GENERAL LINEAR INDEPENDENCE OF A CLASS OF
MULTIPLICATIVE FUNCTIONS
G. MOLTENI
ARCH. MATH. 83, 27-40 (2004).
Abstract. The notion of global non-equivalence of a set of multiplicative functions
is introduced. The linear independence of a set of globally inequivalent multiplicative
functions with respect to the ring C[r] where r is a slowly varying function is proved.
Applications to families of Artin L-functions are given.
AMS 2000 Mathematics Subject Classification:
11M41.
Introduction
Definition. Given a multiplicative arithmetical function f , (f )p is its p-component, i.e.,
the arithmetical function whose values are (f )p (m) := f (pm ). Let f, g : N → C be two
multiplicative arithmetical functions. f is called equivalent to g if (f )p = (g)p for all but
finitely many primes p.
P+∞
−s
converges somewhere and
Suppose that the Dirichlet series F (s) :=
n=1 f (n)n
that F has also a representation asPEuler product, so that the arithmetical function f
−s
so that if one is interested to the
is multiplicative. Then, F 0 (s) = − +∞
n=1 log nf (n)n
linear independence over C of the derivatives of Dirichlet series having Euler product one
has to study the linear independence of multiplicative arithmetical functions over the ring
C[log].
This problem has been treated in [5] by a technic which is essentially based on the
additivity of the function log n. In fact, in that paper the independence of every set of
pairwise inequivalent multiplicative functions with respect to the ring C[r], being r an
arbitrary additive arithmetical function, is proved.
In the present paper we study again the linear independence of multiplicative arithmetical functions with respect to a ring C[r], this time without imposing on r any hypothesis
of arithmetical type but assuming that
r(an)
lim
= 1 for every a ∈ N\{0}.
n→∞ r(n)
It is interesting to remark that such a condition, that will appear to be natural in our
context, is actually strictly related to the notion of slowly varying function introduced
by Karamata [7]. Under this hypothesis Theorem 1 below asserts the independence with
respect to C[r] of every set of globally inequivalent multiplicative functions (see the following definitions). Evidently log n is a good choice for r(n) hence the present result can
be considered as an extension in an “analytical direction” of [5]. However, the global
non-equivalence we assume here is a property much more restrictive than the pairwise
non-equivalence and the result of [5] is not completely included in the present result. In
Date: February, 2003. File name: general_independence.tex.
1
2
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
spite of our inability to prove it, we believe that the weaker pairwise non-equivalence
should be sufficient in order to Theorem 1 holds.
Theorem 2 is essentially the “translation” of Theorem 1 into the context of Dirichlet
series and its Corollaries 1-2 yield some interesting consequences on Artin L-functions
and Dedekind zeta functions. In particular we show that the abelian Artin L-functions
and the Dedekind zeta functions of normal extensions of Q are C[r]-independent.
The results
For future reference we quote as lemma the following Theorem 2 of [6].
Lemma 1. Pairwise non-equivalent multiplicative arithmetical functions are linearly independent over C.
Definition. We say that the multiplicative arithmetical functions f0 . . . , fN are globally
inequivalent if the multiplicative functions
Y
fˆj :=
fl for j = 0, . . . , N
l6=j
are pairwise inequivalent.
Remark 1. Evidently, the functions f0 . . . , fN are globally inequivalent if and only if
(1)
kp
kp
(fi (p ) − fj (p ))
N
Y
fl (pkp ) 6= 0
l=0
l6=i,j
for infinitely many primes p depending on i, j, for some kp depending on i, j, p. By (1)
it is clear that if f0 , . . . , fN are globally inequivalent then they are pairwise inequivalent,
too. When N > 1 the converse is not true.
Let f0 , . . . , fN be multiplicative arithmetical functions and let q1 , . . . , qN ∈ N. Consider
..
1
1
.
1
.
f (q )
.
f
(q
)
.
f
(q
)
0 1
1 1
N 1
..
VN (f0 , . . . , fN ; q1 , . . . , qN ) := f0 (q1 q2 )
f1 (q1 q2 )
.
fN (q1 q2 ) .
..
···
···
.
