# Divisibility conditions on the order
of the reductions of algebraic numbers

Pietro Sgobba  Department of Mathematics, University of Luxembourg, 6 Avenue
de la Fonte, 4364 Esch-sur-Alzette, Luxembourg<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be a number field, and let $G$ be a finitely generated subgroup of
$K^{\times}$. Without relying on the Generalized Riemann Hypothesis we prove
an asymptotic formula for the number of primes $\mathfrak{p}$ of $K$ such that
the order of $(G\bmod\mathfrak{p})$ is divisible by a fixed integer. We also
provide a rational expression for the natural density of this set.
Furthermore, we study the primes $\mathfrak{p}$ for which the order is
$k$-free, and those for which the order has a prescribed $\ell$-adic valuation
for finitely many primes $\ell$. An additional condition on the Frobenius
conjugacy class of $\mathfrak{p}$ may be considered. In order to establish
these results, we prove an unconditional version of the Chebotarev density
theorem for Kummer extensions of number fields.

###### Key words and phrases:

Reductions of algebraic numbers, multiplicative order, distribution of primes,
Chebotarev density theorem, natural density.

###### 1991 Mathematics Subject Classification:

Primary: 11R45; Secondary: 11R44, 11R20

##  1\. Introduction

Consider a number field $K$ and let $G$ be a finitely generated subgroup of
$K^{\times}$. If $\mathfrak{p}$ is a prime of $K$ such that
$v_{\mathfrak{p}}(g)=0$ for all $g\in G$, then the reduction
$(G\bmod\mathfrak{p})$ is a well-defined subgroup of
$k_{\mathfrak{p}}^{\times}$, where $k_{\mathfrak{p}}$ is the residue field at
$\mathfrak{p}$ and $v_{\mathfrak{p}}$ the $\mathfrak{p}$-adic valuation over
$K$. In this paper we investigate the set consisting of the primes
$\mathfrak{p}$ of $K$ such that the order of $(G\bmod\mathfrak{p})$ is well-
defined and it satisfies some divisibility conditions.

More precisely, denote by $\operatorname{ord}_{\mathfrak{p}}(G)$ the order of
$(G\bmod\mathfrak{p})$. In Theorem 1 we prove an asymptotic formula for the
number of primes $\mathfrak{p}$ such that
$m\mid\operatorname{ord}_{\mathfrak{p}}(G)$, where $m$ is any given positive
integer. We also consider the primes $\mathfrak{p}$ such that
$\operatorname{ord}_{\mathfrak{p}}(G)$ is $k$-free, i.e. it is not divisible
by $k$-th powers (greater than $1$), where $k\geqslant 2$. In Theorem 2,
relying on the previous result, we prove an asymptotic formula for the number
of primes satisfying this condition. Given a finite Galois extension of $K$,
an additional condition on the conjugacy class of the Frobenius automorphisms
of the primes lying above $\mathfrak{p}$ may also be considered. Notice that
in this paper we do not rely on the Generalized Riemann Hypothesis (GRH). In
fact, in Theorem 10, which will be essential in the proof of Theorem 1, we
give an unconditional version of the Chebotarev density theorem for
cyclotomic-Kummer extensions of number fields.

The mathematical questions addressed in this paper are closely related to
Artin’s Conjecture on primitive roots, and hence are part of an active
research area; see Moree’s survey [7]. The density of rational primes $p$ such
that $m\mid\operatorname{ord}_{p}(g)$, where
$g\in\mathbb{Q}^{\times}\setminus\\{\pm 1\\}$, has been recently studied by
Pappalardi [10, 11] (also replacing $g$ with a group of rational numbers), by
Moree [6], and previously by Wiertelak [23]. Our paper provides
generalizations and variants of various results by Pappalardi and Moree, as
described in the next sections. Over a number field, Debry and Perucca
considered the density of the primes $\mathfrak{p}$ such that
$\operatorname{ord}_{\mathfrak{p}}(G)$, where $G$ is a group consisting of
algebraic numbers, is not divisible by some fixed prime number (and described
how this permits to treat general divisibility conditions); see [2, 13]. Under
the assumption of GRH, more general results over number fields hold, e.g. for
$\operatorname{ord}_{\mathfrak{p}}(G)$ satisfying a given modular congruence;
see [24] by Ziegler and [14] by Perucca and the author. For more references
and historical background we refer to [7, Sect. 9.2 and 9.3].

###  1.1. Notation

If $m\geqslant 1$ is an integer, then we denote by $\zeta_{m}$ a primitive
$m$-th root of unity, and by $m^{\infty}$ the supernatural number
$\prod_{p\mid m\text{ prime}}p^{\infty}$. As customary, $\mu$ is the Möbius
function.

We fix an algebraic closure $\overline{K}$ of $K$. For $m,n\geqslant 1$ with
$n\mid m$, we write $K_{m,n}:=K(\zeta_{m},G^{1/n})$ for the $n$-th Kummer
extension related to $G$ over $K(\zeta_{m})$, i.e. the subextension of
$\overline{K}/K(\zeta_{m})$ obtained by adding the $n$-th roots of all
elements in $G$. If $F/K$ is a finite Galois extension, and $\mathfrak{p}$ is
a prime of $K$ which does not ramify in $F$, then we denote by
$(\mathfrak{p},F/K)$ the conjugacy class of the Frobenius elements at the
primes of $F$ above $\mathfrak{p}$. If $\mathcal{S}$ is a set of primes of
$K$, then we let $\pi_{\mathcal{S}}(x)$ be the number of primes in
$\mathcal{S}$ with norm up to $x$.

###  1.2. Outline of the main results

The main result is the following.

######  Theorem 1.

Let $F/K$ be a finite Galois extension of number fields, and let $C$ be a
conjugacy-stable subset of $\operatorname{Gal}(F/K)$. Let $G$ be a finitely
generated and torsion-free subgroup of $K^{\times}$ of positive rank $r$. Let
$m$ be a positive integer, and consider the set of primes of $K$ given by

$\mathcal{P}_{m}=\left\\{\mathfrak{p}:m\mid\operatorname{ord}_{\mathfrak{p}}(G),\,\begin{pmatrix}\mathfrak{p}\\\
F/K\end{pmatrix}\subseteq C\right\\}$

(where we are tacitly discarding the finitely many primes $\mathfrak{p}$ that
ramify in $F$ or such that $v_{\mathfrak{p}}(g)\neq 0$ for some $g\in G$).
Then we have

(1) $\pi_{\mathcal{P}_{m}}(x)=\frac{x}{\log
x}\varrho_{C,m}+O\left(x\bigg{(}\frac{(\log\log x)^{2}}{\log
x}\bigg{)}^{1+\frac{1}{r+2}}\right),$

where

(2) $\varrho_{C,m}:=\sum_{n\mid m^{\infty}}\,\sum_{d\mid
m}\frac{\mu(d)c(mn,dn)}{[F_{mn,dn}:K]}$

and where, for all positive integers $a,b$ with $b\mid a$, we set

(3) $c(a,b):=|C\cap\operatorname{Gal}(F/F\cap K_{a,b})|.$

The constant implied by the $O$-term depends only on $F$, $K$, $G$.

The assumption that $G$ is torsion-free allows some simplifications in the
proofs, and in Remark 16 we explain how to deal with the general case. Also
notice that the series in (2) is convergent by Proposition 15. The coefficient
$c(a,b)$ in (3) is always at most $\left\lvert C\right\rvert\leqslant[F:K]$,
and it is equal to $1$ if the condition on the Frobenius is trivial.

For the case $F=K=\mathbb{Q}$, Theorem 1 provides a new version of [11,
Theorem 1] and [6, Lemma 1]. The main challenge for the generalization of
Pappalardi’s method consists in proving a certain unconditional version of the
Chebotarev density theorem for cyclotomic-Kummer extensions of number fields,
as mentioned above. In Section 2 we will argue that this is not difficult if
the base field $K$ is normal over $\mathbb{Q}$. However, for the general case
we need an improvement on the upper bound of a possible Landau–Siegel zero of
the Dedekind zeta function of $K_{m,n}$, which is achieved in Proposition 7.
Notice that Section 2 is self-contained and also of independent interest.

Section 3 is devoted to the proof of Theorem 1, whereas in Section 4 we
justify that the natural density $\varrho_{C,m}$ is a positive rational
number, and for $C=\operatorname{Gal}(F/K)$ (e.g. if $F=K$) we express it in
terms of finite sums and products, see Theorem 18.

Sections 5 and 6 are devoted to proving applications of Theorem 1. In Section
5 we apply Theorem 1 to prove the following result on the primes
$\mathfrak{p}$ of $K$ for which $\operatorname{ord}_{\mathfrak{p}}(G)$ is
$k$-free.

######  Theorem 2.

Let $F/K$ be a finite Galois extension of number fields, and let $C$ be a
conjugacy-stable subset of $\operatorname{Gal}(F/K)$. Let $G$ be a finitely
generated and torsion-free subgroup of $K^{\times}$ of positive rank $r$. Let
$k\geqslant 2$ be an integer, and consider the following set of primes of $K$:

$\mathcal{N}_{k}:=\left\\{\mathfrak{p}:\operatorname{ord}_{\mathfrak{p}}(G)\text{
is $k$-free},\,\begin{pmatrix}\mathfrak{p}\\\ F/K\end{pmatrix}\subseteq
C\right\\}$

(where we are tacitly discarding the primes $\mathfrak{p}$ that ramify in $F$
or such that $v_{\mathfrak{p}}(g)\neq 0$ for some $g\in G$). Then we have

(4) $\pi_{\mathcal{N}_{k}}(x)=\frac{x}{\log x}\sum_{m\geqslant
1}\mu(m)\varrho_{C,m^{k}}+O\left(\frac{x}{(\log
x)^{1+\frac{k-1}{(r+2)k+1}}}\right),$

where $\varrho_{C,m^{k}}$ is as in (2). The set $\mathcal{N}_{k}$ has natural
density

$\beta_{C,k}:=\sum_{m\geqslant 1}\,\sum_{n\mid m^{\infty}}\,\sum_{d\mid
m}\frac{\mu(m)\mu(d)c(nm^{k},dn)}{[F_{nm^{k},dn}:K]},$

where $c(a,b)$ is as in (3). The constant implied by the $O$-term depends only
on $F$, $K$, $G$.

Notice that the convergence of the series $\beta_{C,k}$ follows from
Proposition 15. For the case $F=K=\mathbb{Q}$ we refer to Pappalardi [10,
Theorem 1.2] and [11, Remark (8), p.388]. In Section 6, from Theorem 1 we also
derive Theorem 21, which concerns the primes $\mathfrak{p}$ for which the
$\ell$-adic valuation of $\operatorname{ord}_{\mathfrak{p}}(G)$ has a
prescribed value for finitely many prime numbers $\ell$.

In Section 7, under GRH, we derive some improvements on the error terms of
formulas (1) and (4). Finally, in Section 8 we provide several numerical
examples for the densities considered in this paper.

##  2\. Chebotarev’s density theorem for cyclotomic-Kummer extensions

In this section we prove an effective version of the Chebotarev density
theorem for cyclotomic-Kummer extensions of number fields which is
“unconditional”, i.e. it does not rely on GRH. Let us first introduce some
notation (in addition to the notation of Section 1.1).

###  2.1. Notation

Given a finite Galois extension $L/K$ of number fields and a conju-gacy-stable
subset $C$ of $\operatorname{Gal}(L/K)$, we denote by $\pi_{L/K,C}(x)$ the
number of primes $\mathfrak{p}$ of $K$ with
$\operatorname{N}\mathfrak{p}\leqslant x$ which are unramified in $L$ and such
that $(\mathfrak{p},L/K)\subseteq C$.

Also, $d_{K}$ denotes the absolute discriminant of $K$, $\mathcal{O}_{K}$ the
ring of integers of $K$, and $\zeta_{K}$ the Dedekind zeta function of $K$.
For a finitely generated subgroup $G$ of $K^{\times}$, $\mathcal{P}(G)$ is the
set of primes $\mathfrak{p}$ in $K$ such that $v_{\mathfrak{p}}(g)\neq 0$ for
some $g\in G$ (notice that this set is finite).

Also, we use the following standard notation: $\varphi$ is Euler’s totient
function; given $m\geqslant 1$, $\tau(m)$ is the number of positive divisors
of $m$, and $\operatorname{rad}(m)$ is the radical of $m$, i.e. the largest
squarefree integer dividing $m$;
$\operatorname{Li}(x)=\int_{2}^{x}\frac{dx}{\log x}$ is the logarithmic
integral function.

###  2.2. Chebotarev’s density theorem

We start by stating a recent effective and unconditional version of
Chebotarev’s density theorem by Thorner and Zaman [22]. This result refines
previous versions by Lagarias and Odlyzko [4, Theorem 1.2], Serre [19, Theorem
2], and V.K. Murty [8, Section 4].

######  Theorem 3 ([22, Theorem 1.1]).

Let $L/K$ be a finite Galois extension of number fields with
$L\neq\mathbb{Q}$, and let $C$ be a conjugacy-stable subset of
$\operatorname{Gal}(L/K)$. There exist absolute and effective constants
$c_{1},c_{2}>0$ such that, if

(5) $x\geqslant\big{(}\left\lvert
d_{L}\right\rvert\cdot[L:\mathbb{Q}]^{[L:\mathbb{Q}]}\big{)}^{c_{1}},$

then $\pi_{L/K,C}(x)$ is given by

(6) $\frac{\left\lvert
C\right\rvert}{[L:K]}\big{(}\operatorname{Li}(x)-\theta\operatorname{Li}(x^{\beta})\big{)}\left(1+O\left(e^{-\sqrt{\frac{c_{2}\log
x}{[L:\mathbb{Q}]}}}+e^{-\frac{c_{2}\log x}{\log(\left\lvert
d_{L}\right\rvert[L:\mathbb{Q}]^{[L:\mathbb{Q}]})}}\right)\right)$

where $\theta=\theta(C)\in\\{-1,0,1\\}$, and $\beta$ is a possible
Landau–Siegel zero of the Dedekind zeta function $\zeta_{L}(s)$, i.e. the
unique real zero in the range

$1-\frac{1}{4\log|d_{L}|}\leqslant\alpha\leqslant 1$

(if it exists, otherwise $\theta=0$).

In the formula that we aim to derive, the term
$\frac{|C|}{[L:K]}\operatorname{Li}(x)$ should be dominant. Thus the
difficulty in applying Theorem 3 to cyclotomic-Kummer extensions consists in
estimating the term $\operatorname{Li}(x^{\beta})$, and hence bounding the
value of $\beta$. As of today, the best known bound on $\beta$ is provided by
Stark in [20, Proof of Theorem 1’, p.148]. In fact, if $K/\mathbb{Q}$ is
normal, then that bound is good enough for our purpose, see Remark 8. However,
for the general case we need to deduce from Stark’s results some improvement
which is suitable for our goal.

###  2.3. On a possible Landau–Siegel zero of the Dedekind zeta function

######  Lemma 4 ([20, Lemma 3]).

Let $L\neq\mathbb{Q}$ be a number field. The Dedekind zeta function
$\zeta_{L}(s)$ has at most one zero in the region

(7) $\left\\{s\in\mathbb{C}:1-\frac{1}{4\log\left\lvert
d_{L}\right\rvert}\leqslant\operatorname{Re}(s)<1\text{ and
}\left\lvert\operatorname{Im}(s)\right\rvert\leqslant\frac{1}{4\log\left\lvert
d_{L}\right\rvert}\right\\}.$

If such a zero exists, then it is real and simple.

