The distribution of the multiplicative index of algebraic numbers over residue classes

Max-Planck-Institut für Mathematik Bonn

The distribution of the multiplicative index of algebraic numbers over residue classes

by

Pieter Moree Antonella Perucca

Pietro Sgobba

Max-Planck-Institut für Mathematik Preprint Series 2023 (6)

Date of submission: February 27, 2023

The distribution of the multiplicative index of algebraic numbers over residue classes

by

Pieter Moree Antonella Perucca

Pietro Sgobba

Max-Planck-Institut für Mathematik Vivatsgasse 7

53111 Bonn Germany

University of Luxembourg Department of Mathematics 6 Avenue de la Fonte 4364 Esch-sur-Alzette Luxembourg

MPIM 23-6

OF ALGEBRAIC NUMBERS OVER RESIDUE CLASSES

PIETER MOREE

Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

ANTONELLA PERUCCA, PIETRO SGOBBA

University of Luxembourg, Department of Mathematics, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg

ABSTRACT. LetK be a number field andGa finitely generated torsion-free subgroup of K^×. Given a primepofKwe denote byindp(G)the index of the subgroup(Gmodp)of the multiplicative group of the residue field atp. Under the Generalized Riemann Hypothesis we determine the natural density of primes ofKfor which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extensionF ofK. We study in detail the natural density in caseSis an arithmetic progression, in particular its positivity.

Statements and Declarations

The authors have no conflicts of interest to disclose.

1. INTRODUCTION

The distribution of the multiplicative index of an integer seems to have been first studied by Pappalardi [13] in 1995. Under the Generalized Riemann Hypothesis (GRH) he provided asymptotic formulae forP

p≤xf(ind_p(g)), for f satisfying fairly mild restrictions (here and in the sequel we denote the rational primes byp). This line of investigation was continued in 2012 by Felix and Murty [5] and later by Felix for higher rank in [3]. Given a set of integers Sand a natural numberg, in [5] it was proven that

πg,S(x) :=|{p≤x: indg(p)∈S}|=cg,SLi(x) +O

x (logx)^2−ϵ

, (1)

where c_g,S is a constant defined by a series whose terms depend on the set S, Li(x) :=

Rx

2 dt/logt denotes the logarithmic integral andϵ > 0 is arbitrary. It is a difficult problem to determine whether c_g,S is positive or not, cf. Felix [4]. The special case where S is an arithmetic progression was already considered by Moree [9, Thm. 5] in 2005. For example, he proved Theorem 8 below in caseG=⟨g⟩,F =K =Q.

E-mail addresses:moree@mpim-bonn.mpg.de, antonella.perucca@uni.lu, pietrosgobba1@gmail.com.

2020Mathematics Subject Classification. Primary: 11R45; Secondary: 11A07, 11R44.

Key words and phrases. Reductions of algebraic numbers, multiplicative index and order, primes in arith- metic progression, natural density.

Corresponding author:Pietro Sgobba.

1

In this paper we consider the behavior of π_g,S(x), with Q replaced by a number fieldK andgby a finitely generated torsion-free subgroupGofK^×. Instead of over rational primes, we sum now over primespof norm≤x. Under GRH we establish in Theorem 1, see Section 2, an asymptotic similar to (1), but with a weaker error term depending on the rank ofG. In Section 3 we then restrict to the case whereSconsists of integers in an arithmetic progression amodd. In Theorem 8 we show that in this case the natural density can be expressed as a linear combination of at most φ(d^′) −1 Artin-type constants, with d^′ = d/(a, d). The positivity of the density is studied in Section 4, the numerical evaluation of the Artin-type constants in Section 5. In the final section we demonstrate our results by determining the density for two examples and compare the outcome with an experimental approximation.

We takeGto be fixed, but one can also ask what happens for a “typical”G. Ambrose [1]

considered the average index of the group generated by a finite number of elements in the residue field at a prime of a number field and provided asymptotic formulae for the average order of this quantity.

Likewise we can wonder about the above questions, but for the multiplicative order, rather than the index. As far as the authors know, these were first studied by Chinen and Murata [2] ford = 4, and a little later by Moree by a simpler method. Both Chinen and Murata, and independently Moree, went on to independently write various further papers (he surveyed his results in [12]). Under an appropriate generalization of the Riemann Hypothesis it turns out that the natural density of primesp ≤ x such that the multiplicative order ofg modulopis congruent toamoddexists. Denote it by δ_g(a, d)and the associated counting function by N_g(a, d)(x). The proof of the existence ofδ_g(a, d)by Moree is based on the identity

N_g(a, d)(x) =

∞

X

t=1

|{p≤x : ind_p(g) =t, p≡1 +tamoddt}|.

The average density of elements of order congruent toamoddin a field of prime character- istic also exists, but is a much simpler quantity, see Moree [8]. It has very similar features to δg(a, d).

