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$ the set of elements which
is a generator of $\mathfrak{a}/\mathfrak{q}\mathfrak{a}$ as an
$\mathcal{O}_{K}/\mathfrak{q}$-module. For $\mathcal{O}_{K}=\mathfrak{a}$,
this recovers the set of invertible elements
$(\mathcal{O}_{K}/\mathfrak{q})^{\times}$. Note that by Chinese Remainder
Theorem
$\mathfrak{a}/\mathfrak{q}\mathfrak{a}=\bigoplus_{\mathfrak{p}\mid\mathfrak{q}}\mathfrak{a}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}$,
the condition that $\alpha\in\mathfrak{a}/\mathfrak{q}\mathfrak{a}$ is in
$(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$ is equivalent to the
condition that its image in
$\mathfrak{a}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}$ is in
$(\mathfrak{a}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a})^{\times}$ for all
$\mathfrak{p}\mid\mathfrak{q}$.

Let $v_{\mathfrak{p}}(\mathfrak{a})\in\mathbf{Z}$ be the $\mathfrak{p}$-adic
valuation of a non-zero (fractional) ideal $\mathfrak{a}$. Let us quickly
recall its definition: the scalar extension
$\mathfrak{a}\mathcal{O}_{K,\mathfrak{p}}$ is a free
$\mathcal{O}_{K,\mathfrak{p}}$-module of rank $1$. Take any generator $\alpha$
of $\mathfrak{a}\mathcal{O}_{K,\mathfrak{p}}$ and we write
$v_{\mathfrak{p}}(\mathfrak{a}):=v_{\mathfrak{p}}(\alpha)$, which is
independent of the choice of $\alpha$.

Then $(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$ is equivalently
defined as the set of residue classes
$\overline{\alpha}\in\mathfrak{a}/\mathfrak{q}\mathfrak{a}$ such that some (or
any) lift $\alpha\in\mathfrak{a}$ satisfies

(C.34) $\displaystyle
v_{\mathfrak{p}}(\alpha)=v_{\mathfrak{p}}(\mathfrak{a})\text{ for all
}\mathfrak{p}\mid\mathfrak{q}.$

For each $\mathfrak{p}$, this condition does not depend on the specific lift
$\alpha$.

Consider the map of sets

(C.35)
$\displaystyle(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}\to\left(\bigoplus_{\mathfrak{p}\mid\mathfrak{q}}K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\right)\oplus\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}$

induced by $\mathfrak{a}\setminus\\{0\\}\hookrightarrow K^{\times}$ composed
with $K^{\times}\twoheadrightarrow
K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})$
for each $\mathfrak{p}\mid\mathfrak{q}$ and the valuation
$v_{\mathfrak{p}}\colon K^{\times}\twoheadrightarrow\mathbf{Z}$ for each
$\mathfrak{p}\nmid\mathfrak{q}$. The map (C.35) is well defined on
$(\mathfrak{a}/\mathfrak{q}\mathfrak{a})^{\times}$ because if
$\alpha_{1},\alpha_{2}\in\mathfrak{a}$ are two elements satisfying (C.34) such
that $\varepsilon:=\alpha_{1}-\alpha_{2}\in\mathfrak{q}\mathfrak{a}$, one can
verify $\alpha_{1}/\alpha_{2}\in
1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}}$ for
$\mathfrak{p}\mid\mathfrak{q}$. Indeed, we have
$(\alpha_{1}/\alpha_{2})-1=\varepsilon/\alpha_{2}$ and

(C.36) $\displaystyle
v_{\mathfrak{p}}(\varepsilon/\alpha_{2})\geq(v_{\mathfrak{p}}(\mathfrak{q})+v_{\mathfrak{p}}(\mathfrak{a}))-v_{\mathfrak{p}}(\mathfrak{a})=v_{\mathfrak{p}}(\mathfrak{q})=n_{\mathfrak{p}}.$

######  Proposition C.2.4.

Let $\mathcal{D}\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ be a
fundamental domain for the $\mathcal{O}_{K}^{\times}$-action and
$\\{\mathfrak{a}_{\lambda}\\}_{\lambda\in\operatorname{Cl}(K)}$ be a complete
set of representatives.