######  Lemma 5 ([20, Lemma 8]).

Let $L\neq\mathbb{Q}$ be a number field, and set $c_{0}(L)=4$ if
$L/\mathbb{Q}$ is normal, $c_{0}(L)=4\cdot[L:\mathbb{Q}]!$ otherwise. If
$\zeta_{L}$ has a real zero $\beta$ such that
$1-(c_{0}(L)\cdot\log|d_{L}|)^{-1}\leqslant\beta<1$, then there is a quadratic
number field $M$ inside $L$ such that $\zeta_{M}(\beta)=0$.

######  Lemma 6.

Let $L/K$ be a normal extension of number fields with $L\neq\mathbb{Q}$. If
$\zeta_{L}$ has a real zero $\beta$ such that

(8) $1-\frac{1}{4(2[K:\mathbb{Q}])!\cdot\log\left\lvert
d_{L}\right\rvert}\leqslant\beta<1,$

then there is a quadratic number field $M$ inside $L$ such that
$\zeta_{M}(\beta)=0$.

The Lemma says, in particular, that if $L$ has no quadratic subfields, then
$\zeta_{L}(s)$ has no real zero in the range (8).

###### Proof.

If $K=\mathbb{Q}$, then the statement holds by Lemma 5, hence we suppose
$K\neq\mathbb{Q}$. If $\zeta_{L}(s)$ has a zero with real part lying in the
range (8), then by Lemma 4 it must be real, simple, and unique in that range.
Since $L/K$ is normal, by [20, Theorem 3] there is a subextension $F$ of
$L/K$, which is either trivial or quadratic over $K$, such that
$\zeta_{E}(\beta)=0$ for every field $F\subseteq E\subseteq L$. Therefore we
have that $\zeta_{K}(\beta)=0$ or $\zeta_{F}(\beta)=0$ for some quadratic
extension $F/K$ with $F\subseteq L$. We conclude by applying Lemma 5 either to
$K$ or $F$, noticing that the range of the lemma contains the interval (8). ∎

######  Proposition 7.

Let $L/K$ be a Galois extension of number fields with $L\neq\mathbb{Q}$. Then
the possible unique zero $\beta$ of the Dedekind zeta function $\zeta_{L}(s)$
in the region (7) is real and simple, and we have

(9)
$\frac{1}{2}\leqslant\beta\leqslant\max\left\\{1-\frac{1}{4(2[K:\mathbb{Q}])!\log\left\lvert
d_{L}\right\rvert},1-\frac{1}{c_{3}\left\lvert
d_{L}\right\rvert^{1/[L:\mathbb{Q}]}}\right\\},$

where $c_{3}>0$ is an effective absolute constant.

###### Proof.

Clearly $\beta\geqslant 1/2$ as $4\log\left\lvert d_{L}\right\rvert\geqslant
2$. It suffices to show that if $\beta$ is in the range (8), then $\beta$
satisfies (9). We follow the same argument as in [20, Proof of Theorem 1’,
p.148]. If $L$ has no quadratic subfields, then, as mentioned above,
$\zeta_{L}(s)$ has no real zero in the range (8) by Lemma 6 and hence (9) is
satisfied. If $L$ contains a quadratic field and $\zeta_{L}(\beta)=0$ for some
$\beta$ in the range (8), then by Lemma 6 there must be a quadratic subfield
$M$ of $L$ such that $\zeta_{M}(\beta)=0$. By [20, Lemma 11] we must have
$\beta<1-(c_{3}\left\lvert d_{M}\right\rvert^{1/2})^{-1}$, for an effective
absolute constant $c_{3}>0$. We may conclude because we have $\left\lvert
d_{L}\right\rvert\geqslant\left\lvert d_{M}\right\rvert^{[L:\mathbb{Q}]/2}$. ∎

######  Remark 8.

In fact, if in the proof of Lemma 6 we have $\zeta_{K}(\beta)=0$, then the
lower bound of (8) may be taken as $1-(4[K:\mathbb{Q}]!\log\left\lvert
d_{L}\right\rvert)^{-1}$. Moreover, if $L$ is normal over $\mathbb{Q}$ or if
there is a tower of normal extensions $\mathbb{Q}=k_{0}\subset
k_{1}\subset\cdots\subset k_{m}=K$, then the lower bound of (8) may be taken
to be $1-(4\log\left\lvert d_{L}\right\rvert)^{-1}$ by Lemma 5, or
$1-(16\log\left\lvert d_{L}\right\rvert)^{-1}$ by [20, Lemma 10],
respectively. Accordingly, the factor $4(2[K:\mathbb{Q}])!$ in (9) can be
replaced by $4$ or $16$ in the respective cases, by following the same
argument of the proof of Proposition 7.

###  2.4. Chebotarev’s density theorem for cyclotomic-Kummer extensions

######  Proposition 9.

Let $K$ be a number field, and let $G$ be a finitely generated and torsion-
free subgroup of $K^{\times}$ of positive rank $r$. Then, for $m,n\geqslant 1$
with $n\mid m$, we have

$\frac{\log\left\lvert
d_{K_{m,n}}\right\rvert}{[K_{m,n}:\mathbb{Q}]}\leqslant\log(\varphi(m)mn^{r})+\log\big{(}\sigma_{G}\left\lvert
d_{K}\right\rvert\big{)},$

where $\sigma_{G}$ is the product of the rational primes lying below the
primes in $\mathcal{P}(G)$.

###### Proof.

Given a finite extension $L/K$ we write $d_{L/K}$ for the relative
discriminant. We have

(10)
$d_{K_{m,n/\mathbb{Q}}}=\operatorname{N}_{K/\mathbb{Q}}(d_{K_{m,n}/K})\cdot
d_{K/\mathbb{Q}}^{[K_{m,n}:K]}\,,$

see [9, Ch.III, Corollary 2.10]. By [19, Proposition 5], since $K_{m,n}/K$ is
Galois, we have

(11)
$\log\left\lvert\operatorname{N}_{K/\mathbb{Q}}(d_{K_{m,n}/K})\right\rvert\leqslant[K_{m,n}:\mathbb{Q}]\bigg{(}\log[K_{m,n}:K]+\sum_{p\in
P(K_{m,n}/K)}\log p\bigg{)}$

where $P(K_{m,n}/K)$ is the set of rational primes $p$ lying below the primes
of $K$ that ramify in $K_{m,n}$. These prime numbers divide $m$ or
$\sigma_{G}$, as they lie below the primes $\mathfrak{p}$ that divide
$d_{K_{m,n}/K}$, and an estimate for this relative discriminant is [14,
Formula (4.7)]:

$d_{K_{m,n}/K}\mid\Big{(}mn^{r}\prod_{i=1}^{r}(\alpha_{i}\beta_{i})^{2}\Big{)}^{n^{r}\varphi(m)}\mathcal{O}_{K}\,,$

where $\alpha_{i},\beta_{i}\in\mathcal{O}_{K}$ are such that the elements
$\gamma_{i}:=\alpha_{i}/\beta_{i}$ for $i\in\\{1,\ldots,r\\}$ form a basis of
$G$ as a free $\mathbb{Z}$-module. Since $n\mid m$, we have

(12) $\sum_{p\in P(K_{m,n}/K)}\log p\leqslant\log(m\sigma_{G})\,.$

We conclude by taking the natural logarithm of $\left\lvert
d_{K_{m,n}}\right\rvert$, making use of (10), and by applying the bounds (11),
(12) and $[K_{m,n}:K]\leqslant\varphi(m)n^{r}$. ∎

We are now ready to prove an effective unconditional Chebotarev density
theorem for cyclotomic-Kummer extensions of number fields.

######  Theorem 10.

Let $F/K$ be a Galois extension of number fields, and let $G$ be a finitely
generated and torsion-free subgroup of $K^{\times}$ of positive rank $r$. Let
$C$ be a conjugacy-stable subset of $\operatorname{Gal}(F/K)$, and for all
integers $m,n\geqslant 1$ with $n\mid m$, define

(13) $C_{m,n}:=\\{\sigma\in\operatorname{Gal}(F_{m,n}/K):\sigma|_{F}\in
C,\,\sigma|_{K_{m,n}}=\operatorname{id}\\}\,,$

which is a conjugacy-stable subset of $\operatorname{Gal}(F_{m,n}/K)$. Then
there exist constants $c_{4},c_{5}>0$, which depend only on $F$ and $G$, such
that, for all $b>0$ and uniformly for

(14) $m\leqslant c_{4}\left(\frac{\log x}{(\log\log
x)^{b}}\right)^{\frac{1}{r+2}}\,,$

we have

(15) $\pi_{F_{m,n}/K,C_{m,n}}(x)=\frac{\left\lvert
C_{m,n}\right\rvert}{[F_{m,n}:K]}\operatorname{Li}(x)+O_{F,G}\left(\frac{x}{\log
x\cdot e^{c_{5}(\log\log x)^{b}}}\right).$

We first prove an upper bound for $\pi_{F_{m,n}/K,C_{m,n}}(x)$.

######  Proposition 11.

With the notation of Theorem 10, there exists a constant $c_{6}>0$ (depending
only on $F$ and $G$), such that, for all $b>0$ and uniformly for

(16) $m\leqslant c_{6}\left(\frac{\log x}{(\log\log
x)^{1+b}}\right)^{\frac{1}{r+1}},$

we have

$\pi_{F_{m,n}/K,C_{m,n}}(x)\leqslant(2+o(1))\frac{\left\lvert
C_{m,n}\right\rvert}{[F_{m,n}:K]}\operatorname{Li}(x).$

###### Proof.

We apply Theorem 3 to $F_{m,n}/K$ and $C_{m,n}$. By Proposition 9 and since
$[F_{m,n}:F]\leqslant m^{r+1}$ we have

$\log\left\lvert
d_{F_{m,n}}\right\rvert+[F_{m,n}:\mathbb{Q}]\log[F_{m,n}:\mathbb{Q}]\\\
\leqslant[F:\mathbb{Q}]m^{r+1}\big{(}\log\big{(}[F:\mathbb{Q}]m^{r+2}\big{)}+\log\big{(}\sigma_{G}\left\lvert
d_{F}\right\rvert\big{)}\big{)}\leqslant c_{7}\cdot m^{r+1}\log m,$

where $\sigma_{G}$ is as in Proposition 9. Thus, for (5) to be satisfied, we
should have at most $m\ll(\frac{\log x}{\log\log x})^{\frac{1}{r+1}}$.
However, for the second error term in (6) to be $o(1)$, we require the bound
(16) on $m$, which results in the estimate

$m^{r+1}\log m\ll_{F,G}\frac{\log x}{(\log\log x)^{b}}.$

Notice that the constant $c_{6}$ is chosen in such a way that $c_{1}c_{7}\cdot
m^{r+1}\log m\leqslant\log x$. As for the first error term of (6), we have

$\exp\left(-\sqrt{\frac{c_{2}\log
x}{[F_{m,n}:\mathbb{Q}]}}\right)\leqslant\exp\Big{(}-c_{8}(\log\log
x)^{\frac{1+b}{2}}\Big{)}=o(1)\,.$

Applying the inequality
$0<\operatorname{Li}(x)-\theta\operatorname{Li}(x^{\beta})<2\operatorname{Li}(x)$
yields the result. The constants $c_{1},c_{2}$ are as in Theorem 3, and
$c_{7},c_{8}>0$ depend only on $F$ and $G$. ∎

###### Proof of Theorem 10.

We apply Theorem 3 to $F_{m,n}/K$ and $C_{m,n}$. Since the bound (14) is
smaller than (16), the condition (5) is satisfied by the proof of Proposition
11. The constant $c_{4}$ is chosen similarly as for $c_{6}$. The two error
terms in (6) are bounded by

$\exp\left(-\frac{c_{2}\log x}{c_{7}\,m^{r+1}\log
m}\right)\leqslant\exp\left(-\frac{c_{9}(\log x)^{\frac{1}{r+2}}}{\log\log
x}\right)$

and

$\exp\left(-\sqrt{\frac{c_{2}\log
x}{[F_{m,n}:\mathbb{Q}]}}\right)\leqslant\exp\Big{(}-c_{10}(\log
x)^{\frac{1}{2(r+2)}}\cdot(\log\log x)^{\frac{b}{4}}\Big{)}\,.$

Applying the inequality
$0<\operatorname{Li}(x)-\theta\operatorname{Li}(x^{\beta})<2\operatorname{Li}(x)$,
both $O$-terms are smaller than the one in the statement. We now bound the
term $\operatorname{Li}(x^{\beta})$. Since $\beta\geqslant 1/2$, and denoting
$\alpha:=(1-\beta)^{-1}$, we have

$\operatorname{Li}(x^{\beta})\ll\frac{x^{\beta}}{\log x}=\frac{x}{\log
x}\exp\bigg{(}-\frac{\log x}{\alpha}\bigg{)},$

hence we need to bound $\alpha$ with a function $h(x)$ satisfying
$\exp(-\frac{\log x}{h(x)})=o(1)$. Taking into account the two terms in the
upper bound on $\beta$ from Proposition 7, on the one hand, we have

$\alpha\leqslant 4(2[F:\mathbb{Q}])!\log\left\lvert
d_{F_{m,n}}\right\rvert\ll_{F,G}m^{r+1}\log m,$

so $x^{\beta}/\log x$ is bounded as above. On the other hand, we have

$\displaystyle\alpha\leqslant c_{3}\left\lvert
d_{F_{m,n}}\right\rvert^{1/[F_{m,n}:\mathbb{Q}]}$ $\displaystyle\leqslant
c_{3}\exp\left(\log m^{r+2}+\log\big{(}\sigma_{G}\left\lvert
d_{F}\right\rvert\big{)}\right)$
$\displaystyle\ll_{F,G}m^{r+2}\ll_{F,G}\frac{\log x}{(\log\log x)^{b}},$

which satisfies our requirement and yields the bound

$\frac{x^{\beta}}{\log x}\leqslant\frac{x}{\log x\cdot e^{c_{5}(\log\log
x)^{b}}}.$

Collecting all error terms gives the asymptotic formula in the statement. The
constants $c_{5},c_{9},c_{10}>0$ depend only on $F$ and $G$. ∎

##  3\. The order being divisible by a given integer

In this section we prove Theorem 1. We first set some notation, in addition to
the notation of Sections 1.1 and 2.1.

###  3.1. Notation

Let $F/K$ be a Galois extension of number fields, $C$ a conjugacy-stable
subset of $\operatorname{Gal}(F/K)$, and $G$ a finitely generated and torsion-
free subgroup of $K^{\times}$. We say that $\mathfrak{p}$ is a prime of
_degree_ $1$ in $K$ if its ramification index and residue class degree over
$\mathbb{Q}$ are equal to $1$. For $m,n\geqslant 1$ with $n\mid m$ we define
$\pi^{1}_{m,n}(x)$ to be the number of primes $\mathfrak{p}$ of $K$, with
$\operatorname{N}\mathfrak{p}\leqslant x$, which are of degree $1$, split
completely in $K_{m,n}$, do not ramify in $F$, and satisfy
$(\mathfrak{p},F/K)\subseteq C$. In other words, $\pi^{1}_{m,n}(x)$ is the
number of primes counted by $\pi_{F_{m,n}/K,C_{m,n}}(x)$ which are of degree
$1$, where $C_{m,n}$ is as in (13) (we are fixing $K$, $F$, $G$, and $C$).