In the special case where ddivides a, we are just asking for the density of primes psuch thatddivides the multiplicative order ofgmodulop. This density is much easier to deal with and turns out to be a rational number. This can be proven unconditionally, see for example [11, 18, 19].

Ziegler [20], using the approach of Moree, was the first to study the order in arithmetic progression problem in the setting of number fields. His work was generalized by Perucca and Sgobba in [15, 16], who obtained in particular uniformity results for the distribution of the order. It is expected that, likewise, some uniformity also holds for the distribution of the index into suitably related congruence classes, however at the moment it is not clear how to obtain such a result. For example, it does not follow from [15, Cor. 5.2], in spite of the fact that congruence conditions on both the order and the size of the multiplicative group lead to congruence conditions on the index. We leave this as a research direction and as an open problem to the reader.

2. THE EXISTENCE OF THE DENSITY OF PRIMES WITH PRESCRIBED INDEX AND

FROBENIUS

Let K be a number field, and F/K a finite Galois extension. Let G be a finitely gen- erated and torsion-free subgroup of K^× having positive rank r. Our goal is to determine

the density of the setP of primesp ofK (defined in the next theorem) with prescribed in- dex and Frobenius. The notationF, K, Gandr will be maintained throughout. We also set Km,n :=K(ζm, G^1/n)form |n, and similarly forFm,n. Further we make use of the follow- ing usual notation: ζn denotes ann-th primitive root of unity,µthe M¨obius function, andφ Euler’s totient function. We writelog^axas shorthand for(logx)^a, and(a, b)forgcd(a, b).

We recall that Landau’s prime ideal theorem states that

|{p: Np≤x}|= Li(x) +OK(xe^−cK

√logx

), (2)

wherec_K >0is a constant depending onK.

Theorem 1. (under GRH). Let K be a number field, and let Gbe a finitely generated and torsion-free subgroup ofK^× of positive rank r. Let F/K be a finite Galois extension, and letC be a union of conjugacy classes in Gal(F/K). LetS be a non-empty set of positive integers. Define

P :={p: ind_p(G)∈S, Frob_{F /K}(p)∈C},

wherepranges over the primes ofKunramified inF and for whichind_p(G)is well-defined.

We letP(x)be the number of prime ideals inP of norm≤x. We have the asymptotic estimate P(x) = x

logx X

t∈S

∞

X

v=1

µ(v)c(vt)

[F_vt,vt :K]+O x log^2−r+11 x

! ,

where

c(n) =

{σ ∈Gal(F_n,n/K) :σ|_Kn,n = id, σ|_F ∈C}

. The constant implied by theO-term depends only onK,F andG.

This result in combination with the prime ideal theorem leads to the following corollary.

Corollary 2. (under GRH). LetSbe a non-empty set of positive integers. The natural density of the primespofKsuch thatind_p(G)∈S andFrob_{F /K}(p)∈Cexists and is given by

X

t∈S

∞

X

v=1

µ(v)c(vt) [Fvt,vt :K].

We will now formulate some preliminaries required for the proof of Theorem 1. Our starting point is [15, Proposition 5.1], which was established for rank 1 in [20, Proposition 1].

Theorem 3. (under GRH). Forx ≥ t³, the number R_t(x)of primes p with norm up to x, unramified inF, and such thatind_p(G) = tandFrob_F/K(p)∈Csatisfies

R_t(x) = Li(x)

∞

X

v=1

µ(v)c(vt) [Fvt,vt :K]+O

x log²x

+O

xlog logx φ(t) log²x

.

The constant implied by theO-term depends only onK,F andG.

The following lemma is a straightforward generalisation of [9, Lemma 6], taking into account that for every natural numbern, the ratio

C(n) := φ(n)n^r

[K_n,n :K] (3)

is bounded above by some constantD, depending only onK andG(see [15, Theorem 1.1]).

Lemma 4. For every real numbery≥1we have X

t≤y

∞

X

n=1

µ(n)

[Knt,nt :K] = 1 +O D

y^r

,

where the implied constant is absolute.

Proof. We claim that

X

n>y

1

n^rφ(n) =O1 y^r

.

Forr= 1this is due to Landau [7] who first proved that X

n≤x

1

φ(n) =Alogx+B+O logx

x

, (4)

withAandB explicit constants, and then applied partial integration. The proof for arbitrary ris completely analogous. Sinceφ(nt)≥φ(n)φ(t), we obtain

X

t>y

∞

X

n=1

1

[K_nt,nt :K] ≤DX

t>y

1 t^rφ(t)

∞

X

n=1

1

n^rφ(n) ≪X

t>y

D

t^rφ(t) ≪ D

y^r, (5) where we used that the fourth sum is bounded above by a constant not depending onr. The estimate (5) shows that the double sum in the statement of Lemma 4 is absolutely convergent for ally. Thus, we may rearrange the double sum as follows:

∞

X

t=1

∞

X

n=1

µ(n) [K_nt,nt :K] =

∞

X

m=1

X

s|m

µ(m/s) [K_m,m :K] =

∞

X

m=1

X

d|m

µ(d) [K_m,m :K]

=

∞

X

m=1

1 [K_m,m :K]

X

d|m

µ(d) = 1

[K_1,1 :K] = 1,

completing the proof. □

The following is a generalisation of Ziegler [20, Lemma 13].