···
QN
Q
QN
f0 ( j=1 qj ) f1 ( N
j=1 qj ) · · · fN ( j=1 qj )
Lemma 2. Let f0 , . . . , fN be globally inequivalent multiplicative arithmetical functions.
Then there exist q1 , . . . , qN ∈ N such that VN (f0 , . . . , fN ; q1 , . . . , qN ) 6= 0.
Proof. By contradiction: assume that for some set f0 , . . . , fN of globally inequivalent
multiplicative functions
(2)
VN (f0 , . . . , fN ; q1 , . . . , qN ) = 0
for every choice of q1 , . . . , qN .
Moreover, let N be minimal in order for (2) to be satisfied. Since
1
1
= f1 (q) − f0 (q),
V1 (f0 , f1 ; q) = f0 (q) f1 (q)
GENERAL LINEAR INDEPENDENCE OF A CLASS OF MULTIPLICATIVE FUNCTIONS
3
it is evident that N > 1. Q
Let us suppose that (q1 , N
j=2 qj ) = 1, then calculating VN with respect to the first row
we obtain the recursive identity
(3)
N
N
X
Y
j
ˇ
VN (f0 , . . . , fN ; q1 , . . . , qN ) =
(−1) VN −1 (f0 , . . . , fj , . . . , fN ; q2 , . . . , qN )
fl (q1 ),
j=0
l=0
l6=j
where the accent above fj means that such function is omitted. Let us fix arbitrary values
for q2 , . . . , qN and define, for j = 0, . . . , N

Q
0
if (n, N
j=2 qj ) 6= 1
˜
fj (n) := QN
Q
 l=0 fl (n) if (n, N
j=2 qj ) = 1
l6=j
and
λj := (−1)j VN −1 (f0 , . . . , fˇj , . . . , fN ; q2 , . . . , qN ).
By (2) and (3) we get
N
X
λj f˜j (n) = 0 identically.
j=0
By Remark 1 the multiplicative functions f˜j for j = 0, . . . , N are pairwise inequivalent
hence they are linearly independent by Lemma 1 so that λj = 0 for every j, i.e.,
VN −1 (f0 , . . . , fˇj , . . . , fN ; q2 , . . . , qN ) = 0 for every j = 0, . . . , N .
The argument giving the previous identity can be repeated for every choice of q2 , . . . , qN
therefore we actually proved
VN −1 (f0 , . . . , fˇj , . . . , fN ; q2 , . . . , qN ) = 0 for every q2 , . . . , qN ∈ N, for every j = 0, . . . , N ,
contradicting the minimality of N .
Now we can prove the main result.
Theorem 1.
(i) Let r be an arithmetical function such that
lim r(n) = +∞,
n→∞
lim r(an)/r(n) = 1
n→∞
for every sufficiently large a ∈ R+ .
(ii) Let f0 , . . . , fN be globally inequivalent multiplicative arithmetical functions.
(iii) Suppose that there exist two positive constants c < C and infinitely many primes
p admitting a power pk (with k = k(p)) such that
c < |fj (pk )| < C
for every j, p.
Then, the functions f0 , . . . , fN are linearly independent over the polynomial ring C[r].
4
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
Proof. By contradiction: assume that there exist a set f0 , . . . , fN of globally inequivalent
and multiplicative functions such that
N
X
Pj (r(n))fj (n) = 0 identically
j=0
for some polynomials Pj 6≡ 0. Let M be the largest of the degrees of the polynomials Pj ;
by Remark 1 and Lemma 1 we have M > 0. Without loss of generality we can suppose
that only P0 , . . . , Pk have the largest degree, so that we have
k
X
(4)
M
λj r (n)fj (n) +
M
−1 X
N
X
j=0
βi,j ri (n)fj (n) = 0 identically
i=0 j=0
with λ0 , . . . , λk 6= 0. It is evident that N > 0 (since r(n) diverges and |fj (pk )| > c for
infinitely many pk ).