For $\mathfrak{p}\notin\mathcal{P}(G)$, recall that
$\operatorname{ord}_{\mathfrak{p}}(G)$ is the order of $(G\bmod\mathfrak{p})$,
and we also denote by $\operatorname{ind}_{\mathfrak{p}}(G)$ the index of
$(G\bmod\mathfrak{p})$, namely

$\operatorname{ind}_{\mathfrak{p}}(G)=[k_{\mathfrak{p}}^{\times}:\langle
G\bmod\mathfrak{p}\rangle]=(\operatorname{N}\mathfrak{p}-1)/\operatorname{ord}_{\mathfrak{p}}(G).$

Given integers $m,n\geqslant 1$, we write $(m,n)$ for $\gcd(m,n)$. If $m$ is
replaced by the supernatural number $m^{\infty}$, we write
$(n,m^{\infty})=\prod_{\ell\mid m\text{ prime}}\ell^{v_{\ell}(n)}$, where
$v_{\ell}$ is the $\ell$-adic valuation.

###  3.2. Proof of Theorem 1

Recall that if $\mathfrak{p}$ is a prime of $K$ of degree $1$ such that
$\mathfrak{p}\notin\mathcal{P}(G)$, then $\operatorname{N}\mathfrak{p}\equiv
1\bmod n$ if and only if $\mathfrak{p}$ splits completely in $K(\zeta_{n})$,
and $n\mid\operatorname{ind}_{\mathfrak{p}}(G)$ if and only if $\mathfrak{p}$
splits completely in $K_{n,n}$, where $n\geqslant 1$; see also [24, Lemma 2].
Hence, we easily deduce that, for $m,n\geqslant 1$ with $n\mid m$, we have

(17)
$\pi^{1}_{m,n}(x)=\left\lvert\left\\{\mathfrak{p}:\operatorname{N}\mathfrak{p}\leqslant
x,\,\operatorname{N}\mathfrak{p}\equiv 1\bmod
m,\,n\mid\operatorname{ind}_{\mathfrak{p}}(G),\,\begin{pmatrix}\mathfrak{p}\\\
F/K\end{pmatrix}\subseteq C\right\\}\right\rvert,$

(where we only consider primes of degree $1$). Notice that, since the number
of primes of $K$ not of degree $1$ is of order
$O(\sum_{p\leqslant\sqrt{x}}1)=O\left(\sqrt{x}/\log x\right)$, see e.g. [24,
Lemma 1], the same asymptotic formula (15) gives $\pi_{m,n}^{1}(x)$.

######  Lemma 12.

If $\mathcal{P}_{m}$ is as in Theorem 1, then we have

$\pi_{\mathcal{P}_{m}}(x)=\sum_{n\mid m^{\infty}}\sum_{d\mid
m}\mu(d)\pi^{1}_{mn,dn}(x)+O\left(\frac{\sqrt{x}}{\log x}\right).$

###### Proof.

The proof is a variation of [6, Proof of Proposition 1]. The $O$-term
estimates the primes of $K$ which are not of degree $1$. Let
$\mathfrak{p}\in\mathcal{P}_{m}$ be a prime of degree $1$, and let
$\operatorname{N}\mathfrak{p}=p$. Then we have $m\mid(p-1)$ and there is a
unique $n\mid m^{\infty}$ such that $p\equiv 1\bmod mn$,
$n\mid\operatorname{ind}_{\mathfrak{p}}(G)$ and
$(\frac{\operatorname{ind}_{\mathfrak{p}}(G)}{n},m)=1$ (we must have
$n=(\operatorname{ind}_{\mathfrak{p}}(G),m^{\infty})$). Hence we can write

$\pi_{\mathcal{P}_{m}}(x)=\sum_{n\mid
m^{\infty}}\pi_{\mathcal{B}_{n}}(x)+O\left(\frac{\sqrt{x}}{\log x}\right)\,,$

where for $n\mid m^{\infty}$ we set

(18) $\mathcal{B}_{n}:=\left\\{\mathfrak{p}:p\equiv 1\bmod
mn,\,n\mid\operatorname{ind}_{\mathfrak{p}}(G),\,\Big{(}\frac{\operatorname{ind}_{\mathfrak{p}}(G)}{n},m\Big{)}=1,\,\begin{pmatrix}\mathfrak{p}\\\
F/K\end{pmatrix}\subseteq C\right\\}$

(we are tacitly assuming that the primes in $\mathcal{B}_{n}$ are of degree
$1$, do not lie in $\mathcal{P}(G)$ and do not ramify in $F$). Notice that,
$\mathfrak{p}\in\mathcal{B}_{n}$ satisfies
$m\mid\operatorname{ord}_{\mathfrak{p}}(G)$ because of the two conditions
$p\equiv 1\bmod mn$ and
$\big{(}\frac{\operatorname{ind}_{\mathfrak{p}}(G)}{n},m\big{)}=1$ and the
identity
$\operatorname{ord}_{\mathfrak{p}}(G){\cdot}\operatorname{ind}_{\mathfrak{p}}(G)=p-1$.

Next we apply the inclusion-exclusion principle to the condition
$\big{(}\frac{\operatorname{ind}_{\mathfrak{p}}(G)}{n},m\big{)}=1$, which
amounts to $n\mid\operatorname{ind}_{\mathfrak{p}}(G)$ and
$n\ell\nmid\operatorname{ind}_{\mathfrak{p}}(G)$ for all primes $\ell\mid m$,
and so we obtain

$\pi_{\mathcal{B}_{n}}(x)=\sum_{d\mid
m}\mu(d)\left\lvert\left\\{\mathfrak{p}:p\leqslant x,\,p\equiv 1\bmod
mn,\,dn\mid\operatorname{ind}_{\mathfrak{p}}(G),\,\begin{pmatrix}\mathfrak{p}\\\
F/K\end{pmatrix}\subseteq C\right\\}\right\rvert\,.$

We conclude by (17) that

(19) $\pi_{\mathcal{B}_{n}}(x)=\sum_{d\mid m}\mu(d)\pi^{1}_{mn,dn}(x)\,.\qed$

######  Remark 13.

It follows from the definition of $\mathcal{B}_{n}$ (18) that

$\pi_{\mathcal{B}_{n}}(x)\leqslant[K:\mathbb{Q}]\cdot\left\lvert\left\\{p\leqslant
x:p\equiv 1\bmod mn\right\\}\right\rvert,$

and $\pi_{\mathcal{B}_{n}}(x)\leqslant\pi_{mn,n}^{1}(x)$. From this last
inequality, identity (19), and by the Chebotarev density theorem we deduce

$0\leqslant\sum_{d\mid m}\frac{\mu(d)\left\lvert
C_{mn,dn}\right\rvert}{[F_{mn,dn}:K]}\leqslant\frac{\left\lvert
C_{mn,n}\right\rvert}{[F_{mn,n}:K]}\leqslant\frac{1}{[K_{mn,n}:K]}.$

We are now ready to prove Theorem 1.

###### Proof of Theorem 1.

Notice that, for $m,n\geqslant 1$ with $n\mid m$, the coefficient
$c(m,n)=|C\cap\operatorname{Gal}(F/F\cap K_{m,n})|$ defined in (3) is equal to
$|C_{m,n}|$, because if $\sigma\in C$ fixes $F\cap K_{m,n}$, then $\sigma$ can
be lifted to a unique element of $C_{m,n}$. We are going to apply Lemma 12 and
Theorem 10 with $b=2$ (in fact, $b>1$ suffices). For $n\mid m^{\infty}$ let
$\mathcal{B}_{n}$ be as in (18), and recall (19). Set $y:=c_{4}(\log
x/(\log\log x)^{2})^{1/(r+2)}$, where $c_{4}>0$ is the constant of Theorem 10.
Thus, we have

$\displaystyle\pi_{\mathcal{P}_{m}}(x)=$
$\displaystyle\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\ nm\leqslant
y\end{subarray}}\sum_{d\mid
m}\mu(d)\pi^{1}_{mn,dn}(x)+O\Bigg{(}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\
nm>y\end{subarray}}\pi_{\mathcal{B}_{n}}(x)\Bigg{)}+O\left(\frac{\sqrt{x}}{\log
x}\right)$ $\displaystyle=$
$\displaystyle\operatorname{Li}(x)\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm\leqslant y\end{subarray}}\sum_{d\mid
m}\frac{\mu(d)c(mn,dn)}{[F_{mn,dn}:K]}+O_{F,G}\Bigg{(}\frac{\tau(m)}{m}\frac{x\cdot
y}{\log x\cdot e^{c_{5}(\log\log x)^{2}}}\Bigg{)}$
$\displaystyle+O\Bigg{(}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm>y\end{subarray}}\pi_{\mathcal{B}_{n}}(x)\Bigg{)}+O\left(\frac{\sqrt{x}}{\log
x}\right).$

In order to estimate the tail of the series in the main term we make use of
Remark 13 and obtain

(20) $\displaystyle\pi_{\mathcal{P}_{m}}(x)=$
$\displaystyle\operatorname{Li}(x)\sum_{n\mid m^{\infty}}\sum_{d\mid
m}\frac{\mu(d)c(mn,dn)}{[F_{mn,dn}:K]}+O\Bigg{(}\frac{x}{\log
x}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
mn>y\end{subarray}}\frac{1}{[K_{mn,n}:K]}\Bigg{)}$ (21)
$\displaystyle+O_{F,G}\Bigg{(}\frac{x\cdot y}{\log x\cdot e^{c_{5}(\log\log
x)^{2}}}\Bigg{)}+O\Bigg{(}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm>y\end{subarray}}\pi_{\mathcal{B}_{n}}(x)\Bigg{)}.$

The first error term in (21) is negligible with respect to the error term in
the statement. Let us estimate the error term in (20). Since
$[K_{mn,n}:K]\gg_{K,G}\varphi(mn)n$ by [14, Theorem 3.1] and
$\varphi(mn)=\varphi(m)n$ (as $\operatorname{rad}(n)\mid m$), we can bound

$\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
mn>y\end{subarray}}\frac{1}{[K_{mn,n}:K]}\ll_{K,G}\frac{1}{\varphi(m)}\sum_{n>y/m}\frac{1}{n^{2}}\ll_{K,G}\frac{m}{\varphi(m)y}.$

As $m/\varphi(m)=O\left(\log\log m\right)$, see e.g. [17, Theorem 15], and
$m\leqslant x$ without loss of generality, we then have

(22) $\frac{x}{\log x}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
mn>y\end{subarray}}\frac{1}{[K_{mn,n}:K]}\ll_{K,G}\frac{x\log\log x}{y\log
x}.$

Next we focus on the second error term in (21). First notice that, since
$\frac{mn}{\varphi(mn)}=\frac{m}{\varphi(m)}$, by [10, Lemma 3.3] for
$T\geqslant 1$ and $0<c<1$, we have

(23) $\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm>T\end{subarray}}\frac{1}{\varphi(mn)}=\frac{m}{\varphi(m)}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\
nm>T\end{subarray}}\frac{1}{mn}\ll_{c}\frac{m}{\varphi(m)}\frac{1}{T^{c}}.$

We bound $\pi_{\mathcal{B}_{n}}(x)$ first with the number of primes
$\mathfrak{p}$ with $\operatorname{N}\mathfrak{p}\leqslant x$ splitting
completely in $K(\zeta_{mn},\alpha^{1/n})$ for some $\alpha\in G$, $\alpha\neq
1$, and then with $[K:\mathbb{Q}]$ times the number of multiples of $mn$ up to
$x$ (see Remark 13). Hence, setting $z:=(\log x)^{2/c}$, we have

(24) $\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm>y\end{subarray}}\pi_{\mathcal{B}_{n}}(x)\ll_{K}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\ y<nm\leqslant
z\end{subarray}}\pi_{K(\zeta_{mn},\alpha^{1/n})/K,\operatorname{id}}(x)+\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\ nm>z\end{subarray}}\left\lvert\left\\{k\leqslant x:mn\mid
k\right\\}\right\rvert$

Applying (23), the second term in this last expression is bounded by

$\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
nm>z\end{subarray}}\frac{x}{mn}\ll\frac{x}{\log^{2}x}.$

Setting $w:=c_{6}\frac{\sqrt{\log x}}{\log\log x}$, where $c_{6}>0$ is as in
(16), we split the first sum in the right-hand side of (24) into two parts.
Thanks to Proposition 11 (applied with $r=b=1$) we bound the sum with range
$y<mn\leqslant w$ by

$\frac{x}{\log x}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\ y<nm\leqslant
w\end{subarray}}\frac{1}{[K(\zeta_{mn},\alpha^{1/n}):K]}\ll\frac{x}{\varphi(m)\log
x}\sum_{n>y/m}\frac{1}{n^{2}},$

and this is estimated as in (22). Invoking Remark 13, and applying the
Brun–Titchmarsh Theorem, we estimate the sum with range $w<mn\leqslant z$ by

$\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\ w<nm\leqslant
z\end{subarray}}\frac{x}{\varphi(mn)\log(x/mn)}\leqslant\frac{x}{\log(x/z)}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\
mn>w\end{subarray}}\frac{1}{\varphi(mn)}\ll_{K,G}\frac{x(\log\log
x)^{9/5}}{(\log x)^{7/5}},$

where we used (23) with $c=4/5$. This term is bounded by the error term in the
statement.

Notice that we may replace $\operatorname{Li}(x)$ with $x/\log x$ in the main
term of (20) because $\operatorname{Li}(x)=x/\log x+O(x/\log^{2}x)$ and the
series by which it is multiplied is convergent (see Proposition 15). ∎

###  3.3. Properties and remarks

######  Remark 14.

One can see from [10, Proof of Lemma 3.3] that the constant depending on $c$
arising in (23) can be taken equal to

$\prod_{\begin{subarray}{c}p\text{ prime}\\\ p^{1-c}\leqslant
2\end{subarray}}\frac{1}{p^{1-c}-1}.$

Thus, in [11, Proof of Theorem 1] the parameter $0<c<1$ should be independent
of $x$, and this would result in an error term having the exponent
$1+\frac{c}{3(r+1)}$. Although our technique avoids this problem, it is still
useful to see how to deal with $c$ depending on $x$. Let $m\geqslant 1$, and
$\omega(m)$ the number of prime factors of $m$. By the Mean value theorem we
have $1-1/p^{b}>bp^{-b}\log p$, with $0<b<1$ and $p$ any prime number. Thus,
from [10, Proof of Lemma 3.3] we obtain

$\sum_{\begin{subarray}{c}k>T\\\ m\mid k\mid
m^{\infty}\end{subarray}}\frac{1}{k}\leqslant\frac{1}{m^{1-c}T^{c}}\prod_{\begin{subarray}{c}p\mid
m\\\
\text{prime}\end{subarray}}\left(1-\frac{1}{p^{1-c}}\right)^{-1}\leqslant\frac{2(1-c)^{-\omega(m)}}{T^{c}}.$

Taking $c=1-1/\log\log x$ and making use of this inequality in [11, Proof of
Theorem 1] reduces the final error term to

$x(\log\log x)^{\omega(m)}\bigg{(}\frac{(\log\log x)^{2}}{\log
x}\bigg{)}^{1+\frac{1}{3(r+1)}}.$

######  Proposition 15.

The series $\varrho_{C,m}$ from Theorem 1 is convergent, and we have

$\varrho_{C,m}\ll_{K}\frac{\zeta(r+1)}{\varphi(m)},$

where $\zeta$ is the Riemann zeta function.