Lemma 5. (under GRH). We have

n

p: Np≤x, ind_p(G)>(logx)^r+11 o

=O x log^2−r+11 x

! ,

where the primes p of K are restricted to those for which ind_p(G) is well-defined. The constant implied by theO-term depends only onK andG.

Proof. The number of primes with ramification index or residue class degree at least 2is of orderO(P

p≤√

x1) = O(√

x/logx). We make use of the functionsRt(x)from Theorem 3 withF =K. For any real numbery≥ 1, letE_y(x)be the number of primespwithNp≤x and such thatind_p(G)> y. Notice that

E_y(x) = |{p: Np≤x}| −X

t≤y

R_t(x) +O √

x logx

.

Landau’s prime ideal theorem (2) implies the (much) weaker estimate

|{p: Np≤x}|= Li(x) +O x

log²x

= x

logx+O x

log²x

, (6)

which is all we need. By Theorem 3 and Lemma 4 we obtain X

t≤y

R_t(x) = Li(x)X

t≤y

∞

X

n=1

µ(n)

[K_nt,nt :K] +O xy

log²x

+O xlog logx log²x

X

t≤y

1 φ(t)

!

= Li(x) +O

x y^rlogx

+O

xy log²x

+O xlog logx log²x

X

t≤y

1 φ(t)

! .

On takingy= (logx)^1/(r+1) we now obtain on invoking (6) and (4), the estimate E_y(x) =O

x y^rlogx

+O

xy log²x

+O

x(log logx)² log²x

=O x

log^2−r+11 x

! ,

completing the proof. □

Proof of Theorem 1. Setρ= 2− _r+1¹ . Lemma 5 withy= (logx)^r+11 yields P(x) =X

t≤y t∈S

R_t(x) +O x

log^ρx

.

Estimating the sum as in the proof of Lemma 5 we obtain P(x) = Li(x)X

t≤y t∈S

∞

X

v=1

µ(v)c(vt) [F_vt,vt :K] +O

x log^ρx

.

Now we focus on the main term. We have

X

t∈S

∞

X

v=1

µ(v)c(vt)

[Fvt,vt :K] −X

t≤y t∈S

∞

X

v=1

µ(v)c(vt) [Fvt,vt :K]

≤X

t>y

∞

X

v=1

1 [Fvt,vt:F]

By (5) the right-hand side is bounded by ≪ y^−r. Using this estimate the proof is easily

completed. □

3. THE DISTRIBUTION OF THE INDEX OVER RESIDUE CLASSES

Leta, dbe integers withd≥2. We study the density of primespofKsuch thatind_p(G)≡ amodd. Under GRH, by Theorem 1 this density exists and equals

densG(a, d) := X

t≡amodd

X

v≥1

µ(v)

[K_vt,vt :K]. (7)

The goal of this section is to prove Theorem 8, which expressesdens_G(a, d)as a finite sum of terms depending on Dirichlet charactersχof modulusd. These terms involve Artin-type constantsBχ(r)that can be evaluated with multi-precision using Theorem 15, thus allowing one to evaluatedensG(a, d)with multi-precision.

We start by explaining our notation. Given an integer n ≥ 1 we let G_n be the group of characters defined on (Z/nZ)^×, so that G_n ∼= (Z/nZ)^×. For a Dirichlet character χ we denote byh_χ the (Dirichlet) convolutionµ∗χ of the M¨obius functionµwithχ, that is µ∗χ(n) = P

d|nµ(d)χ(n/d). Recall that the Dirichlet convolution of two multiplicative functions is again a multiplicative function.

Putw= gcd(a, d), a^′ =a/wandd^′ =d/w. The integerst≡amoddare of the formwt^′ witht^′ ≡a^′ modd^′. Thus we can rewritedens_G(a, d)as

dens_G(a, d) = X

t≡a^′modd^′

X

v≥1

µ(v) [K_vwt,vwt :K]. This expression on its turn can be rewritten as

dens_G(a, d) = X

t≡a^′modd^′

X

v1≥1 t|v₁

µ(v₁/t) [K_v1w,v1w :K]

= X

v1≥1

X

t≡a^′modd^′ t|v1

µ(v₁/t) [K_v1w,v1w :K]

= X

v1≥1



 1 φ(d^′)

X

χ∈G_d′

χ(a^′)h_χ(v₁)





1 [K_v1w,v1w :K]

= 1

φ(d^′) X

χ∈G_d′

χ(a^′)X

v1≥1

h_χ(v₁)

[K_v1w,v1w :K]. (8) In the second step we used that the double series is absolutely convergent (see the proof of Lemma 4). In the third step we used [9, Lemma 9], whereχruns over the Dirichlet characters modulod^′.