Every subset of a set of globally inequivalent multiplicative functions is globally inequivalent as well, hence by Lemma 2 there exist q1 , . . . , qk such that Vk (f0 , . . . , fk ; q1 , . . . , qk ) 6=
Qk
0. Let q0 = 1. Moreover, let {su }∞
u=1 be a divergent sequence such that (su ,
j=0 qj ) = 1
for every u and such that
c < |fj (su )| < C
(5)
for every j whose existence is stated in (iii). Considering (4) for n = su q0 , n = su q0 q1 ,
Q
n = su q0 q1 q2 ,. . . , n = su kj=0 qj we obtain a set of k + 1 linear equations for λj and βi,j
k
X
λj rM (su
l
Y
qj )fj (su )fj (
qj ) +
M
−1 X
N
X
u=0
u=0
j=0
l
Y
βi,j ri (su
i=0 j=0
l
Y
qj )fj (su )fj (
l
Y
qj ) = 0
u=0
u=0
for 0 ≤ l ≤ k, with λj 6= 0 for every j, hence for every u the system
(6)
k
X
j=0
(u)
λj rM (su
l
Y
qj )fj (
u=0
l
Y
qj ) +
M
−1 X
N
X
u=0
(u)
i=0 j=0
(u)
βi,j ri (su
l
Y
qj )fj (
u=0
l
Y
qj ) = 0
u=0
(u)
(u)
for 0 ≤ l ≤ k, has a solution λj , βi,j ((λ(u) , β (u) ), in brief) with λj
(u)
βi,j := βi,j fj (su ) for every i, j. From (5) we have that
(u)
0 < c|λj | ≤ |λj | ≤ C|λj |,
:= λj fj (su ),
(u)
c|βi,j | ≤ |βi,j | ≤ C|βi,j |,
for every u, i and j, so that (λ(u) , β (u) ) runs in a compact subset of an Euclidean space
when u → ∞. Considering a suitable subsequence of u if necessary, we can assume that
(∞)
(λ(u) , β (u) ) → (λ(∞) , β (∞) ), with λ1 6= 0.
Dividing (6) by rM (su ) and taking the limit for u → ∞ from (i) it follows that
k
X
l
Y
(∞)
λj fj ( qj )
u=0
j=0
(∞)
= 0 for 0 ≤ l ≤ k,
but this equation is impossible since λ1 6= 0 and the determinant of the matrix of such
a linear system is Vk (f0 , . . . , fk ; q1 , . . . , qk ) which is not zero.
GENERAL LINEAR INDEPENDENCE OF A CLASS OF MULTIPLICATIVE FUNCTIONS
5
P
−s
Let us suppose that log r(n) log n, so that the Dirichlet series R(s) := ∞
n=1 r(n)n
converges when <s is sufficiently large.
by Φr the operator acting on every
P Let us denote
−s
converging Dirichlet series F (s) := ∞
f
(n)n
by
n=1
∞
X
(7)
(Φr ∗ F )(s) :=
r(n)f (n)n−s .
n=1
This operator has two more representations; the first one relies on the general theory of
almost periodic functions (but can be proved directly) and is
Z c+iT
1
(8)
(Φr ∗ F )(s) = lim
R(w)F (s − w)dw,
T →+∞ 2iT c−iT
when c is sufficiently large.
The second one is valid when r(n) = φ(log n) where φ(x) is a measurable, positive function
which is supported on [0, +∞) and satisfies the bound log φ(x) x when x → ∞. In
this case we have
Z c+i∞
1
(9)
(Φr ∗ F )(s) =
φ̂(w)F (s − w) dw,
2πi c−i∞
when c, <z and <s are sufficiently large, where φ̂(w) is the Laplace transform of φ
Z +∞
φ̂(w) :=
φ(x)e−xw dx.
0
Using this notation from Theorem 1 we immediately get the following result.
Theorem 2.
(i) Let φ : [0, +∞) → [0, +∞) be a measurable function such that
lim φ(x) = +∞,
x→∞
lim φ(x + a)/φ(x) = 1
x→∞
for every a > 0 sufficiently large.
(ii) Let f0 , . . . , fN be globally inequivalent multiplicative arithmetical functions.
(iii) Suppose that there exist two positive constants c < C and infinitely many primes
p admitting a power pk (with k = k(p)) such that
c < |fj (pk )| < C for every j, p.