The inclusions $\mathfrak{a}_{\lambda}\setminus\\{0\\}\hookrightarrow
K^{\times}$ induce the following bijections of sets

(C.37) $\displaystyle
C(\mathfrak{q})\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\frac{(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}}{\mathcal{O}_{K}^{\times}}\cong\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\mathcal{D}\times(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}.$

###### Proof.

Recall $C(\mathfrak{q})=\widetilde{C}(\mathfrak{q})/K^{\times}$. Consider the
following maps

(C.38)
$\displaystyle\widetilde{C}(\mathfrak{q})\twoheadrightarrow\bigoplus_{\mathfrak{p}\text{
all}}\mathbf{Z}\twoheadrightarrow\operatorname{Cl}(K)$

and for each $\lambda\in\operatorname{Cl}(K)$ let
$\widetilde{C}(\mathfrak{q})_{\lambda}$ be the fiber of this map over
$\lambda$:

(C.39)
$\displaystyle\widetilde{C}(\mathfrak{q})_{\lambda}:=\widetilde{C}(\mathfrak{q})\times_{\operatorname{Cl}(K)}\\{\lambda\\}.$

Then we get a decomposition
$\widetilde{C}(\mathfrak{q})=\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\widetilde{C}(\mathfrak{q})_{\lambda}$.
Since the action of $K^{\times}$ preserves each
$\widetilde{C}(\mathfrak{q})_{\lambda}$, we get

(C.40) $\displaystyle
C(\mathfrak{q})=\bigsqcup_{\lambda\in\operatorname{Cl}(K)}\widetilde{C}(\mathfrak{q})_{\lambda}/K^{\times}.$

For a non-zero (fractional) ideal $\mathfrak{a}$ we consider the following
constructions. We define subsets

(C.41) $\displaystyle K_{\mathfrak{q},\mathfrak{a}}$
$\displaystyle:=\Bigl{\\{}((x_{\mathfrak{p}})_{\mathfrak{p}\mid\mathfrak{q}},(a_{\mathfrak{p}})_{\mathfrak{p}\nmid\mathfrak{q}})\in\left(\bigoplus_{\mathfrak{p}\mid\mathfrak{q}}K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\times\bigoplus_{\mathfrak{p}\nmid\mathfrak{q}}\mathbf{Z}\middle)\right|$
(C.42) $\displaystyle\hskip
85.35826pt\forall\mathfrak{p}\mid\mathfrak{q}:v_{\mathfrak{p}}(x_{\mathfrak{p}})=v_{\mathfrak{p}}(\mathfrak{a}),\quad\forall\mathfrak{p}\nmid\mathfrak{q}:a_{\mathfrak{p}}=v_{\mathfrak{p}}(\mathfrak{a})\Bigr{\\}},$
(C.43) $\displaystyle\widetilde{C}(\mathfrak{q})_{\mathfrak{a}}$
$\displaystyle:=(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}\times
K_{\mathfrak{q},\mathfrak{a}}\subset\widetilde{C}(\mathfrak{q})_{[\mathfrak{a}]},\text{
in particular
}\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}\subset\widetilde{C}(\mathfrak{q})_{\lambda}.$

One sees that $\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}$ is stable
under the $\mathcal{O}_{K}^{\times}$-action. The induced map

(C.44)
$\displaystyle\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}/\mathcal{O}_{K}^{\times}\to\widetilde{C}(\mathfrak{q})_{\lambda}/K^{\times}$

is a bijection. The inverse is described as follows: let
$(x;(x_{\mathfrak{p}})_{\mathfrak{p}\mid\mathfrak{q}};(a_{\mathfrak{p}})_{\mathfrak{p}\nmid\mathfrak{q}})\in\widetilde{C}(\mathfrak{q})_{\lambda}$.
By the definitions of the sets
$\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}\subset\widetilde{C}(\mathfrak{q})_{\lambda}$
and
$\operatorname{Cl}(K)=\operatorname{cok}(K^{\times}\to\bigoplus_{\mathfrak{p}\text{:
all}}\mathbf{Z})$ there is an $\alpha\in K^{\times}$ such that if we let
$\operatorname{\mathbf{diag}}(\alpha)\in\widetilde{C}(\mathfrak{q})$ be its
diagonal image

(C.45)
$\displaystyle\operatorname{\mathbf{diag}}(\alpha)\cdot(x;(x_{\mathfrak{p}})_{\mathfrak{p}\mid\mathfrak{q}};(a_{\mathfrak{p}})_{\mathfrak{p}\nmid\mathfrak{q}})\in\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}.$

This gives an element of
$\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}$ whose class in
$\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}/\mathcal{O}_{K}^{\times}$
is independent of the choice of $\alpha\in K^{\times}$.