We now focus on the final sum in (8). Recall the definition (3) of C(n). By Perucca et al. [17, Theorem 1.1] there exists an integern₀ (depending only onGandK) such that

C(n) =C(gcd(n, n₀)). (9)

One can easily show that form | n, one hasC(m)| C(n), and hence n₀ can be taken to be the minimal integer satisfying

C(n₀) = max

n≥1

φ(n)n^r [K_n,n :K]. By (9) we have

1

[K_n,n :K] = C(gcd(n, n₀)) φ(n)n^r , and therefore

X

n≥1

1

[K_n,n :K] =X

g|n₀

X

n≥1 (n,n0)=g

C(g) φ(n)n^r .

In our case,

X

v≥1

hχ(v)

[K_vw,vw :K] =X

g|n0

X

v≥1 (vw,n0)=g

C(g)hχ(v)

φ(vw)v^rw^r. (10) IfP

v≥1f(v)is some absolute convergent series, we have X

v≥1 (vw,n0)=g

f(v) = X

v≥1 (^vw_g ,ⁿ_g⁰)=1

f(v) = X

v≥1

f(v) X

n|ⁿ_g⁰, n|^vw_g

µ(n)

= X

n|ⁿ⁰

g

µ(n)X

v≥1 n|^vw_g

f(v) = X

n|ⁿ⁰

g

µ(n) X

v≥1gn (gn,w)|v

f(v),

where we used thatn divides the integervw/g if and only ifgn/(gn, w)dividesv. Thus, in particular,

X

v≥1 (vw,n0)=g

C(g)h_χ(v)

φ(vw)v^rw^r = C(g) w^r

X

n|ⁿ_g⁰

µ(n) X

v≥1

gn (gn,w)|v

h_χ(v) φ(vw)v^r .

Inserting the right hand side into (10) and inserting the resulting expression into (8) yields dens_G(a, d) = 1

φ(d^′) X

χ∈G_d′

χ(a^′)X

g|n₀

C(g) w^r

X

n|ⁿ⁰

g

µ(n) X

v≥1gn (gn,w)|v

h_χ(v) φ(vw)v^r .

Denoting

C_χ(N, w, r) =X

v≥1 N|v

h_χ(v) φ(vw)v^r, we can write this as

dens_G(a, d) = 1 φ(d^′)

X

χ∈G_d′

χ(a^′)X

g|n₀

C(g) w^r

X

n|ⁿ_g⁰

µ(n)C_χ gn

(gn, w), w, r

. (11) Letκ(n) =Q

p|npdenote the squarefree kernel ofn.Recall thath_χ =µ∗χ. The following result is a special case of [9, Lemma 10] and expressesC_χ(N, w, r)as an Euler product.

Lemma 6. We have

C_χ(N, w, r) = c_χ(N, w, r)B_χ(r), where

c_χ(N, w, r) = h_χ(N)κ(N w) N^r+1w

Y

p|N

p^r+1

p^r+2−p^r+1−p+χ(p) Y

p∤N p|w

p^r+1−1

p^r+2−p^r+1−p+χ(p), and

B_χ(r) =Y

p

1 + p(χ(p)−1) (p−1)(p^r+1−χ(p))

, (12)

wherepruns over all rational prime numbers.

Corollary 7. We haveC_χ(1,1, r) = P

v≥1 hχ(v)

vφ(v) =B_χ(r).

Proof of Lemma 6. We distinguish two cases:

a) The case whereh_χ(N) = 0.

We have to verify thatC_χ(N, w, r) = 0.Sinceh_χ is multiplicative and we haveh_χ(p^k) = χ(p)^k−1(χ(p)−1), it follows that ifh_χ(N) = 0,then there is a prime divisor pofN with χ(p) = 1.Hence,h_χ(v) = 0for allvthat are divisible byN and soC_χ(N, w, r) = 0.

b) The case whereh_χ(N)̸= 0.

We rewriteC_χ(N, w, r)as

Cχ(N, w, r) = h_χ(N) φ(N w)N^r

X

v≥1

h_χ(N v)φ(N w)

h_χ(N)φ(N vw)v^r, (13) and note that the argument is a multiplicative function in v. We apply the Euler product identity to evaluate the sum and obtain

Y

p|N

p^r+1 p^r+1−χ(p)

Y

p∤N p|w

1 + (χ(p)−1) p^r+1−χ(p)

Y

p∤N w

1 + p(χ(p)−1) (p−1)(p^r+1−χ(p))

,

which can be rewritten as B_χ(r)Y

p|N

p^r+1(p−1) p^r+2−p^r+1−p+χ(p)

Y

p∤N p|w

(p−1)(p^r+1−1) p^r+2−p^r+1−p+χ(p). On inserting this in (13) and noting that

φ(κ(N w)) =Y

p|N

(p−1)Y

p∤N p|w

(p−1), φ(κ(N w))

φ(N w) = κ(N w) N w ,

the proof is completed. □

The densities we are interested in can be expressed as a finite linear combination involving the constantsB_χ(r). Our result generalizes [9, Thm. 5] by Moree, who dealt with the case F =K =QandGof rank1.