P
−s
(iv) Suppose that Fj (s) := ∞
converges when <s is sufficiently large.
n=1 fj (n)n
Then, the functions F0 , . . . , FN are linearly independent with respect to the polynomial
ring C[Φr ], where the actionPof Φr on F is given by (7) (in terms of r(x) := φ(log x))
−s
or (8) (in terms of R(s) := ∞
n=1 r(n)n ) or directly by (9).
Proof. By (i) r(x) := φ(log x) is a slowly varying function, i.e.,
r(ax)
= 1 for every a ∈ R+
n→∞ r(x)
lim
and Karamata (see [7], [2] and [9]) characterized these functions as those ones for which
there exists a constant d ∈ R+ and a continuous function δ : [0, ∞) → [0, ∞) such that
Z x
dt δ(x) → 0, and r(x) ∼ d exp
δ(t)
, when x → +∞.
t
1
6
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
As a consequence r(x) xkδk∞ , i.e., φ(x) ekδk∞ x , so that the claim follows now from
Theorem 1.
When φ(x) = x, Φr acts as the (opposite of the) derivation and Theorem 2 states
the linear independence of the derivatives of the involved Dirichlet series. For Artin Lfunctions a similar but weaker result has recently been proved by Nicolae [8]. In the
remaining sections we will show suitable subsets of Artin L-functions satisfy the hypotheses of Theorem 2 so that the linear independence of their derivatives follows. Actually, as
we already recalled, the linear independence of the derivatives of a much ampler set of Lfunctions (containing the totality of Artin L-functions, cuspidal automorphic L-functions
for GL(m) and the Selberg class) has been proved in [5].
Other elementary choices for φ(x) are exp(xβ ) (with 0 < β < 1) or φ(x) = xα (with
α > 0); in both such cases the claim of Theorem 2 is new and is not in the range of the
technic employed in [5].
Remark 2. Representations (7)-(9) are equivalent, but (9) is more appropriate when analytical properties of Φr ∗ F are studied. When φ(x) = xα , α > 0, the Laplace transform
φ̂(w) is Γ(α + 1)w−α−1 and, on deforming the path in (9) to a path encircling the real
negative half-line, it is possible to prove that for α 6∈ N,
P[α] F n (s)
Z
Γ(α + 1) sin(απ) +∞ F (s + w) − n=0 n! wn
(Φr ∗ F )(s) = −
dw,
π
wα+1
0
where [α] is the integral part of α. For example, when α = 1/2 we get the identity
Z +∞
∞ p
X
log(n)f (n)
1
F (s + w) − F (s)
= (Φr ∗ F )(s) = − √
dw.
s
n
w3/2
2 π 0
n=1
Hypotheses (ii) and (iii) in Theorems 1 and 2 deserve some further discussion. In the
following section we will show that (iii) is satisfied by every finite set of Artin L-functions.
Assuming the Langlands conjectures (LH, in brief) it is easy to prove that (iii) holds also
for every finite
automorphic L-functions. In fact, for j = 0, . . . , N , let
P∞ set of cuspidal
−s
be the L-function related to a cuspidal representation πj of
L(s, πj ) =
n=1 fj (n)n
GL(m). For almost every prime the p-component of L(s, πj ) is the inverse of a polynomial
of degree m in the variable p−s ; under LH the roots of such polynomial have modulus
1 (i.e., the Ramanujan conjecture holds) so that |fj (p)| ≤ m for almost every prime:
this fact proves the validity of the right-hand side of (iii). The
L(s, π0 ⊗ · · · ⊗
P function −s
πN ⊗ π0 ⊗ · · · ⊗ πN ) has a representation as Dirichlet series ∞
a(n)n
with a(p) =
n=1
QN
2
| j=0 fj (p)| for almost every prime p. Under LH L(s, π0 ⊗ · · · ⊗ πN ⊗ π0 ⊗ · · · ⊗ πN )
is automorphic, as well, in particular it has a meromorphic continuation to C with an
unique pole at s = 1 and no zeros on the line <s = 1, so that by a standard argument of
analytical number theory we get
Q
2
X| N
X a(p)
j=0 fj (p)|
(10)
=
∼ M ln ln x
p
p
p<x
p<x
P
when x → +∞, for some M > 0. Since p<x p1 ∼ ln ln x by prime number theorem, (10)
Q
implies that N
j=0 fj (p) 6→ 0 when p → ∞, therefore there exists l > 0 and an infinite
Q
set of primes P such that N
j=0 |fj (p)| → l when p runs over P. We already know that
GENERAL LINEAR INDEPENDENCE OF A CLASS OF MULTIPLICATIVE FUNCTIONS
7
|fj (p)| < m for every j, therefore from the previous fact we get |fj (p)| > lm−N /2 for every
j and every sufficiently large p ∈ P: also the lower-bound to the left-hand side of (iii) is
proved.