###### Proof.

Applying Remark 13 and the estimate
$[K_{mn,n}:K]\gg_{K,G}\varphi(mn)n^{r}=\varphi(m)n^{r+1}$ (see [14, Theorem
3.1]), we have

$\varrho_{C,m}\leqslant\sum_{n\mid
m^{\infty}}\frac{1}{[K_{mn,n}:K]}\ll_{K,G}\frac{1}{\varphi(m)}\sum_{n\mid
m^{\infty}}\frac{1}{n^{r+1}}\leqslant\frac{1}{\varphi(m)}\zeta(r+1).\qed$

######  Remark 16.

The assumption that the group $G$ is torsion-free allows some simplifications
throughout the proof of Theorem 1. However, the general case can be treated
easily. Let $G^{\prime}$ be a finitely generated subgroup of $K^{\times}$ with
torsion, and write $G^{\prime}=G\times\langle\zeta_{t}\rangle$, where
$\zeta_{t}\in K^{\times}$, $t\geqslant 2$, and $G\subseteq K^{\times}$ is
torsion-free. Then, for all primes $\mathfrak{p}$ of $K$ of norm large enough,
we have

$m\mid\operatorname{ord}_{\mathfrak{p}}(G^{\prime})\qquad\text{ if and only if
}\qquad\prod_{\begin{subarray}{c}\ell\mid m\text{ prime}\\\
v_{\ell}(m)>v_{\ell}(t)\end{subarray}}\ell^{v_{\ell}(m)}\mid\operatorname{ord}_{\mathfrak{p}}(G).$

######  Remark 17.

Let us consider the expression of the density $\varrho_{C,m}$ for some special
cases of $C$. If $C=\operatorname{Gal}(F/K)$, then the condition on the
Frobenius becomes trivial and $c(a,b)=[F:F\cap K_{a,b}]$. Therefore, we obtain

$\varrho_{m}:=\varrho_{\operatorname{Gal}(F/K),m}=\sum_{n\mid
m^{\infty}}\sum_{d\mid m}\frac{\mu(d)}{[K_{mn,dn}:K]}.$

If $C=\left\\{\operatorname{id}\right\\}$, then the condition
$(\mathfrak{p},F/K)=\operatorname{id}$ is equivalent to $\mathfrak{p}$
splitting completely in $F$. In this case $c(a,b)=1$, and hence
$\varrho_{\left\\{\operatorname{id}\right\\},m}$ equals $1/[F:K]$ times the
density of primes $\mathfrak{P}$ of $F$ such that
$m\mid\operatorname{ord}_{\mathfrak{P}}(G)$.

Finally, if $F$ is linearly disjoint over $K$ from $K_{a,b}$, then
$c(a,b)=\left\lvert C\right\rvert$. Hence, if this holds for all $a,b$ we
obtain $\varrho_{C,m}=\varrho_{m}\cdot\left\lvert C\right\rvert/[F:K]$.

Clearly, analogous statements hold for the densities $\beta_{C,k}$ of Theorem
2 and $\gamma_{C,k,m}$ of Theorem 21.

##  4\. A rational formula for the density

As a special case of [15, Corollary 7], the natural density $\varrho_{C,m}$ of
the set $\mathcal{P}_{m}$ from Theorem 1 is a positive rational number. In
this section we also provide an explicit closed formula for $\varrho_{C,m}$
when the condition on the Frobenius is trivial (in this case we write
$\varrho_{m}$ for $\varrho_{C,m}$, as in Remark 17). In the rest of the paper,
$\ell$ will always represent a prime number (also when not mentioned
explicitly). Notice that, over $\mathbb{Q}$, Pappalardi [11] provided an
explicit rational formula for $\varrho_{m}$ for $G$ consisting of positive
rationals, whereas the case of groups with negative rationals was considered
in Abdullah et al. [1].

######  Theorem 18.

Let $K$ be a number field and let $G$ be a finitely generated and torsion-free
subgroup of $K^{\times}$ of positive rank $r$. Let $m\geqslant 1$ be an
integer and let $\varrho_{m}$ be the natural density of the set of primes
$\mathfrak{p}$ of $K$ such that $m\mid\operatorname{ord}_{\mathfrak{p}}(G)$
(where $\mathfrak{p}\notin\mathcal{P}(G)$). Then there is an integer $z$,
which depends only on $K$ and $G$, such that

(25) $\varrho_{m}=\frac{1}{\varphi(m)}\prod_{\begin{subarray}{c}\ell\mid m\\\
\ell\nmid
z\end{subarray}}\frac{\ell(\ell^{r}-1)}{\ell^{r+1}-1}\cdot\sum_{\begin{subarray}{c}g\mid
z\\\ \operatorname{rad}(g)\mid(m,z)\mid g\end{subarray}}\sum_{h\mid
g}\frac{p(g,h)}{[K_{g,h}:K]},$

where we set $p(g,h)=0$ if and only if at least one of the following
conditions holds:

  * •

there is $\ell\mid g$, $\ell\nmid h$ such that $v_{\ell}(g/(m,z))>0$,

  * •

there is $\ell\mid h$ such that $v_{\ell}(z/g)>0$ and
$v_{\ell}(g/h)\notin\\{v_{\ell}(m),v_{\ell}(m)-1\\}$,

  * •

there is $\ell\mid h$ such that $v_{\ell}(z/g)=0$ and
$v_{\ell}(g/h)>v_{\ell}(m)$;

else we define

$p(g,h)=\frac{\varphi(g)}{h}\cdot\prod_{\begin{subarray}{c}\ell\mid h,\
v_{\ell}(\frac{z}{g})>0\\\
v_{\ell}(\frac{g}{h})=v_{\ell}(m)-1\end{subarray}}-\ell\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\ v_{\ell}(\frac{z}{g})=0\\\ 1\leqslant
v_{\ell}(\frac{g}{h})<v_{\ell}(m)\end{subarray}}-(\ell-1)\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\
v_{\ell}(\frac{z}{h})=0\end{subarray}}\frac{-\ell^{r+1}(\ell-1)}{\ell^{r+1}-1}.$

Notice that the formula for $\varrho_{m}$ involves only finite sums and
products. Moreover, for a general number field $K$ and a finitely generated
and torsion-free group $G\subseteq K^{\times}$, the integer $z$ is explicitly
described by the results of [16] (see e.g. [16, Theorem 1.2 and its proof]).
Also, notice that for all $m$ such that $(m,z)=1$ we have

$\varrho_{m}=\frac{\operatorname{rad}(m)}{\varphi(m)}\prod_{\ell\mid
m}\frac{\ell^{r}-1}{\ell^{r+1}-1}.$

The rational number characterized by $z$ in (25), i.e. obtained by isolating
the dependence on $z$ of $\varrho_{m}$, should be interpreted as a correction
factor due to the entanglement between the radicals of elements in $G$ and the
roots of unity in cyclotomic extensions of $K$. The same remark applies to the
case of $k$-free orders in Theorem 20. The correction factors arising in
problems related to Artin’s primitive root conjecture have been the object of
study of several articles; see, for instance, [3, 5, 21].

###### Proof of Theorem 18.

By [16, Theorem 1.1] there is an integer $z$, which depends only on $K$ and
$G$, such that for $n\mid m$ we have

$[K_{m,n}:K]=\frac{\varphi(m)n^{r}}{\varphi((m,z))(n,z)^{r}}\cdot[K_{(m,z),(n,z)}:K]\,.$

Therefore, we have

(26) $\varrho_{m}=\sum_{n\mid m^{\infty}}\sum_{d\mid
m}\frac{\mu(d)}{[K_{mn,dn}:K]}=\frac{1}{\varphi(m)}\sum_{\begin{subarray}{c}g\mid
z\\\ h\mid
g\end{subarray}}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\ (mn,z)=g\end{subarray}}\ \sum_{\begin{subarray}{c}d\mid m\\\
(dn,z)=h\end{subarray}}\frac{\mu(d)}{n^{r+1}d^{r}}.$

First of all, for all $n\mid m^{\infty}$ we have that
$\operatorname{rad}(mn,z)=\operatorname{rad}(m,z)$, hence we may restrict the
sum on $g\mid z$ to the divisors such that $(m,z)\mid g$ and
$\operatorname{rad}(g)=\operatorname{rad}((m,z))$ both hold. To simplify the
notation, let us write $m_{\ell}=v_{\ell}(m)$, and similarly for
$z_{\ell},g_{\ell},h_{\ell}$. Then, by properties of the multiplicative
functions, from (26) we obtain

$\varrho_{m}=\frac{1}{\varphi(m)}\cdot\sum_{\begin{subarray}{c}g\mid z\\\
\operatorname{rad}(g)\mid(m,z)\mid g\end{subarray}}\sum_{h\mid
g}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\cdot\prod_{\ell\mid m}p_{\ell}(g,h)$

where for $\ell\mid m$ we define

(27) $p_{\ell}(g,h):=\sum_{\begin{subarray}{c}s\geqslant 0\\\
\min(m_{\ell}+s,z_{\ell})=g_{\ell}\end{subarray}}\
\sum_{\begin{subarray}{c}e\in\\{0,1\\}\\\
\min(s+e,z_{\ell})=h_{\ell}\end{subarray}}\frac{\mu(\ell^{e})}{\ell^{s(r+1)}\ell^{er}}.$

If $\ell\mid m$ and $\ell\nmid z$, then the two conditions on the indices are
trivial and we have

$p_{\ell}(g,h)=\frac{\ell(\ell^{r}-1)}{\ell^{r+1}-1}.$

This computation already justifies the first product in (25). Next, we take

$p(g,h):=\varphi(g)h^{r}\prod_{\ell\mid g}p_{\ell}(g,h),$

and compute $p_{\ell}(g,h)$ depending on the prime factors $\ell$ of $g$
(equivalently, of $(m,z)$).

_Case 1: $\ell\mid g$ and $\ell\nmid h$._ Since $\ell\nmid h$, the conditions
on the indices in (27) hold only for $s=e=0$, and so $p_{\ell}(g,h)=1$ if
$\min(m_{\ell},z_{\ell})=g_{\ell}$, and $p_{\ell}(g,h)=0$ otherwise.

_Case 2: $\ell\mid h$ and $g_{\ell}<z_{\ell}$._ Since $1\leqslant
h_{\ell}<z_{\ell}$, the conditions on the indices hold only for
$s+e=h_{\ell}$. Therefore, if $g_{\ell}=m_{\ell}+h_{\ell}$, then
$p_{\ell}(g,h)=1/\ell^{h_{\ell}(r+1)}$; if $g_{\ell}=m_{\ell}+h_{\ell}-1$,
then $p_{\ell}(g,h)=-\ell/\ell^{h_{\ell}(r+1)}$; otherwise $p_{\ell}(g,h)=0$.

_Case 3: $\ell\mid h$ and $h_{\ell}<g_{\ell}=z_{\ell}$._ The conditions on the
indices hold only for $s+e=h_{\ell}$ and $m_{\ell}+s\geqslant
z_{\ell}=g_{\ell}$. Therefore, if $m_{\ell}+h_{\ell}-1\geqslant z_{\ell}$,
then $p_{\ell}(g,h)=-(\ell-1)/\ell^{h_{\ell}(r+1)}$; if
$m_{\ell}+h_{\ell}=z_{\ell}$, then $p_{\ell}(g,h)=1/\ell^{h_{\ell}(r+1)}$;
otherwise $p_{\ell}(g,h)=0$.

_Case 4: $\ell\mid h$ and $h_{\ell}=z_{\ell}$._ Since
$h_{\ell}=g_{\ell}=z_{\ell}\geqslant 1$, the conditions on the indices hold if
and only if $s+e\geqslant h_{\ell}$. Therefore, we obtain

$p_{\ell}(g,h)=-\frac{1}{\ell^{h_{\ell}(r+1)}}\frac{\ell^{r+1}(\ell-1)}{\ell^{r+1}-1}.\qed$

##  5\. The order being $k$-free

In this section we prove Theorem 2.

###### Proof of Theorem 2.

If $\mathfrak{p}$ is a prime of $K$ with $\mathfrak{p}\notin\mathcal{P}(G)$,
then $\operatorname{ord}_{\mathfrak{p}}(G)$ is $k$-free if and only if for
every rational prime $q$ we have
$q^{k}\nmid\operatorname{ord}_{\mathfrak{p}}(G)$. Therefore, by the inclusion-
exclusion principle we have

$\pi_{\mathcal{N}_{k}}(x)=\sum_{m\geqslant
1}\mu(m)\pi_{\mathcal{P}_{m^{k}}}^{1}(x)+O\left(\frac{\sqrt{x}}{\log
x}\right),$

where $\pi_{\mathcal{P}_{m}}^{1}(x)$ counts the primes of degree $1$ in
$\mathcal{P}_{m}$ with norm up to $x$ (the $O$-term estimates the primes not
of degree 1). Notice that for $\pi_{\mathcal{P}_{m}}^{1}(x)$ we may take the
same asymptotic formula (1) as for $\pi_{\mathcal{P}_{m}}(x)$. Then, for
$0<a<1$ and $z:=\log^{a}x$ we have

$\displaystyle\pi_{\mathcal{N}_{k}}(x)$ $\displaystyle=\sum_{m\leqslant
z}\mu(m)\pi_{\mathcal{P}_{m^{k}}}^{1}(x)+\bigg{(}\sum_{m>z}\pi_{\mathcal{P}_{m^{k}}}^{1}(x)\bigg{)}+O\left(\frac{\sqrt{x}}{\log
x}\right)$ (28) $\displaystyle=\frac{x}{\log
x}\beta_{C,k}+\bigg{(}\frac{x}{\log
x}\sum_{m>z}\varrho_{C,m^{k}}\bigg{)}+\bigg{(}\sum_{m>z}\pi_{\mathcal{P}_{m^{k}}}^{1}(x)\bigg{)}$
(29) $\displaystyle\quad+O\Bigg{(}\sum_{m\leqslant z}x\bigg{(}\frac{(\log\log
x)^{2}}{\log x}\bigg{)}^{1+\frac{1}{r+2}}\Bigg{)}+O\left(\frac{\sqrt{x}}{\log
x}\right)\,.$

By Theorem 1 we have

$\sum_{m>z}\pi_{\mathcal{P}_{m^{k}}}^{1}(x)\ll_{F,K,G}\frac{x}{\log
x}\sum_{m>z}\varrho_{C,m^{k}}.$

By Proposition 15 and recalling that $m/\varphi(m)=O_{\eta}(m^{\eta})$, we
have $\varrho_{C,m^{k}}\ll_{\eta}1/m^{k-\eta}$ for every $0<\eta<1$. Hence
both $O$-terms in (28) are bounded by

(30) $\frac{x}{\log x}\sum_{m>z}\frac{1}{m^{k-\eta}}\ll_{\eta}\frac{x}{(\log
x)^{1+a(k-1-\eta)}}.$

Next, we can bound the first error term in (29) by

(31) $(\log x)^{a}\cdot x\bigg{(}\frac{(\log\log x)^{2}}{\log
x}\bigg{)}^{1+\frac{1}{r+2}}\leqslant\frac{x(\log\log x)^{3}}{(\log
x)^{1+\frac{1}{r+2}-a}},$

and we may choose $a<\frac{1}{(r+2)(k-\eta)}$ so that (31) can be bounded by
(30). Thus, we may decrease the exponent in the denominator of (30) until
$1+\frac{k-1}{(r+2)k+1}$, taking, e.g. $\eta=1/(r+4)$. Collecting the errors
yields the result. ∎

######  Remark 19.

In the context of Theorem 2, the case of groups with torsion is
straightforward: if $G^{\prime}$ is a finitely generated subgroup of
$K^{\times}$ with torsion of order $t$, and
$G=G^{\prime}/\langle\zeta_{t}\rangle$, then the density of the set
$\mathcal{N}_{k,G^{\prime}}$ (i.e. $\mathcal{N}_{k}$ defined for the group
$G^{\prime}$) is equal to the density of $\mathcal{N}_{k,G}$ if $t$ is
$k$-free, and it is $0$ otherwise.