Theorem 8. (under GRH). Letaanddbe two natural numbers. Putd^′ =d/(a, d). Assuming that the functionC(n), defined in(3), is explicitly given, we can write

dens_G(a, d) = X

χ∈G_d′

d_χB_χ(r),

with thed_χ explicit complex numbers (they can be determined using(11)and Lemma 6).

Proof. In the identity (11) fordens_G(a, d)we make the substitution C_χ

gn

(gn, w), w, r

=c_χ

gn

(gn, w), w, r

B_χ(r)

(which is allowed by Lemma 6). The constants d_χ are obtained by factoring out the terms B_χ(r), so that for eachχ∈G_d^′ we have

d_χ = χ(a^′) φ(d^′)

X

g|n₀

C(g) w^r

X

n|ⁿ⁰

g

µ(n)c_χ gn

(gn, w), w, r

. □

3.1. Generic aspects of the behaviour ofdens_G(a, d). Generically the degree[K_vt,vt :K] equalsvtφ(vt)ifGhas rank1. If every degree occurring in (7) would satisfy this, then we would obtain

ρ(a, d) := X

t≡amodd

X

v≥1

µ(v) vtφ(vt). The inner sum is easily seen to equalA·r(t), with

r(t) = 1 t²

Y

p|t

p²−1 p²−p−1. Thus we can alternatively write

ρ(a, d) = A₁ X

t≡amodd

r(t) with

Ar :=Y

p

1− 1

p^r(p−1)

(14) therankrArtin constant. The “incomplete” rankrArtin constant, defined by restricting top odd, appears also in other works, such as in Pappalardi [14]. For everyB > 0we have, see [8, Theorem 4],

X

p≤x

ρ(p;a, d) =ρ(a, d) Li(x) +O x

log^Bx

,

withρ(p;a, d)the density of elements of inF^∗p having index congruent toamodd. Thus on average a finite field of prime order hasρ(a, d)elements having index congruent toamodd.

Two cases are particularly easy.

Proposition 9([8, Proposition 4]). One has ρ(0, d) = 1

dφ(d) and ρ(d,2d) =

(ρ(0,2d) if d is odd;

3ρ(0,2d) if d is even.

In the remaining cases it is not difficult to expressρ(a, d)in terms of theB_χ(1)’s, see [8, Prop. 6]. When(a, d) = 1this expression takes a particularly simple form, namely

ρ(a, d) = 1 φ(d)

X

χmodd

χ(a)B_χ(1). (15)

In the examples in Section 6 we will meetρ(a, d)again.

4. THE POSITIVITY OFdens_G(a, d)

As in the previous sections we consider a number fieldK, a finitely generated and torsion- free subgroupGofK^×, and the natural densitydensG(a, d)of the primespofK such that ind_p(G)≡amodd. We are interested in characterizing when this density is positive.

Recall that for a prime p of K of degree 1 such that ind_p(G) is well-defined, we have d| ind_p(G)if and only ifpsplits completely inK_d,d(cf. [20, Lem. 2]). So by Chebotarev’s density theorem we have

dens_G(0, d) = 1

[Kd,d:K] >0, (16)

and thus we may suppose in the following that0< a < d.

We denote bydens_G(h)the density of primespsuch thatind_p(G) = hwithha prescribed integer, and byn₀ an integer satisfyingC(n) = C(gcd(n, n₀)), whereC(n)was defined in (3). With this notation we are ready to recall the following result by J¨arviniemi and Perucca:

Theorem 10 ([6, Main Thm. and Rem. 4.2], under GRH). The density dens_G(h) is well- defined for allh≥ 1, and we havedens_G(h) >0if and only ifdens_G(gcd(h, n₀))> 0. For any setS of positive integers the following holds: if the density of primes p ofK such that ind_p(G)∈Sis positive, then there is someh∈S such thatdens_G(h)>0.

Proposition 11. (under GRH). Ifd≥2is coprime ton₀, thendens_G(a, d)>0.

Proof. By Theorem 10 (taking S to be the set of positive integers) we know that there is someh ≥ 1such thatdens_G(h) > 0. Moreover, we deduce that there is an integer h₀ | n₀ such that for every integertcoprime to n₀ we havedens_G(th₀)>0. We conclude by taking

t≡1 modn₀andt≡ah⁻¹₀ modd. □

The following result tells us in particular that for every prime numberℓand for everye≫0 there is a positive density of primespofK such thatv_ℓ(ind_p(G)) =e.

Proposition 12. For every prime numberℓthere is some non-negative integere_ℓ(and we can takee_ℓ = 0for all but finitely manyℓ) such that for everye⩾e_ℓ we have

densG(0, ℓ^e)>densG(0, ℓ^e+1).