Since $\mathcal{D}\subset(K\otimes_{\mathbf{Q}}\mathbf{R})^{\times}$ is a
fundamental domain for the $\mathcal{O}_{K}^{\times}$-action we see that the
composite

(C.46) $\displaystyle\mathcal{D}\times
K_{\mathfrak{q},\mathfrak{a}_{\lambda}}\hookrightarrow\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}\twoheadrightarrow\widetilde{C}(\mathfrak{q})_{\mathfrak{a}_{\lambda}}/\mathcal{O}_{K}^{\times}$

is a bijection.

In view of the bijections (C.40) (C.44) (C.46), it remains to verify that the
map (C.35) induces a bijection

(C.47)
$\displaystyle(\mathfrak{a}_{\lambda}/\mathfrak{q}\mathfrak{a}_{\lambda})^{\times}\xrightarrow{\cong}K_{\mathfrak{q},\mathfrak{a}_{\lambda}}.$

Both sides naturally decompose into products
$\prod_{\mathfrak{p}\mid\mathfrak{q}}$. It suffices to show the following
bijection

(C.48)
$\displaystyle(\mathfrak{a}_{\lambda}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}_{\lambda})^{\times}\xrightarrow{\cong}\\{x\in
K^{\times}/(1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}})\mid
v_{\mathfrak{p}}(x)=v_{\mathfrak{p}}(\mathfrak{a}_{\lambda})\\}$

for each $\mathfrak{p}\mid\mathfrak{q}$.

The inverse map can be constructed as follows. Let $\alpha\in K^{\times}$ be
an element satisfying
$v_{\mathfrak{p}}(\alpha)=v_{\mathfrak{p}}(\mathfrak{a}_{\lambda})$. This
implies $\alpha\in\mathfrak{a}_{\lambda}\mathcal{O}_{K,\mathfrak{p}}$ and that
it generates $\mathfrak{a}_{\lambda}\mathcal{O}_{K,\mathfrak{p}}$ as an
$\mathcal{O}_{K,\mathfrak{p}}$-module. So it determines a class in
$(\mathfrak{a}_{\lambda}\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}_{\lambda}\mathcal{O}_{K,\mathfrak{p}})^{\times}=(\mathfrak{a}_{\lambda}/\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}_{\lambda})^{\times}$.
Moreover, it is obvious that if the ratio of two such $\alpha,\alpha^{\prime}$
is in $1+\mathfrak{p}^{n_{\mathfrak{p}}}\mathcal{O}_{K,\mathfrak{p}}$, their
difference is in
$\mathfrak{p}^{n_{\mathfrak{p}}}\mathfrak{a}_{\lambda}\mathcal{O}_{K,\mathfrak{p}}$

It is routine to check that this is actually the inverse. ∎

##  Appendix D Euclid’s algorithm

The following statement is used in §C.2.1.

######  Proposition D.1.

Let $\Lambda\subset\mathbf{Z}^{n}$ be a subgroup of finite index
$\\#(\mathbf{Z}^{n}/\Lambda)=D$. Then there is a basis
$\bm{v}_{1},\dots,\bm{v}_{n}$ of $\Lambda$ whose entries have absolute values
$\leq D$.

###### Proof.

Take an arbitrary basis of $\Lambda$ and write it as a set of column vectors.

The assertion is translated to the following assertion on matrices: let $A\in
M_{n\times n}(\mathbf{Z})$ be an $n\times n$ matrix with positive determinant
$D$. Then there is an invertible matrix
$U\in\operatorname{GL}_{n}(\mathbf{Z})$ such that $AU$ has entries whose
absolute values are $\leq D$.

When $n=1$ (or $n=0$), the assertion is trivial. We proceed by induction on
$n$.

Let $d_{1}>0$ be the greatest common divisor of the entries in the first row
of $A$. By Euclid’s algorithm, which can be realized as the right
multiplication by invertible matrices, the matrix $A$ can be transformed into
the form

(D.1) $\displaystyle\begin{pmatrix}d_{1}&0\dots 0\\\
*&A^{\prime}\end{pmatrix},\quad\text{ with }A^{\prime}\in
M_{(n-1)\times(n-1)}(\mathbf{Z}).$

This also shows $d_{1}\mid D$. Set $\det(A^{\prime})=D^{\prime}=D/d_{1}$.

From the induction hypothesis, by the right multiplication of invertible
matrices which leaves the first column and row unchanged, one can transform
$A^{\prime}$ so that it has entries whose absolute values are $\leq
D^{\prime}$.

The entries $*$ in (D.1) might a priori have absolute values $>D$, but one can
use the entries of the modified $A^{\prime}$ to make them smaller than
$D^{\prime}$. This completes the proof. ∎

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