Next we prove an explicit formula for the density $\beta_{C,k}$ of Theorem 2
if the condition on the Frobenius is trivial, and in this case we simply write
$\beta_{k}$.

######  Theorem 20.

Let $K$ be a number field and let $G$ be a finitely generated and torsion-free
subgroup of $K^{\times}$ of positive rank $r$. Let $k\geqslant 2$ and let
$\beta_{k}$ be the natural density of the set of primes $\mathfrak{p}$ of $K$
such that $\operatorname{ord}_{\mathfrak{p}}(G)$ is $k$-free (where
$\mathfrak{p}\notin\mathcal{P}(G)$). Then there is an integer $z$, which
depends only on $K$ and $G$, such that

(32) $\beta_{k}=\prod_{\ell\nmid
z}\Big{(}1-\frac{\ell^{r}-1}{(\ell-1)(\ell^{r+1}-1)\ell^{k-2}}\Big{)}\cdot\sum_{\begin{subarray}{c}g\mid
z\\\ (\operatorname{rad}(g)^{k},z)\mid g\end{subarray}}\sum_{h\mid
g}\frac{p(g,h)}{[K_{g,h}:K]},$

where we set $p(g,h)=0$ if and only if at least one of the following
conditions is satisfied:

  * •

there is $\ell\mid g$, $\ell\nmid h$ and $v_{\ell}(g)\neq
v_{\ell}((\ell^{k},z))$,

  * •

there is $\ell\mid h$ such that $v_{\ell}(g/h)>k$, or $v_{\ell}(z/g)>0$ and
$v_{\ell}(g/h)<k-1$;

else we define $p(g,h)$ to be

$\frac{g}{h\operatorname{rad}(g)^{k}}\cdot\prod_{\begin{subarray}{c}\ell\mid
g,\,\ell\nmid h\\\ v_{\ell}(\frac{g}{(\ell^{k},z)})=0\\\ \text{or }\ell\mid
h,\,v_{\ell}(\frac{g}{h})=k\end{subarray}}(-1)\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\ v_{\ell}(\frac{z}{g})>0\\\
v_{\ell}(\frac{g}{h})=k-1\end{subarray}}\ell\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\ v_{\ell}(\frac{z}{g})=0\\\
0<v_{\ell}(\frac{g}{h})<k\end{subarray}}(\ell-1)\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\
v_{\ell}(\frac{z}{h})=0\end{subarray}}\frac{\ell^{r+1}(\ell-1)}{\ell^{r+1}-1}.$

In Section 8 we point out that the obtained formula for $\beta_{k}$ can be
rewritten as a rational multiple of a certain absolute constant $A_{k,r}$, see
(35), depending only on the parameters $k$ and $r$, which we evaluate in Table
5.

###### Proof.

Applying [16, Theorem 1.1] as in the proof of Theorem 18 we obtain

$\displaystyle\beta_{k}$ $\displaystyle=\sum_{m\geqslant
1}\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\ d\mid
m\end{subarray}}\frac{\mu(m)\mu(d)}{[K_{nm^{k},dn}:K]}$
$\displaystyle=\sum_{\begin{subarray}{c}g\mid z\\\ h\mid
g\end{subarray}}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\sum_{m\geqslant
1}\,\sum_{\begin{subarray}{c}n\mid m^{\infty}\\\
(nm^{k},z)=g\end{subarray}}\,\sum_{\begin{subarray}{c}d\mid m\\\
(dn,z)=h\end{subarray}}\frac{\mu(m)\mu(d)}{n^{r+1}d^{r}\varphi(m^{k})}$ (33)
$\displaystyle=\sum_{\begin{subarray}{c}g\mid z\\\ h\mid
g\end{subarray}}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\prod_{\ell}p_{\ell}(g,h)$

where for $\ell\nmid z$ (and hence $\ell\nmid g$) we have

$p_{\ell}(g,h)=1-\frac{1}{\varphi(\ell^{k})}\sum_{s\geqslant 0}\
\sum_{e\in\\{0,1\\}}\frac{\mu(\ell^{e})}{\ell^{s(r+1)+er}}=1-\frac{1}{\varphi(\ell^{k})}\frac{\ell(\ell^{r}-1)}{\ell^{r+1}-1},$

and for $\ell\mid z$, $\ell\nmid g$ we have $p_{\ell}(g,h)=1$, whereas for
$\ell\mid g$, setting $z_{\ell}:=v_{\ell}(z)$ and similarly for
$g_{\ell},h_{\ell}$, we have

$p_{\ell}(g,h)=-\frac{1}{\varphi(\ell^{k})}\sum_{\begin{subarray}{c}s\geqslant
0\\\ \min(k+s,z_{\ell})=g_{\ell}\end{subarray}}\
\sum_{\begin{subarray}{c}e\in\\{0,1\\}\\\
\min(s+e,z_{\ell})=h_{\ell}\end{subarray}}\frac{\mu(\ell^{e})}{\ell^{s(r+1)+er}}.$

We take $p(g,h):=\varphi(g)h^{r}\prod_{\ell\mid g}p_{\ell}(g,h)$ (and we will
make use of the identity
$\varphi(g)/\varphi(\operatorname{rad}(g)^{k})=g/\operatorname{rad}(g)^{k}$).
Let us compute $p_{\ell}(g,h)$ depending on the prime $\ell\mid g$. If
$g_{\ell}<\min(k,z_{\ell})$, then $p_{\ell}(g,h)=0$, so we may restrict the
sum in (33) to the divisors $g$ such that $(\operatorname{rad}(g)^{k},z)\mid
g$.

_Case 1: $\ell\nmid h$._ The conditions on the indices hold only for $s=e=0$.
Thus, if $g_{\ell}=\min(k,z_{\ell})$, then
$p_{\ell}(g,h)=-1/\varphi(\ell^{k})$, otherwise $p_{\ell}(g,h)=0$.

_Case 2: $\ell\mid h$ and $h_{\ell}=g_{\ell}=z_{\ell}$._ The sums reduce to
the indices $s,e$ such that $s+e\geqslant h_{\ell}$ (recall that $k\geqslant
2$). Hence, we have

$p_{\ell}(g,h)=\frac{1}{\varphi(\ell^{k})\ell^{(r+1)h_{\ell}}}\frac{\ell^{r+1}(\ell-1)}{\ell^{r+1}-1}.$

_Case 3: $\ell\mid h$ and $h_{\ell}<g_{\ell}=z_{\ell}$._ The conditions on the
indices become $s+e=h_{\ell}$ and $k+s\geqslant g_{\ell}$. Hence, we have:
$p_{\ell}(g,h)=0$ if $g_{\ell}-h_{\ell}>k$;
$p_{\ell}(g,h)=-1/(\varphi(\ell^{k})\ell^{(r+1)h_{\ell}})$ if
$g_{\ell}-h_{\ell}=k$;
$p_{\ell}(g,h)=(\ell-1)/(\varphi(\ell^{k})\ell^{(r+1)h_{\ell}})$ if
$g_{\ell}-h_{\ell}<k$.

_Case 4: $\ell\mid h$ and $g_{\ell}<z_{\ell}$._ The conditions on the indices
become $s+e=h_{\ell}$ and $k+s=g_{\ell}$. Thus, we have: if
$g_{\ell}-h_{\ell}=k$, then
$p_{\ell}(g,h)=-1/(\varphi(\ell^{k})\ell^{(r+1)h_{\ell}})$; if
$g_{\ell}-h_{\ell}=k-1$, then
$p_{\ell}(g,h)=\ell/(\varphi(\ell^{k})\ell^{(r+1)h_{\ell}})$; otherwise,
$p_{\ell}(g,h)=0$. ∎

##  6\. Prescribing valuations for the order

In this section we apply Theorem 1 to prove an asymptotic formula for the
number of primes $\mathfrak{p}$ of $K$ for which the order of
$(G\bmod\mathfrak{p})$ has prescribed $\ell$-adic valuations for finitely many
given primes $\ell$.

######  Theorem 21.

Let $K$ be a number field and let $G$ be a finitely generated and torsion-free
subgroup of $K^{\times}$ of positive rank $r$. Let $F/K$ be a finite Galois
extension, and let $C$ be a conjugacy-stable subset of
$\operatorname{Gal}(F/K)$. Consider finitely many prime numbers $\ell$, and
for each of them fix a nonnegative integer $a_{\ell}$. Set $k=\prod\ell$ and
$m=\prod\ell^{a_{\ell}}$, where $\ell$ runs through the considered primes.
Consider the set of primes of $K$ given by

$\mathcal{V}=\left\\{\mathfrak{p}:v_{\ell}(\operatorname{ord}_{\mathfrak{p}}(G))=a_{\ell}\
\forall\ell\mid k,\,\begin{pmatrix}\mathfrak{p}\\\ F/K\end{pmatrix}\subseteq
C\right\\},$

where we are assuming that $\mathfrak{p}\notin\mathcal{P}(G)$ and
$\mathfrak{p}$ does not ramify in $F$. Then we have

$\pi_{\mathcal{V}}(x)=\frac{x}{\log x}\sum_{f\mid
k}\mu(f)\varrho_{C,mf}+O\left(\tau(k)x\bigg{(}\frac{(\log\log x)^{2}}{\log
x}\bigg{)}^{1+\frac{1}{r+2}}\right),$

where, for $t\geqslant 1$, $\varrho_{C,t}$ is as in (2), hence the set
$\mathcal{V}$ has natural density

$\gamma_{C,k,m}:=\sum_{f\mid k}\,\sum_{n\mid(fm)^{\infty}}\,\sum_{d\mid
fm}\frac{\mu(f)\mu(d)c(fmn,dn)}{[F_{fmn,dn}:K]}$

(with $c(a,b)$ as in (3)). The constant implied by the $O$-term depends only
on $F$, $K$, $G$.

It follows from Proposition 15 that the series $\gamma_{C,k,m}$ is convergent.

###### Proof.

We must have that $\operatorname{ord}_{\mathfrak{p}}(G)$ is divisible by $m$
and not by $m\ell$ for any prime factor $\ell$ of $k$. Hence applying the
inclusion-exclusion principle and Theorem 1 we obtain the desired formula. ∎

Notice that $\gamma_{C,k,m}$ is given by a finite sum of terms of the form
$\pm\varrho_{C,t}$. In the following we provide an explicit formula for the
special case of trivial condition on the Frobenius.

######  Theorem 22.

Let $K$ be a number field and let $G$ be a torsion-free subgroup of
$K^{\times}$ of positive rank $r$. Let $k,m\geqslant 1$ be integers with $k$
squarefree and $\operatorname{rad}(m)\mid k$. Let $\gamma_{k,m}$ be the
natural density of the set of primes $\mathfrak{p}$ of $K$ such that
$v_{\ell}(\operatorname{ord}_{\mathfrak{p}}(G))=v_{\ell}(m)$ for all $\ell\mid
k$ (where $\mathfrak{p}\notin\mathcal{P}(G)$). Then there is an integer $z$,
which depends only on $K$ and $G$, such that

$\gamma_{k,m}=\frac{1}{\varphi(m)}\prod_{\begin{subarray}{c}\ell\mid m\\\
\ell\nmid
z\end{subarray}}\frac{(\ell-1)(\ell^{r}-1)}{\ell^{r+1}-1}\prod_{\begin{subarray}{c}\ell\mid
k\\\ \ell\nmid
mz\end{subarray}}\bigg{(}1-\frac{\ell(\ell^{r}-1)}{(\ell^{r+1}-1)(\ell-1)}\bigg{)}\sum_{\begin{subarray}{c}g\mid
z\\\ (m,z)\mid g\\\ \operatorname{rad}(g)\mid k\end{subarray}}\sum_{h\mid
g}\frac{p(g,h)}{[K_{g,h}:K]},$

where, for $h\mid g$, we set $p(g,h)=0$ if and only if at least one of the
following conditions holds:

  * •

there is $\ell\mid(k,g)$, $\ell\nmid m$, such that $v_{\ell}(g/h)>1$,

  * •

there is $\ell\mid(g,m)$, $\ell\nmid h$, such that we have
$v_{\ell}(g)\notin\\{v_{\ell}(z),v_{\ell}(m),v_{\ell}(m)+1\\}$, or
$v_{\ell}(g)=v_{\ell}(z)>v_{\ell}(m)+1$,

  * •

there is $\ell\mid(h,m)$, such that we have $v_{\ell}(g/h)>v_{\ell}(m)+1$, or
we have $v_{\ell}(z/g)>0$ and $v_{\ell}(g/h)<v_{\ell}(m)-1$;

else we define $p(g,h)=\frac{\varphi(g)}{h}\cdot q_{1}(g,h)q_{2}(g,h)$, with

$q_{1}(g,h)=\prod_{\begin{subarray}{c}\ell\mid\frac{g}{h}\\\ \ell\nmid
m\end{subarray}}\frac{-1}{\ell-1}\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\,\ell\nmid m\\\
v_{\ell}(\frac{z}{h})=0\end{subarray}}\frac{\ell^{r+1}}{\ell^{r+1}-1}\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\,\ell\nmid m\\\
v_{\ell}(h)=v_{\ell}(g)<v_{\ell}(z)\end{subarray}}\frac{\ell}{\ell-1}$

(the primes involved in these products are coprime with $m$), and

$q_{2}(g,h)=\prod_{\begin{subarray}{c}\ell\mid(h,m)\\\
v_{\ell}(\frac{z}{h})=0\end{subarray}}\frac{-\ell^{r}(\ell-1)^{2}}{\ell^{r+1}-1}\cdot\prod_{\begin{subarray}{c}\ell\mid(g,m)\\\
v_{\ell}(\frac{g}{h})=v_{\ell}(m)+1\end{subarray}}\frac{-1}{\ell}\cdot\prod_{\begin{subarray}{c}\ell\mid
g,\,\ell\nmid h\\\ v_{\ell}(g)=v_{\ell}(z)\leqslant
v_{\ell}(m)\end{subarray}}\frac{\ell-1}{\ell}\,\cdot\\\
\cdot\prod_{\begin{subarray}{c}\ell\mid(h,m)\\\ v_{\ell}(\frac{z}{g})>0\\\
v_{\ell}(\frac{g}{h})=v_{\ell}(m)\end{subarray}}2\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\,v_{\ell}(\frac{z}{g})=0\\\
0<v_{\ell}(\frac{g}{h})<v_{\ell}(m)\end{subarray}}\frac{-(\ell-1)^{2}}{\ell}\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\,v_{\ell}(\frac{z}{g})=0\\\
v_{\ell}(\frac{g}{h})=v_{\ell}(m)>0\end{subarray}}\frac{2\ell-1}{\ell}\cdot\prod_{\begin{subarray}{c}\ell\mid
h,\,v_{\ell}(\frac{z}{g})>0\\\
v_{\ell}(\frac{g}{h})=v_{\ell}(m)-1\end{subarray}}-\ell$

(the primes involved in these products are prime factors of $m$).