Since it is generally believed that every Dirichlet series occurring in number theory is
actually a cuspidal automorphic L-function, from the previous argument appears that
hypothesis (iii) is limiting but not unnatural. Hypothesis (ii), however, is more critical
since also in the context of L-functions arising in number theory it can fail: we will see
in the next section an example involving the Artin L-functions.
In spite of such difficulties, there are some subsets of Artin L-functions satisfying the
hypotheses of our theorems: some results in this sense are collected in the next two
sections.
Consequence for Artin L-functions
Let K/H be a finite Galois extension of a number field H. Let G := Gal(K, H) and
let χ be the character of a representation π of G. Let L(s, χ) be the associated Artin
L-function
∞
X
L(s, χ) =
f (n)n−s .
n=1
From the definition of L(s, χ) as Euler product it follows that
f (N (p)) = χ(Frob(p))
when p is a non-ramified prime ideal of H, where Frob is the Frobenius map: H → G and
N is the absolute ideal norm K → Q. By Chebotarev’s density theorem there are infinitely
many prime ideals p such that Frob(p) = id. For such primes χ(Frob(p)) = χ(id) = dim π,
therefore for Artin L-functions hypothesis (iii) in Theorems 1-2 is always satisfied.
Hypothesis (ii), however, is not always satisfied since not every set of Artin L-functions
is a set of multiplicative globally inequivalent functions, not even when the involved
representations are distinct and irreducible.
Remark 3. An example: let K be the splitting field of x3 − 2, so that Gal(K, Q) = S3 (the
permutation group over 3 objects). The table of the characters of S3 is
χ1
χ2
χ3
< e > < (1, 2) > < (1, 2, 3) >
1
1
1
1
−1
1
2
0
−1
so that for non-ramified primes p the p-components of L(s, χj ) are respectively
Lp (s, χ1 )
Lp (s, χ2 )
Lp (s, χ3 )
−s −1
−s −1
Frob(p) ∈< e >
(1 − p )
(1 − p )
(1 − p−s )−2
−s −1
−s −1
(1 + p )
(1 − p−2s )−1
Frob(p) ∈< (1, 2) > (1 − p )
Frob(p) ∈< (1, 2, 3) > (1 − p−s )−1 (1 − p−s )−1 (1 + p−s + p−2s )−1
From this table it is evident that (f1 (pk ) − f2 (pk ))f3 (pk ) = 0 for every non-ramified prime
and for every k. Therefore the L-functions L(s, χ1 ), L(s, χ2 ), L(s, χ3 ) are not globally
inequivalent.
In spite of this fact, the following result holds.
8
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
Corollary 1. Let K/H be a finite Galois extension of a number field H. Let G :=
Gal(K, H) and for j = 0, . . . , N let πj be representations of G with distinct characters
χj . Let fj be the arithmetical function giving the coefficients of the Dirichlet series of
L(s, χj ). In each of the following cases the functions f0 , . . . , fN are globally inequivalent
and Theorems 1-2 hold.
(i) π̂i 6= π̂j whenever i 6= j, where π̂k := ⊗N
l=0 πl .
l6=k
(ii)
(iii)
(iv)
(v)
N = 1, i.e., only two representations are considered.
deg πi 6= deg πj for i 6= j.
G is abelian and πj is irreducible for every j.
The L-functions are Dirichlet L-functions whose Dirichlet characters are induced
by primitive and pairwise distinct characters.
Remark 4. We note that only in case (iv) the irreducibility of the involved representations
is assumed and a simple example shows that the irreducibility cannot be avoided in this
case: let K be a number field whose Galois group G is the cyclic group of order four.