Under GRH, for everyn >0and for every integerzwe havedens_G(zℓ^eℓ, ℓⁿ)>0.

Proof. By Chebotarev’s density theorem the density of primespofKsuch thatv_ℓ(ind_p(G)) = e is given by 1/[K_ℓ^e_,ℓ^e : K] −1/[K_ℓ^e+1_,ℓ^e+1 : K], so the first assertion follows from the eventual maximal growth of the Kummer degrees, see [15, Lem. 3.2]. By the first assertion (and by applying Theorem 10 to the setS of positive integers havingℓ-adic valuation equal to e) for every e ≥ e_ℓ there is some b coprime to ℓ such that dens_G(bℓ^e) > 0. Then for every prime q coprime to n0 we have densG(qbℓ^e) > 0 so we may conclude by selecting q ≡ b⁻¹zℓ^−vℓ(z) modℓⁿ, which is possible by Dirichlet’s theorem on primes in arithmetic

progressions. □

If x, y are positive integers, then we use the notation gcd(x, y^∞) to denote the positive integer obtained fromxby removing the prime factors that do not dividey.

Theorem 13. (under GRH). We havedens_G(a, d)>0if and only if dens_G(a,gcd(d, n^∞₀ ))>0.

Proof. Setd₀ = gcd(d, n^∞₀ ). The former inequality in the statement clearly implies the latter because the integers congruent toa moddare also congruent toamodd0. Now suppose that there is a positive density of primespofK such thatindp(G) ≡ amodd0. From Theorem 10 we deduce that there existsh≥ 1such thath ≡ amodd₀anddens_G(h) >0. For every positive integerst, s coprime to n₀ such that s | th we then have dens_G(th/s) > 0. If we chooset ≡ smodd₀, then th/s ≡ amodd₀. We claim that we may also choose t, s so thatth/s ≡ a modd/d₀. Because of the Chinese remainder theorem it will be possible to simultaneously ensure the two conditions and hencedens_G(th/s)>0impliesdens_G(a, d)>

0. To prove the claim, we first chooset, s so thatgcd(a, d/d₀) = gcd(th/s, d/d₀)and then multiplytby an integer invertible modulod/d₀ to obtain the requested congruence. □ Theorem 14. (under GRH). The following conditions are equivalent:

(1) the densitydens_G(a, d)is positive;

(2) there is an integerAsuch thatA≡amoddanddens_G(gcd(A, n₀))is positive.

Notice that it suffices to considerAmodulolcm(d, n₀).

Proof. WriteD:= lcm(d, n0). By Theorem 10 the densitydensG(a, d)is positive if and only if there is an integerA≡ amoddfor which densG(A, D) >0. The latter holds if and only if there is an indexh such that h ≡ AmodD and dens_G(h) > 0. Since n₀ | D, we have gcd(h, n₀) = gcd(A, n₀), and hence by Theorem 10 we have thatdens_G(h)is positive if and only ifdens_G(gcd(A, n₀))is positive.

The final assertion follows from the fact that the properties in (2) only depend onAmodulo

lcm(d, n₀). □

5. THEARTIN-TYPE CONSTANTSB_χ(r)

Let r ≥ 1 be an integer. Recall the Euler product definition (12) of B_χ(r). For r = 1 this was introduced in [8, Sec. 6] and denoted by B_χ, along with a variant A_χ, where pis restricted to those primes for whichχ(p)̸= 0. We have

B_χ(1) =A_χY

p|d

1− 1

p(p−1)

,

wheredis the modulus of the character. Note thatA_χ= 1in caseχis the principal character.

If χ₀ is the principal character, then B_χ0(r) is a rational number. This leaves at most φ(d^′)− 1 linearly independent Artin-type constants, withd^′ = d/(a, d). For example, in cased^′ = 3andd^′ = 4only one Artin-type constant is involved. They are real numbers. As an illustration we point out the result that the average density of elements of multiplicative order±1 mod 3equals ₁₆⁵ ±₁₀³ B_χ3(1), whereχ₃is the non-principal character modulo3and B_χ3(1) = ⁵₆A_χ3 = 0.1449809353580. . ., see [8].

Approximating the numerical value of Bχ(r) by computing partial Euler products, gives a quite poor accuracy. The following result allows us to do rather better and generalizes [8, Thm. 6] to arbitrary r. It involves special values of Dirichlet L-series. Recall that for ℜ(s)>1andχa Dirichlet character, we have

L(s, χ) =

∞

X

n=1

χ(n) n^s =Y

p

1−χ(p) p^s

⁻¹ .

Theorem 15. Letp₁(= 2), p₂, . . . denote the sequence of consecutive primes and χbe any Dirichlet character. Put

Λ_r=A_rL(r+ 1, χ)L(r+ 2, χ)L(r+ 3, χ).