The proof is similar to that of Theorem 18: one needs to first apply [16,
Theorem 1.1], then transform the obtained inner sums into a product on the
prime factors $\ell$ of $k$, and compute these through a certain case
distinction. For the convenience of the reader, we give all details below.

###### Proof.

Applying [16, Theorem 1.1] as in the proof of Theorem 18, we obtain

$\displaystyle\gamma_{k,m}$ $\displaystyle=\sum_{f\mid k}\
\sum_{n\mid(fm)^{\infty}}\ \sum_{d\mid fm}\frac{\mu(f)\mu(d)}{[K_{fmn,dn}:K]}$
$\displaystyle=\sum_{\begin{subarray}{c}g\mid z\\\ h\mid
g\end{subarray}}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\ \sum_{f\mid k}\
\sum_{\begin{subarray}{c}n\mid(fm)^{\infty}\\\ (fmn,z)=g\end{subarray}}\
\sum_{\begin{subarray}{c}d\mid fm\\\
(dn,z)=h\end{subarray}}\frac{\mu(f)\mu(d)}{\varphi(fmn)(nd)^{r}}.$

Write $m=\prod_{\ell\mid k}\ell^{a_{\ell}}$ with $a_{\ell}\geqslant 0$. Notice
that the inner sums do not vanish only if $(m,z)\mid g$ and
$\operatorname{rad}(g)\mid k$. Thus we reduce the formula to

$\gamma_{k,m}=\frac{1}{\varphi(m)}\cdot\sum_{\begin{subarray}{c}g\mid z\\\
(m,z)\mid g\\\ \operatorname{rad}(g)\mid k\end{subarray}}\sum_{h\mid
g}\frac{\varphi(g)h^{r}}{[K_{g,h}:K]}\cdot\prod_{\ell\mid k}p_{\ell}(g,h),$

where for $\ell\mid k$ and $a_{\ell}=0$ we define $p_{\ell}(g,h)$ according to
a case distinction on the prime factors $\ell$ of $k$. Set
$z_{\ell}:=v_{\ell}(z)$, and similarly for $g_{\ell},h_{\ell}$.

_Case 1: $\ell\mid k$ and $a_{\ell}=0$._ If $\ell\nmid z$, then we set

$p_{\ell}(g,h):=1-\frac{1}{\ell-1}\sum_{s\geqslant 0}\
\sum_{t\in\\{0,1\\}}\frac{\mu(\ell^{t})}{\ell^{s(r+1)+tr}}=1-\frac{\ell(\ell^{r}-1)}{(\ell-1)(\ell^{r+1}-1)};$

if $\ell\mid z$ and $\ell\nmid g$, then $p_{\ell}(g,h):=1$; if $\ell\mid g$,
then we set

$p_{\ell}(g,h):=-\frac{1}{\ell-1}\sum_{\begin{subarray}{c}s\geqslant 0\\\
\min(s+1,z_{\ell})=g_{\ell}\end{subarray}}\
\sum_{\begin{subarray}{c}t\in\\{0,1\\}\\\
\min(s+t,z_{\ell})=h_{\ell}\end{subarray}}\frac{\mu(\ell^{t})}{\ell^{s(r+1)+tr}}.$

We take
$q_{1}(g,h):=\prod_{\ell\mid(k,g),\,a_{\ell}=0}p_{\ell}(g,h)\ell^{h_{\ell}(r+1)}$,
and we compute $p_{\ell}(g,h)$ depending on the prime factor $\ell$ of
$(k,g)$.

_Case 1.1: $\ell\nmid h$._ We must have $s=t=0$, hence
$p_{\ell}(g,h)=-1/(\ell-1)$ if $g_{\ell}=1$, otherwise $p_{\ell}(g,h)=0$.

_Case 1.2: $\ell\mid h$ and $h_{\ell}=g_{\ell}=z_{\ell}$._ The conditions on
the indices hold only for $s+t\geqslant z_{\ell}$. We obtain
$p_{\ell}(g,h)=\frac{1}{\ell^{h_{\ell}(r+1)}}\frac{\ell^{r+1}}{\ell^{r+1}-1}$.

_Case 1.3: $\ell\mid h$ and $h_{\ell}<g_{\ell}=z_{\ell}$._ We must have
$s+t=h_{\ell}$ and $s+1\geqslant z_{\ell}$, hence
$p_{\ell}(g,h)=-1/(\ell^{h_{\ell}(r+1)}(\ell-1))$ if $g_{\ell}-h_{\ell}=1$,
and $p_{\ell}(g,h)=0$ otherwise.

_Case 1.4: $\ell\mid h$ and $g_{\ell}<z_{\ell}$._ We must have $s=g_{\ell}-1$.
Thus, we obtain: $p_{\ell}(g,h)=-1/(\ell^{h_{\ell}(r+1)}(\ell-1))$ if
$g_{\ell}-h_{\ell}=1$; $p_{\ell}(g,h)=\ell/(\ell^{h_{\ell}(r+1)}(\ell-1))$ if
$g_{\ell}=h_{\ell}$; $p_{\ell}(g,h)=0$ otherwise.

_Case 2: $\ell\mid k$ and $a_{\ell}>0$._ Recall that $g$ is such that
$g_{\ell}\geqslant\min(a_{\ell},z_{\ell})$. We set

$p_{\ell}(g,h):=\sum_{e\in\\{0,1\\}}\ \sum_{\begin{subarray}{c}s\geqslant 0\\\
\min(e+a_{\ell}+s,z_{\ell})=g_{\ell}\end{subarray}}\
\sum_{\begin{subarray}{c}t\in\\{0,1\\}\\\
\min(s+t,z_{\ell})=h_{\ell}\end{subarray}}\frac{\mu(\ell^{e})\mu(\ell^{t})}{\ell^{s(r+1)+e+tr}}.$

If $\ell\nmid z$, then we have
$p_{\ell}(g,h)=\frac{(\ell-1)(\ell^{r}-1)}{\ell^{r+1}-1}$. Before dealing with
the case $\ell\mid z$, we set
$q_{2}(g,h):=\prod_{\ell\mid(k,z),\,a_{\ell}>0}p_{\ell}(g,h)\ell^{h_{\ell}(r+1)}$,
and we compute $p_{\ell}(g,h)$ depending on the prime factor $\ell$ of
$(k,z)$.

_Case 2.1: $\ell\nmid h$._ We must have $s=t=0$ and hence we have:
$p_{\ell}(g,h)=0$ if $g_{\ell}\notin\\{z_{\ell},a_{\ell},a_{\ell}+1\\}$, or
$g_{\ell}=z_{\ell}>a_{\ell}+1$; $p_{\ell}(g,h)=1$ if
$g_{\ell}=a_{\ell}<z_{\ell}$; $p_{\ell}(g,h)=-1/\ell$ if
$g_{\ell}=a_{\ell}+1$; $p_{\ell}(g,h)=1-1/\ell$ if $g_{\ell}=z_{\ell}\leqslant
a_{\ell}$.

_Case 2.2: $\ell\mid h$ and $h_{\ell}=g_{\ell}=z_{\ell}$._ The conditions on
the indices hold only for $s\geqslant h_{\ell}$, or $s=h_{\ell}-1$ and $t=1$
(as $a_{\ell}\geqslant 1$). We obtain

$p_{\ell}(g,h)=\Big{(}1-\frac{1}{\ell}\Big{)}\Big{(}\Big{(}1-\frac{1}{\ell^{r}}\Big{)}\frac{1}{\ell^{h_{\ell}(r+1)}}\sum_{s\geqslant
0}\frac{1}{\ell^{s(r+1)}}-\frac{\ell}{\ell^{h_{\ell}}}\Big{)}=-\frac{\ell^{r}(\ell-1)^{2}}{\ell^{h_{\ell}(r+1)}(\ell^{r+1}-1)}.$

_Case 2.3: $\ell\mid h$ and $h_{\ell}<g_{\ell}=z_{\ell}$_. Then the conditions
are satisfied only if $s+e+a_{\ell}\geqslant z_{\ell}$. We deduce that
$p_{\ell}(g,h)$ equals: $-(\ell-1)^{2}/\ell^{h_{\ell}(r+1)+1}$ if
$g_{\ell}-h_{\ell}<a_{\ell}$; $(2\ell-1)/\ell^{h_{\ell}(r+1)+1}$ if
$g_{\ell}-h_{\ell}=a_{\ell}$; $-1/\ell^{h_{\ell}(r+1)+1}$ if
$g_{\ell}-h_{\ell}=a_{\ell}+1$; $0$ if $g_{\ell}-h_{\ell}>a_{\ell}+1$.

_Case 2.4: $\ell\mid h$ and $g_{\ell}<z_{\ell}$_. The conditions are satisfied
only if $a_{\ell}+s+e=g_{\ell}$. Thus, $p_{\ell}(g,h)$ equals: $0$ if
$g_{\ell}-h_{\ell}\notin\\{a_{\ell}-1,a_{\ell},a_{\ell}+1\\}$;
$-\ell/\ell^{h_{\ell}(r+1)}$ if $g_{\ell}-h_{\ell}=a_{\ell}-1$;
$2/\ell^{h_{\ell}(r+1)}$ if $g_{\ell}-h_{\ell}=a_{\ell}$;
$-1/\ell^{h_{\ell}(r+1)+1}$ if $g_{\ell}-h_{\ell}=a_{\ell}+1$.

Finally, take $p(g,h):=\frac{\varphi(g)}{h}\cdot q_{1}(g,h)q_{2}(g,h)$. ∎

######  Corollary 23.

Let $K$ be a number field and let $G$ be a torsion-free subgroup of
$K^{\times}$ of positive rank $r$. Let $k\geqslant 1$ be a squarefree integer.
The natural density of the set of primes $\mathfrak{p}$ of $K$ such that
$\operatorname{ord}_{\mathfrak{p}}(G)$ is coprime to $k$ and
$(\mathfrak{p},F/K)\subseteq C$ (where $\mathfrak{p}\notin\mathcal{P}(G)$ and
$\mathfrak{p}$ does not ramify in $F$) is given by

$\sum_{f\mid k}\mu(f)\varrho_{C,f},$

where, for $t\geqslant 1$, $\varrho_{C,t}$ is as in (2). Moreover, supposing
that the condition on the Frobenius is trivial, there is an integer $z$, which
depends only on $K$ and $G$, such that this density equals

$\prod_{\begin{subarray}{c}\ell\mid k\\\ \ell\nmid
z\end{subarray}}\bigg{(}1-\frac{\ell(\ell^{r}-1)}{(\ell^{r+1}-1)(\ell-1)}\bigg{)}\cdot\sum_{\begin{subarray}{c}g\mid
z\\\ \operatorname{rad}(g)\mid k\end{subarray}}\sum_{h\mid
g}\mu\left(\frac{g}{h}\right)\frac{p(g,h)}{[K_{g,h}:K]},$

where, for $h\mid g$, we set

$p(g,h)=\frac{\varphi(g)}{\varphi(g/h)h}\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\
\ell\nmid\frac{z}{h}\end{subarray}}\frac{\ell^{r+1}}{\ell^{r+1}-1}\cdot\prod_{\begin{subarray}{c}\ell\mid
h\\\ v_{\ell}(h)=v_{\ell}(g)<v_{\ell}(z)\end{subarray}}\frac{\ell}{\ell-1}.$

###### Proof.

The first part of the statement follows directly from Theorem 1 and it is a
special case of Theorem 21. As for the second part, it suffices to take $m=1$
in Theorem 22. Clearly we have $q_{2}(g,h)=1$. We obtain that $p(g,h)=0$ if
and only if $g/h$ is not squarefree, and this together with the factor
$\prod_{\ell\mid\frac{g}{h}}(-1)$ in $q_{1}(g,h)$ yields the term $\mu(g/h)$
in the formula. Also, the factor $\prod_{\ell\mid\frac{g}{h}}1/(\ell-1)$ in
$q_{1}(g,h)$ is equal to $1/\varphi(g/h)$. ∎

##  7\. Conditional results assuming GRH

In this section we show how Theorems 1 and 2 can be improved if we assume GRH
for the Dedekind zeta functions of number fields of the type $K_{m,n}$. In
fact, in this case we can make use of the stronger version of the Chebotarev
density theorem, namely [19, Théorème 4] or [24, Theorem 2], and we obtain
smaller error terms. Let us first apply this theorem to cyclotomic-Kummer
extensions of $K$.