Then

B_χ(r) =E_r,nΛ_r

n

Y

k=1

1 + χ(p_k) p_k(p^r+1_k −p^r_k−1)

1− χ(p_k) p^r+2_k

1− χ(p_k) p^r+3_k

with

1− 1

p^r+2_n+1 ≤ |E_r,n| ≤1 + 1 p^r+2_n+1, provided thatr = 1andp_n+1 ≥5, orr = 2andp_n+1 ≥3.

Proof. Recall the definition (14) ofA_r. Noting that (1−yt^r+1)





1 + _(1−yt^(y−1)tr+1^r+1)(1−t)

1− ^t_1−t^r+1



= 1 + yt^r+2 1−t−t^r+1, we obtain

B_χ(r) = A_rL(r+ 1, χ)

∞

Y

k=1

1 + χ(p_k) p_k(p^r+1_k −p^r_k−1)

, (17)

on settingy=χ(p_k)andt= _p¹

k. We rewrite the infinite product as L(r+ 2, χ)L(r+ 3, χ)

∞

Y

k=1

1 + χ(p_k) p_k(p^r+1_k −p^r_k−1)

1−χ(p_k) p^r+2_k

1−χ(p_k) p^r+3_k

in order to improve its convergence. Denoting thek-th term in the infinite product byPr,k, we see that (17) holds withE_r,n=Q

k≥n+1P_r,k.

It remains to estimate the relative error E_r,n (which in general is a complex number).

Multiplying out

(1−t−t^r+1+yt^r+2)(1−yt^r+2)(1−yt^r+3)/(1−t−t^r+1) gives

1 + yt^r+4 1−t−t^r+1

1 +t^r−1+ (1−y)t^r−yt^r+2−yt^2r+2+y²t^2r+3 , and leads to the estimate

|P_r,k| ≤1 +t^r+3G(t, r), with

G_r(t) = t(1 +t^r−1 + 2t^r+t^r+2+t^2r+2+t^2r+3) 1−t−t^r+1

andt= _p¹

k. Note thatG_r(t)is increasing intand decreasing inrin the region0< t <1and r≥1. Thus

|P_r,k| ≤1 +p^−r−3_k G_r(p⁻¹_k )≤1 +p^−r−3_k G_r(p⁻¹_n+1)for everyk ≥n+ 1.

Ast tends to zero,G_r(t)tends to zero, and so we can choosen so large thatG_r(p⁻¹_n+1)≤ 1.

Now

|E_r,n|= Y

k≥n+1

|P_r,k|< Y

p>pn

1 + 1 p^r+3

<1 + X

m>pn

1 m^r+3. Comparing the sum with an integral leads to the final estimate

|E_r,n| ≤1 + 1 p^r+3_n+1 +

Z ∞ pn+1

dz

z^r+3 ≤1 + 1 p^r+2_n+1, where the sum is over the integersm > p_n. Similarly,

|E_r,n|> Y

p>pn

1− 1 p^r+3

>1− X

m>pn

1

m^r+3 >1− 1 p^r+2_n+1.

Some calculus shows thatG_r(¹_p)≤1if and only ifr= 1andp≥5orr ≥2andp≥3. The proof is now completed on invoking the if-part of this statement. □

Remark 16. In the proof of[8, Thm. 6]there are a few typos:

For “2 + 2t+t³+t⁵” read “2 + 2t+t³+t⁴+t⁵”.

For “t≥127” read “t≤1/127”.

For “pn+!” read “pn+1”.

6. TWO EXAMPLES

In this section we demonstrate our results by two relatively easy, but illustrative, examples forK = Q(√

5), r = 1and d = 5. Some examples for the same r andd values, but with K =Qare given in Moree [10, Table 2].

G dens_G(0,5) dens_G(1,5) dens_G(2,5) dens_G(3,5) dens_G(4,5)

P0,5(10⁶) πK(10⁶)

P1,5(10⁶) πK(10⁶)

P2,5(10⁶) πK(10⁶)

P3,5(10⁶) πK(10⁶)

P4,5(10⁶) πK(10⁶)

⟨¹⁺

√ 5

2 ⟩ 0.100000 0.418205 0.296724 0.0950872 0.0899840 0.100093 0.419351 0.296954 0.0947177 0.0888838

⟨−⁵⁺

√ 5

2 ⟩ 0.100000 0.451872 0.266393 0.0995570 0.0821785 0.099787 0.450979 0.267518 0.0996599 0.0820564 TABLE1. Examples of densitiesdens_G(a,5)withK =Q(√

5)

In Table 1 we denote by Pa,d(x) the number of primespofK of norm up toxsuch that ind_p(G)≡amodd, and byπ_K(x)the number of primespofK with norm up tox. The top row gives the theoretical density, the second row an experimental approximation (both with rounding of the final decimal).