######  Lemma 24.

Assume GRH. Let $F/K$, $G$, $C$ and $\pi_{F_{m,n}/K,C_{m,n}}(x)$ be as in
Theorem 10. Then we have

(34) $\pi_{F_{m,n}/K,C_{m,n}}(x)=\frac{\left\lvert
C_{m,n}\right\rvert}{[F_{m,n}:K]}\operatorname{Li}(x)+O_{F,G}\left(\sqrt{x}\log(mx)\right).$

###### Proof.

Applying [19, Théorème 4] we have

$\frac{\left\lvert
C_{m,n}\right\rvert}{[F_{m,n}:K]}\Big{(}\operatorname{Li}(x)+O\left(\sqrt{x}\log\big{(}\left\lvert
d_{F_{m,n}}\right\rvert x^{[F_{m,n}:\mathbb{Q}]}\big{)}\right)\Big{)}.$

Recalling that $\left\lvert C_{m,n}\right\rvert\leqslant[F:K]$ and applying
Proposition 9, we can reduce the error term to

$O_{F}\bigg{(}\sqrt{x}\cdot\frac{\log\left\lvert
d_{F_{m,n}}\right\rvert}{[F_{m,n}:\mathbb{Q}]}+\sqrt{x}\log
x\bigg{)}=O_{F,G}\left(\sqrt{x}\log m+\sqrt{x}\log x\right).\qed$

######  Theorem 25.

Assume GRH. With the setup of Theorem 1, we have

$\pi_{\mathcal{P}_{m}}(x)=\operatorname{Li}(x)\varrho_{C,m}+O_{F,K,G}(x^{3/4}\log
x).$

###### Proof.

We start as in the proof of Theorem 1. Recall (19), where $\mathcal{B}_{n}$
was defined in (18). Applying first Lemma 12, and then Lemma 24 to the
functions $\pi^{1}_{mn,dn}(x)$ (notice that (34) also holds if we restrict to
the primes of $K$ of degree $1$), setting $y:=x^{1/4}$ we obtain:

$\displaystyle\pi_{\mathcal{P}_{m}}(x)=$ $\displaystyle\
\operatorname{Li}(x)\varrho_{C,m}+O\Bigg{(}\sum_{n\leqslant y/m}\,\sum_{d\mid
m}\sqrt{x}\log(mnx)\Bigg{)}+O\left(\frac{\sqrt{x}}{\log x}\right)$
$\displaystyle\ +O\Bigg{(}\operatorname{Li}(x)\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\ nm>y\end{subarray}}\,\sum_{d\mid
m}\frac{\mu(d)c(mn,dn)}{[F_{mn,dn}:K]}\Bigg{)}+O\Bigg{(}\sum_{\begin{subarray}{c}n\mid
m^{\infty}\\\ nm>y\end{subarray}}\pi_{\mathcal{B}_{n}}(x)\Bigg{)}$
$\displaystyle=$ $\displaystyle\
\operatorname{Li}(x)\varrho_{C,m}+O\left(y\sqrt{x}\log
x\right)+O\left(\frac{x\log\log x}{y\log
x}\right)+O\Bigg{(}\frac{\operatorname{Li}(x)}{\varphi(m)}\sum_{n>y/m}\frac{1}{n^{2}}\Bigg{)}.$

For the last error term, we used that

$\pi_{\mathcal{B}_{n}}(x)\leqslant\pi_{K_{mn,n}/K,\operatorname{id}}(x)\ll_{K,G}\frac{1}{[K_{mn,n}:K]}\operatorname{Li}(x)$

by Lemma 24, and the estimate $[K_{mn,n}:K]\gg_{K,G}\varphi(m)n^{2}$; see [14,
Theorem 3.1]. All error terms are hence estimated by $x^{3/4}\log x$. ∎

######  Corollary 26.

Assume GRH. With the setup of Theorem 2, we have

$\pi_{\mathcal{N}_{k}}(x)=\operatorname{Li}(x)\beta_{C,k}+O_{F,K,G}\Bigg{(}\frac{x^{1-\frac{k-1}{4k+1}}}{\log
x}\Bigg{)}.$

Moreover, with the setup of Theorem 21, we have

$\pi_{\mathcal{V}}(x)=\operatorname{Li}(x)\gamma_{C,k,m}+O_{F,K,G}\big{(}\tau(k)x^{3/4}\log
x\big{)}.$

###### Proof.

As for the first assertion, it is sufficient to follow the proof of Theorem 2,
making use of Theorem 25 instead of Theorem 1. Here we set $z=x^{a}$ for some
$0<a<1/4$. This yields

$\pi_{\mathcal{N}_{k}}(x)=\operatorname{Li}(x)\beta_{C,k}+O\left(\frac{x}{x^{a(k-1-\eta)}\log
x}+x^{3/4+a}\log x\right).$

Taking $a<\frac{1}{4(k-\eta)}$ we can bound the second error term with the
first one. Hence we may decrease the exponent $a(k-1-\eta)$ to
$\frac{k-1}{4k+1}$ (e.g. taking $\eta=1/10$), resulting in the error term of
the statement. The second assertion is a direct consequence of Theorem 25. ∎

##  8\. Numerical examples

In this section we provide several examples of densities computed with the
formulas of Theorems 18, 20 and 22. All values have been verified with
SageMath [18] by computing the approximated density that considers only primes
up to a certain bound. In particular, we have tested these formulas for $K$
and $G$ as in the several numerical examples from [12, 2, 13, 11, 10] (notice
that in [2, Table 3, left side] the density for the fifth and seventh entries
should both read $121/960$).

Let $K$ be a number field and $G$ a finitely generated subgroup of
$K^{\times}$. Recall the notation $\varrho_{m}$ introduced in Theorem 18. In
Tables 1–4 we provide several examples of densities $\varrho_{m}$.

Table 1. Examples of densities $\varrho_{m}$ with $K=\mathbb{Q}$ $G$ | $\varrho_{2}$ | $\varrho_{3}$ | $\varrho_{4}$ | $\varrho_{6}$ | $\varrho_{9}$ | $\varrho_{12}$ | $\varrho_{16}$ | $\varrho_{27}$
---|---|---|---|---|---|---|---|---
$\langle 2\rangle$ | $17/24$ | $3/8$ | $5/12$ | $17/64$ | $1/8$ | $5/32$ | $1/24$ | $1/24$
$\langle 16\rangle$ | $1/12$ | $3/8$ | $1/24$ | $1/32$ | $1/8$ | $1/64$ | $1/96$ | $1/24$
$\langle 3\rangle$ | $2/3$ | $3/8$ | $1/3$ | $5/16$ | $1/8$ | $1/16$ | $1/12$ | $1/24$
$\langle 27\rangle$ | $2/3$ | $1/8$ | $1/3$ | $5/48$ | $1/24$ | $1/48$ | $1/12$ | $1/72$
$\langle 2,3\rangle$ | $195/224$ | $6/13$ | $27/56$ | $333/728$ | $2/13$ | $3/14$ | $5/56$ | $2/39$
$\langle 16,27\rangle$ | $75/112$ | $5/13$ | $75/224$ | $235/728$ | $5/39$ | $95/1456$ | $75/896$ | $5/117$
$\langle 2,27,25\rangle$ | $839/960$ | $37/80$ | $59/120$ | $17723/38400$ | $37/240$ | $1073/4800$ | $11/120$ | $37/720$
Table 2. Examples of densities $\varrho_{m}$ with $K=\mathbb{Q}(\zeta_{3})$ $G$ | $\varrho_{2}$ | $\varrho_{3}$ | $\varrho_{4}$ | $\varrho_{6}$ | $\varrho_{9}$ | $\varrho_{12}$ | $\varrho_{16}$ | $\varrho_{27}$
---|---|---|---|---|---|---|---|---
$\langle 2\rangle$ | $17/24$ | $3/4$ | $5/12$ | $17/32$ | $1/4$ | $5/16$ | $1/24$ | $1/12$
$\langle 16\rangle$ | $1/12$ | $3/4$ | $1/24$ | $1/16$ | $1/4$ | $1/32$ | $1/96$ | $1/12$
$\langle 3\rangle$ | $5/6$ | $3/4$ | $1/6$ | $5/8$ | $1/4$ | $1/8$ | $1/24$ | $1/12$
$\langle 27\rangle$ | $5/6$ | $1/4$ | $1/6$ | $5/24$ | $1/12$ | $1/24$ | $1/24$ | $1/36$
$\langle 2,3\rangle$ | $111/112$ | $12/13$ | $13/28$ | $333/364$ | $4/13$ | $3/7$ | $3/56$ | $4/39$
$\langle 16,27\rangle$ | $47/56$ | $10/13$ | $19/112$ | $235/364$ | $10/39$ | $95/728$ | $19/448$ | $10/117$
$\langle 2,27,25\rangle$ | $479/480$ | $37/40$ | $29/60$ | $17723/19200$ | $37/120$ | $1073/2400$ | $7/120$ | $37/360$
Table 3. Examples of densities $\varrho_{m}$ with $K=\mathbb{Q}(\zeta_{4},\sqrt{3})$ $G$ | $\varrho_{2}$ | $\varrho_{3}$ | $\varrho_{4}$ | $\varrho_{6}$ | $\varrho_{9}$ | $\varrho_{12}$ | $\varrho_{16}$ | $\varrho_{27}$
---|---|---|---|---|---|---|---|---
$\langle 2\rangle$ | $11/12$ | $3/4$ | $5/6$ | $11/16$ | $1/4$ | $5/8$ | $1/12$ | $1/12$
$\langle 16\rangle$ | $1/6$ | $3/4$ | $1/12$ | $1/8$ | $1/4$ | $1/16$ | $1/48$ | $1/12$
$\langle 3\rangle$ | $2/3$ | $3/4$ | $1/3$ | $1/2$ | $1/4$ | $1/4$ | $1/12$ | $1/12$
$\langle 27\rangle$ | $2/3$ | $1/4$ | $1/3$ | $1/6$ | $1/12$ | $1/12$ | $1/12$ | $1/36$
$\langle 2,3\rangle$ | $55/56$ | $12/13$ | $13/14$ | $165/182$ | $4/13$ | $6/7$ | $3/28$ | $4/39$
$\langle 16,27\rangle$ | $19/28$ | $10/13$ | $19/56$ | $95/182$ | $10/39$ | $95/364$ | $19/224$ | $10/117$
$\langle 2,27,25\rangle$ | $239/240$ | $37/40$ | $29/30$ | $8843/9600$ | $37/120$ | $1073/1200$ | $7/60$ | $37/360$
Table 4. Examples of densities $\varrho_{m}$ with $K=\mathbb{Q}(\zeta_{4})$ $G$ | $\varrho_{2}$ | $\varrho_{3}$ | $\varrho_{4}$ | $\varrho_{6}$ | $\varrho_{9}$ | $\varrho_{12}$ | $\varrho_{16}$ | $\varrho_{27}$
---|---|---|---|---|---|---|---|---
$\langle 2\zeta_{4}\rangle$ | $2/3$ | $3/8$ | $1/3$ | $1/4$ | $1/8$ | $1/8$ | $1/12$ | $1/24$
$\langle 16\zeta_{4}\rangle$ | $47/48$ | $3/8$ | $23/24$ | $47/128$ | $1/8$ | $23/64$ | $1/48$ | $1/24$
$\langle 3\zeta_{4}\rangle$ | $5/6$ | $3/8$ | $2/3$ | $11/32$ | $1/8$ | $5/16$ | $1/6$ | $1/24$
$\langle 27\zeta_{4}\rangle$ | $5/6$ | $1/8$ | $2/3$ | $11/96$ | $1/24$ | $5/48$ | $1/6$ | $1/72$
$\langle 2\zeta_{4},3\zeta_{4}\rangle$ | $13/14$ | $6/13$ | $5/7$ | $165/364$ | $2/13$ | $3/7$ | $5/28$ | $2/39$
$\langle 16\zeta_{4},27\rangle$ | $1791/1792$ | $5/13$ | $447/448$ | $4475/11648$ | $5/39$ | $1115/2912$ | $75/448$ | $5/117$
$\langle 2\zeta_{4},27,25\rangle$ | $29/30$ | $37/80$ | $11/15$ | $259/600$ | $37/240$ | $259/1200$ | $11/60$ | $37/720$

Recall the notation $\beta_{k}$ from Theorem 20 (which is the density of
primes $\mathfrak{p}$ of $K$ such that $\operatorname{ord}_{\mathfrak{p}}(G)$
is $k$-free). Notice that, setting

(35) $A_{k,r}:=\prod_{\ell\text{
prime}}\Big{(}1-\frac{\ell^{r}-1}{(\ell-1)(\ell^{r+1}-1)\ell^{k-2}}\Big{)},$

a constant which only depends on the parameters $k\geqslant 2$ and $r\geqslant
1$, the infinite product in (32) is equal to

$A_{k,r}\cdot\prod_{\ell\mid
z}\Big{(}1-\frac{\ell^{r}-1}{(\ell-1)(\ell^{r+1}-1)\ell^{k-2}}\Big{)}^{-1},$

so $\beta_{k}$ can be expressed as rational multiple of $A_{k,r}$. In Table 5
we show some values for the $A_{k,r}$’s, approximated by considering only the
primes $\ell$ up to $10^{5}$. In Tables 6 and 7 we provide some examples of
densities $\beta_{k}$, expressed both as rational multiples of $A_{k,r}$ and
as approximated value.

Table 5. Examples of constants $A_{k,r}$ approximated ($\ell<10^{5}$) $A_{k,r}$ | $k=2$ | $k=3$ | $k=4$ | $k=5$ | $k=6$ | $k=7$ | $k=8$
---|---|---|---|---|---|---|---
$r=1$ | $0.530712$ | $0.788163$ | $0.901926$ | $0.953511$ | $0.977581$ | $0.989060$ | $0.994618$
$r=2$ | $0.434934$ | $0.734313$ | $0.875354$ | $0.940597$ | $0.971280$ | $0.985966$ | $0.993091$
$r=3$ | $0.401045$ | $0.714103$ | $0.865118$ | $0.935552$ | $0.968798$ | $0.984741$ | $0.992484$
$r=4$ | $0.386687$ | $0.705354$ | $0.860624$ | $0.933316$ | $0.967691$ | $0.984192$ | $0.992211$
$r=5$ | $0.380106$ | $0.701307$ | $0.858528$ | $0.932267$ | $0.967169$ | $0.983932$ | $0.992082$
Table 6. Examples of densities $\beta_{k}$ over $K=\mathbb{Q}(\zeta_{3})$ $G$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$
---|---|---|---|---
$\langle 2\rangle$ | $\frac{3}{4}A_{2,1}\approx 0.398$ | $\frac{121}{115}A_{3,1}\approx 0.829$ | $\frac{805}{781}A_{4,1}\approx 0.930$ | $\frac{5029}{4945}A_{5,1}\approx 0.970$
$\langle 16\rangle$ | $\frac{69}{56}A_{2,1}\approx 0.654$ | $\frac{517}{460}A_{3,1}\approx 0.886$ | $\frac{3325}{3124}A_{4,1}\approx 0.960$ | $\frac{20437}{19780}A_{5,1}\approx 0.985$
$\langle 3\rangle$ | $\frac{15}{14}A_{2,1}\approx 0.569$ | $\frac{121}{115}A_{3,1}\approx 0.829$ | $\frac{805}{781}A_{4,1}\approx 0.930$ | $\frac{5029}{4945}A_{5,1}\approx 0.970$
$\langle 27\rangle$ | $\frac{55}{42}A_{2,1}\approx 0.695$ | $\frac{77}{69}A_{3,1}\approx 0.880$ | $\frac{2461}{2343}A_{4,1}\approx 0.947$ | $\frac{15181}{14835}A_{5,1}\approx 0.976$
$\langle 2,3\rangle$ | $\frac{135}{176}A_{2,2}\approx 0.334$ | $\frac{875}{814}A_{3,2}\approx 0.789$ | $\frac{5989}{5750}A_{4,2}\approx 0.912$ | $\frac{37823}{36994}A_{5,2}\approx 0.962$
$\langle 16,27\rangle$ | $\frac{899}{704}A_{2,2}\approx 0.555$ | $\frac{21935}{19536}A_{3,2}\approx 0.824$ | $\frac{48763}{46000}A_{4,2}\approx 0.928$ | $\frac{914711}{887856}A_{5,2}\approx 0.969$
$\langle 2,27,25\rangle$ | $\frac{95201}{119193}A_{2,3}$ | $\frac{105751169}{96766014}A_{3,3}$ | $\frac{524265887}{500045142}A_{4,3}$ | $\frac{116376274169}{113496822354}A_{5,3}$
| $\approx 0.320$ | $\approx 0.780$ | $\approx 0.907$ | $\approx 0.959$
Table 7. Examples of densities $\beta_{k}$ over $K=\mathbb{Q}(\zeta_{4})$ $G$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$
---|---|---|---|---
$\langle 2\rangle$ | $\frac{1}{4}A_{2,1}\approx 0.133$ | $A_{3,1}\approx 0.788$ | $A_{4,1}\approx 0.902$ | $A_{5,1}\approx 0.953$
$\langle 16\rangle$ | $\frac{11}{8}A_{2,1}\approx 0.730$ | $\frac{23}{20}A_{3,1}\approx 0.906$ | $\frac{47}{44}A_{4,1}\approx 0.963$ | $\frac{95}{92}A_{5,1}\approx 0.985$
$\langle 3\rangle$ | $\frac{3}{7}A_{2,1}\approx 0.227$ | $\frac{91}{115}A_{3,1}\approx 0.624$ | $\frac{709}{781}A_{4,1}\approx 0.819$ | $\frac{4729}{4945}A_{5,1}\approx 0.912$
$\langle 27\rangle$ | $\frac{11}{21}A_{2,1}\approx 0.278$ | $\frac{283}{345}A_{3,1}\approx 0.647$ | $\frac{2149}{2343}A_{4,1}\approx 0.827$ | $\frac{331}{345}A_{5,1}\approx 0.915$
$\langle 2,3\rangle$ | $\frac{9}{176}A_{2,2}\approx 0.0222$ | $\frac{329}{407}A_{3,2}\approx 0.594$ | $\frac{2641}{2875}A_{4,2}\approx 0.804$ | $\frac{17795}{18497}A_{5,2}\approx 0.905$
$\langle 16,27\rangle$ | $\frac{1073}{2112}A_{2,2}\approx 0.221$ | $\frac{5501}{6512}A_{3,2}\approx 0.620$ | $\frac{128873}{138000}A_{4,2}\approx 0.817$ | $\frac{286741}{295952}A_{5,2}\approx 0.911$
$\langle 2,27,25\rangle$ | $\frac{23323}{953544}A_{2,3}$ | $\frac{79247549}{96766014}A_{3,3}$ | $\frac{3234551969}{3500315994}A_{4,3}$ | $\frac{109490052089}{113496822354}A_{5,3}$
| $\approx 0.00981$ | $\approx 0.585$ | $\approx 0.799$ | $\approx 0.903$

Finally, recall the notation $\gamma_{k,m}$ from Corollary 22 (which is the
density of primes $\mathfrak{p}$ of $K$ such that
$\operatorname{ord}_{\mathfrak{p}}(G)$ has $\ell$-adic valuation equal to
$v_{\ell}(m)$ for every prime $\ell\mid k$). In Table 8 we provide some
examples of these densities.