We will now treat these two examples without using the machinery of Section 3 (however, with complicated enough examples this becomes unavoidable). Our approach requires some further notation. Given a divisorδof an integerd₁, we put

ρ_δ,d1(a, d) := X

t≡amodd

X

v≥1 (v,d1)=δ

µ(v) vtφ(vt).

6.1. First example.

Proposition 17. SetK =Q(√

5)andG=⟨¹⁺

√ 5

2 ⟩. We have dens_G(0,5) = 1

10 = 2ρ(0,5) and, for1≤a≤4,

dens_G(a,5) = 18

19ρ(a,5) = 9 38

19

20+ψ(a)B_ψ(1) +ψ(a)B_ψ³(1) +ψ²(a)B_ψ²(1) . Proof. The first claim follows from (16) and Proposition 9. Next assume that1≤a≤4. We haveρ(a,5) =ρ_1,5(a,5) +ρ_5,5(a,5). If5∤t, then

X

5|v

µ(v)

vtφ(vt) =X

w

µ(5w)

5wtφ(5wt) =− 1 20

X

5∤w

µ(w) wtφ(wt).

We conclude thatρ5,5(a,5) = −₂₀¹ρ1,5(a,5). It thus follows thatρ1,5(a,5) = ²⁰₁₉ρ(a,5)and ρ_5,5(a,5) = −₁₉¹ρ(a,5). Since the degree [K_n,n : K] equals φ(n)n if 5 ∤ n and ¹₂nφ(n)

otherwise, we infer that dens_G(a,5) = ρ_1,5(a,5) + 2ρ_5,5(a,5) = ¹⁸₁₉ρ(a,5). The proof is completed on invoking (15) and noting thatψ²(a)is real andψ³(a) =ψ(a). □ Approximations toBχ(1) can be found in Table 2, whereψ denotes the character modulo 5determined uniquely byψ(2) =i.

χ Bχ(1)

ψ 0.34645514515465. . .+i·0.21283903970350. . . ψ² 0.12284254160167. . .

ψ³ 0.34645514515465. . .−i·0.21283903970350. . .

ψ⁴ 0.95

TABLE 2. The constantsB_χ(1)ford= 5

The character group has ψ, ψ², ψ³ andψ⁴ as elements, with ψ⁴ being the principal char- acter. The table is taken from [8, Table 3], where ford ≤ 12further approximations can be found. It was kindly verified by Alessandro Languasco using Theorem 15 withn = 10⁶. 6.2. Second example.

Proposition 18. SetK =Q(√

5)andG=⟨−⁵⁺

√ 5

2 ⟩. Let1 ≤a ≤4. One ofaanda+ 5is even. Denoting this number bya1, we have

dens_G(a,5) = 20

19ρ(a,5)− 4

19ρ(a₁,10).

Furthermore,densG(0,5) = ₁₀¹.

Proof. Using (16) we see that dens_G(0,5) = ₁₀¹ . We will determine dens_G(a,10) in case 5∤a. The result then follows on addingdensG(a,10)anddensG(a+ 5,10).

Since Q

q−⁵⁺

√5 2

= Q(ζ₅), the degree [K_n,n : K] equals φ(n)n if 5 ∤ n, it equals

1

2nφ(n)if(n,10) = 5, and it equals¹₄nφ(n)if10|n. These degree considerations lead to dens_G(a,10) =

(ρ_1,5(a,10) + 4ρ_5,5(a,10) if2|a;

ρ_1,5(a,10) + 2ρ_5,10(a,10) + 4ρ_10,10(a,10) if2∤a.

Reasoning as in the proof of Proposition 17 we deduce thatρ_5,5(a,10) = −₂₀¹ρ_1,5(a,10) andρ_1,5(a,10) = ²⁰₁₉ρ(a,10). It follows that dens_G(a,10) = ⁴₅ρ_1,5(a,10) = ¹⁶₁₉ρ(a,10) in caseais even.

If ais odd, then so are the integers t ≡ amod 10 and soρ_10,10(a,10) = −¹₂ρ_5,10(a,10), leading todens_G(a,10) = ρ_1,5(a,10). Reasoning as in the proof of Proposition 17 we then

deduce thatdens_G(a,10) = ²⁰₁₉ρ(a,10). □

For reasons of space we refrain here from explicitly writing outdens_G(a,5)as a linear sum in theB_χ’s, but we will indicate how this is done. For ρ(a,5)we use (15). Fora with5∤ a we have by [8, Proposition 6] withw= 5andδ = 2,

ρ(2a,10) = 3 8

X

χmod 5

χ(a) B_χ(1) 2 +χ(2) .

ACKNOWLEDGMENTS

This project was started when the first author gave a talk in the Luxembourg Number Theory Day 2019. He thanks the other authors for the invitation and for several subsequent invitations. The second and third author thank the Max Planck Institut f¨ur Mathematik and the first author for organizing a short visit in October 2022. Thanks are also due to Alessandro Languasco for verifying Table 2 using Theorem 15. We thank Valentin Blomer for pointing out reference [1].

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