Table 8. Examples of densities $\gamma_{k,m}$ over $K=\mathbb{Q}(\sqrt{-5})$ $G$ | $\gamma_{6,1}$ | $\gamma_{6,2}$ | $\gamma_{6,3}$ | $\gamma_{6,4}$ | $\gamma_{6,6}$ | $\gamma_{6,9}$ | $\gamma_{6,12}$
---|---|---|---|---|---|---|---
$\langle 2\rangle$ | $35/192$ | $35/192$ | $7/96$ | $5/24$ | $7/96$ | $7/288$ | $1/12$
$\langle 16\rangle$ | $55/96$ | $5/192$ | $11/48$ | $5/384$ | $1/96$ | $11/144$ | $1/192$
$\langle 3\rangle$ | $13/48$ | $1/12$ | $1/24$ | $13/96$ | $1/6$ | $1/72$ | $1/48$
$\langle 27\rangle$ | $5/16$ | $1/4$ | $1/72$ | $5/32$ | $1/18$ | $1/216$ | $1/144$
$\langle 2,3\rangle$ | $365/2912$ | $423/2912$ | $1/364$ | $101/728$ | $59/364$ | $1/1092$ | $10/91$
$\langle 16,27\rangle$ | $391/1456$ | $225/2912$ | $15/364$ | $785/5824$ | $125/728$ | $5/364$ | $95/4368$
$\langle 2,27,25\rangle$ | $801/6400$ | $927/6400$ | $37/28800$ | $443/3200$ | $4699/28800$ | $37/86400$ | $1591/14400$

## Acknowledgments

The author would like to thank Francesco Pappalardi for suggesting the problem
and for helpful discussions (Remark 14 is due to him). Also many thanks to
Antonella Perucca for her continuous support and valuable feedback. I thank
the reviewers for many helpful comments, and, in particular, one of them for
suggesting to use the results of [22], leading to an improvement of the error
terms in Theorems 1 and 2. I thank Pieter Moree for some useful remarks. The
Sage code used for the examples has been partly adapted from Sebastiano
Tronto’s code _kummer-degrees_ available on his GitHub page.

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[a,1]Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for
Astroparticle Physics, Erwin-Rommel-Straße 1, 91058 Erlangen, Germany
[b]Nikhef, National Institute for Subatomic Physics, PO Box 41882, Amsterdam,
1009 DB Netherlands<EMAIL_ADDRESS>
# GiBUU based neutrino interaction simulations in KM3NeT

J. Schumann,Corresponding author  B. Jung

###### Abstract

The simulation of the neutrino interaction is a crucial step in the simulation
chain of a neutrino experiment. The different processes taking part in the
neutrino scattering on a nucleus require several approximations in order to
make the simulation possible and to realize reasonable computation times. This
can be realised in different ways, e.g. by parametrised models for the
different scattering processes and energy regimes as it is implemented in
GENIE. The GiBUU neutrino generator utilises the Boltzmann-Uehling-Uhlenbeck
equation to simulate the particle flow after the neutrino interactions, the
so-called final state interactions. The detector-specific results in form of
the visible energy in the detector after the light propagation simulation and
the KM3NeT event reconstruction are presented. In addition to that, the
comparison to the GENIE based simulation environment in KM3NeT (gSeaGen) is
drawn.

Neutrino detectors, Simulation methods and programs [c]on behalf of the KM3NeT
collaboration VLVnT 2021 - Very Large Volume Neutrino Telescope Workshop
18-21 May 2021
Valencia (Online)

##  1 Introduction

The KM3NeT neutrino telescope is currently being built in the depths of the
Mediterranean Sea. It will host two water Cherenkov detectors comprising
3D-arrays of photomultiplier tubes (PMT) with different layouts. The detector
designed for the low energy regime is called ORCA (Oscillation Research with
Cosmics in the Abyss) in order to tackle the determination of the neutrino
mass hierarchy. In this lower energy regime the secondary particles from the
neutrino interaction are distributed more isotropically and deviate further
from the direction of the primary neutrino. Also the reconstruction of the
energy depends stronger on energy thresholds for the emission of the secondary
particles, thus information about the interaction is crucial in order to gain
knowledge about the light distribution of low-energy event topologies and
subsequently improve the energy and direction reconstruction of the primary
neutrino [1].

##  2 Neutrino Interaction Simulation

In order to make a numerical simulation of neutrino scattering processes with
so-called neutrino generators possible within reasonable computation times
several approximations are necessary. The neutrino generators GiBUU and GENIE
implement the neutrino interaction via a factorised model, i.e. splitting the
scattering event into the initial electroweak scattering and the propagation
of the generated particles through the nucleus [2, 3]. In GiBUU this
propagation of final states is done via propagating phase space densities
utilising the Boltzmann-Uehling-Uhlenbeck equation. In order to adapt the
simulated neutrino interactions to the KM3NeT environment, the neutrino
generators are wrapped by custom software packages. For GENIE, the KM3NeT
collaboration has developed a wrapper called gSeaGen [4] and for GiBUU the
KM3BUU software package (see Section 3) has been developed.

###  2.1 Final State Particle Picture

The resulting particles after the propagation through the nucleus are written
out by the generator and are referred to as final state particles. The
contribution of the different particle types over energy is compared via the
production cross section $\frac{d\sigma_{X}}{dE_{\nu}}$, where $X$ stands for
the individual secondary particle type.

\begin{overpic}[width=476.98463pt,trim=0.0pt 400.49625pt 0.0pt
100.37498pt,clip]{fig1.pdf} \put(53.0,10.0){\footnotesize KM3NeT Preliminary}
\end{overpic} Figure 1: Difference of the production cross sections, i.e. the
GENIE cross section subtracted from the GiBUU cross section:
$\left.\frac{d\sigma_{X}}{dE_{\nu}}\right|_{\text{GiBUU}}-\left.\frac{d\sigma_{X}}{dE_{\nu}}\right|_{\text{GENIE}}$.

In the regime below $10\text{\,}\mathrm{GeV}$ the final state particle types
consist of photons and
$\pi$-mesons (besides the out going lepton from the neutrino vertex itself)
and the emission of secondary particles from GENIE is higher (see Figure 1).
At higher energies also heavier mesons, i.e. Kaons, are generated and their
production cross section from GiBUU data exceeds the one from GENIE. The GiBUU
dataset does not contain photons, which yields the dominant contribution of
photons for the GENIE dataset. The difference in the production cross section
for protons and neutrons is currently under further investigation. The mean
energy of those secondary nucleons is
$\bar{E}_{p,n}=$1.08\text{\,}\mathrm{GeV}$$
($\sigma_{p,n}=$0.99\text{\,}\mathrm{GeV}$$) which is close to the rest energy
of the particles. Thus, the contribution to the visible light yield in the
detector is expected to be neglectable.

###  2.2 Visible Energy

The visible energy of a given final state is defined as the energy contained
in an electromagnetic shower which is converted to photons. The definition was
applied and the values for the parametrisation of the used rational function
are taken from [5]. The visible energy distribution for both generators is
shown in Figure 2.

\begin{overpic}[width=433.62pt,trim=0.0pt 0.0pt 0.0pt 40.15pt,clip]{fig2a.png}
\put(20.0,80.0){\tiny KM3NeT Preliminary} \end{overpic}

\begin{overpic}[width=433.62pt,trim=0.0pt 15.05624pt 85.31874pt
50.18748pt,clip]{fig2b.png} \put(40.0,58.0){\tiny KM3NeT Preliminary}
\end{overpic}

Figure 2: Visible energy $E_{\text{vis}}$ over the primary neutrino energy
$E_{\nu}$ for GiBUU (left) and the ratio $E_{\text{vis}}/E_{\nu}$ for
$\nu_{e,\tau}$-CC-interactions for GiBUU and gSeaGen in the energy range
$E_{\nu}\in[$1\text{\,}\mathrm{GeV}$,$10\text{\,}\mathrm{GeV}$]$ (right).

In the used configuration the distribution of the ratio
$E_{\text{vis}}/E_{\nu}$ for the KM3BUU data has a slight shift to higher
values. This seems to contradict the results from Section 2.1, but those are
given in units of production cross section which is independent from the
simulated flux $\Phi(E)\approx E^{-1}$.

###  2.3 Event Trigger & Reconstruction

In order to get a realistic detector picture including the environment
properties and geometry effects, in the next step the neutrino generator
output is fed into a light generation and propagation simulation resulting in
photosensor readings (hits). With the hit information the simulation data
contains the same shape compared to the DAQ data from the detector and can
subsequently be fed into the subsequent data processing tools, i.e. triggering
and reconstruction. The normalisation of the number of events over energy can
be compared via the effective volume $V_{\text{eff}}$ defined as
$V_{\text{gen}}\cdot{N_{\text{det}}}/{N_{\text{gen}}}$, which is shown in
Figure 3. It stands out, that the distribution for GiBUU follows the expected
distribution in the energy regime between $1\text{\,}\mathrm{GeV}$ and
$10\text{\,}\mathrm{GeV}$, but exceeds the expected upper limit at
$7\text{\,}{\mathrm{Mm}}^{3}$ and continues rising at energies higher than
$10\text{\,}\mathrm{GeV}$. This deviation from the expectation is currently
further investigated.

\begin{overpic}[width=433.62pt]{fig3a.pdf} \put(14.0,60.0){\footnotesize
KM3NeT Preliminary} \end{overpic}

Figure 3: Effective volume for the reconstructed events for the GiBUU dataset
(left) and offical estimate for KM3NeT taken from [1] (right).

In Figure 4 the comparison between KM3BUU and gSeaGen is shown for the
reconstructed energy over the true Monte-Carlo energy.

\begin{overpic}[width=433.62pt]{fig4a.pdf} \put(14.0,62.0){\tiny KM3NeT
Preliminary} \end{overpic}

\begin{overpic}[width=433.62pt]{fig4b.pdf} \put(14.0,62.0){\tiny KM3NeT
Preliminary} \end{overpic}

Figure 4: Comparison of the reconstructed event shower energy and the
simulated (MC) energy of GiBUU (left) and gSeaGen (right) normalised to a
uniform distribution of neutrino events in the used energy range.

The ridge of the events lies for both on the expected straight line of
$E_{\text{reco}}/E_{\text{MC}}=1$, which indicates similar magnitude of
systematic effects from the processed KM3BUU dataset compared to the gSeaGen
dataset. The origin of systematics effects have not been studied yet. The
distributions yield a lower reconstruction efficiency for gSeaGen at higher
energies above $10\text{\,}\mathrm{GeV}$ compared to KM3BUU. This difference
can be explained by the normalisation comparing the excess of effective volume
above $10\text{\,}\mathrm{GeV}$ for KM3BUU (see Figure 3).

##  3 KM3BUU

The KM3BUU framework is a python based wrapper for the GiBUU Fortran
environment, which is installed and operated inside a singularity container.
The general GiBUU workflow for running the simulation is unchanged, but can be
completely steered via the KM3BUU python environment. The GiBUU output
consists of multiple files which carry the full information about the
simulated neutrino scatterings. The output directory containing those GiBUU
output files can be parsed with the KM3BUU framework and converted to multiple
data formats, e.g. awkward arrays [6] and pandas dataframes [7] for the pure
neutrino generator output or converted to the KM3NeT dataformat. This KM3NeT
dataformat is then completely compatible to the KM3NeT (simulation) tools .

##  4 Conclusion

The KM3BUU package is a quite new software package and the technical aspect of
software errors is non-neglectable, but the distributions along the simulation
steps do not show any major discrepancies compared to GENIE. The difference in
the $\pi$-meson production rate in the lower $\mathrm{GeV}$ regime is expected
to be higher compared to previous studies, e.g. Coloma et al [8], but this
depends on the GENIE input cross section and the simulation configuration of
GiBUU. This difference affects also the event topology starting in the
analysis step of the visible energy. In the final step the event
reconstruction shows no systematic offsets or deviations with focus on the
reconstructed energy. The excess of effective volume visible for the KM3BUU
data follows the trend of a higher visible energy fraction $R$, but as this
exceeds the estimate of previous studies significantly this is currently
further investigated.

## References

  * [1] KM3NeT collaboration, _Determining the Neutrino Mass Ordering and Oscillation Parameters with KM3NeT/ORCA_ , _arXiv:2103.09885 [hep-ex]_ (2021) .
  * [2] O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich et al., _Transport-theoretical description of nuclear reactions_ , _Physics Reports_ 512 (2012) 1.
  * [3] C. Andreopoulos, C. Barry, S. Dytman, H. Gallagher, T. Golan, R. Hatcher et al., _The GENIE Neutrino Monte Carlo Generator: Physics and User Manual_ , _arXiv:1510.05494 [hep-ex, physics:hep-ph]_ (2015) .
  * [4] KM3NeT collaboration, _gSeaGen: the KM3NeT GENIE-based code for neutrino telescopes_ , _Computer Physics Communications_ 256 (2020) 107477.
  * [5] M. Dentler, _Investigation of the One-Particle Approximation in the ANTARES simulation package KM3_ , Oct., 2012.
  * [6] J. Pivarski et al., _scikit-hep/awkward-0.x: 0.15.5_ , Feb., 2021.  10.5281/ZENODO.4520562.
  * [7] J. Reback et al., _pandas-dev/pandas: Pandas 1.3.0_ , July, 2021.  10.5281/ZENODO.5060318.
  * [8] P. Coloma et al., _Neutrino-nucleus interaction models and their impact on oscillation analyses_ , _Physical Review D_ 89 (2014) 073015.