Let g be a generator fo G and consider the linear representations whose values at g are
π1 (g) = 1 and π2 (g) = −1 respectively and let π3 := π1 ⊕ π2 . Then it is immediate to
check that L(s, χ1 ), L(s, χ2 ), L(s, χ3 ) are not globally inequivalent.
Proof. (i) Since fj (N p) = χj (Frob(p)) when p is not ramified, in order to prove that
f0 , . . . , fN are globally inequivalent, i.e., that (1) holds, it is sufficient to prove that for
every choice of i 6= j,
(11)
N
Y
l=0
l6=i
χl (Frob(p)) 6=
N
Y
χl (Frob(p)) for infinitely many prime ideals p.
l=0
l6=j
By Chebotarev’s density theorem every conjugacy class C in G is the image of the Frobenius map for infinitely many prime ideals, therefore in order to (11) holds it is sufficient
to prove that for every i 6= j there exists a class C˜ such that
(12)
N
Y
˜ 6=
χl (C)
l=0
l6=i
The claim follows on noting that
N
Y
˜
χl (C).
l=0
l6=j
QN
l=0
l6=i
χl is the character of the representation π̂i and
that two representations are similar if and only if their characters are equal.
(ii) This follows from (i) since π̂0 = π1 and π̂1 = π0 , when N = 1.
(iii) This is immediate from (i); in fact, if deg πi 6= deg πj then deg π̂i 6= deg π̂j as well,
so that π̂i 6= π̂j whenever i 6= j.
(iv) The irreducible characters of an abelian group are automorphisms over a finite
group of complex roots of unity, in particular they do not assume the value 0. As a
consequence, in (12) we can cancel all the representations which appear both on the left
and on the right-hand side, obtaining that in the abelian case π̂i = π̂j if and only if
πi = πj . Hence, the claim follows from (i).
(v) Let L(s, χj ) for j = 0, . . . , N be the set of Dirichlet L-functions. Let χ0j be the primitive character inducing χj . Since χj and χ0j differ only on a finite set of p-components,
GENERAL LINEAR INDEPENDENCE OF A CLASS OF MULTIPLICATIVE FUNCTIONS
9
the arithmetical functions χ0 , . . . , χN are globally inequivalent if and only if the functions χ00 , . . . , χ0N are globally inequivalent. The claim follows by (iv), since the primitive
Dirichlet characters are characters of abelian extensions of Q.
The following question arises: does there exist a finite non-abelian group G such that
the set of all its irreducible representations satisfies (i)? This problem appears to be quite
difficult; we conjecture that the answer is no. A good test for the conjecture is given
by the simple groups: Livio Di Martino (personal communication) proved that if a such
group exists, it cannot be a PSL(2, pm ), p odd. His argument probably can be extended
to the entire family PSL(n, pm ). It is interesting to remark that the argument that was
used in (iv) to prove the pairwise non-equality of the representations π̂j does not hold
for all irreducible representations of a non-abelian group: Burnside proved (Theorem 3.15
in [4] and Theorem 21.1 in [1]) that when G is not abelian, every non-linear irreducible
character of G assumes the value zero somewhere.
Remark 5. The group S4 has five irreducible representations: π1 and π2 of degree 1, π3 of
degree 2 and π4 and π5 of degree 3, say: the set {π1 , π2 , π4 , π5 } satisfies (i).
The group A5 has five irreducible representations: π1 of degree 1, π2 and π3 of degree
3, π4 of degree 4 and π5 of degree 5, say: the set {π1 , π2 , π3 , π4 } satisfies (i).
The group A6 has seven irreducible representations: π1 of degree 1, π2 and π3 of degree
5, π4 and π5 of degree 8, π6 of degree 9 and π7 of degree 10, say: the sets {π1 , π2 , π3 , π5 , π7 }
and {π1 , π2 , π3 , π4 , π7 } satisfy (i).
Consequence for Dedekind zeta-functions
In this section we specialize the previous argument to the set of Dedekind zeta-functions.
The main result we prove here, case (iv) of Corollary 2 below, shows that our argument
is sufficiently strong to deal with the case of normal extensions of Q. The other cases
considered in that corollary show that the normality of the fields is probably unnecessary
and that the non-equality of the Dedekind zeta functions involved should be sufficient:
we did not succeed in proving the claim under an hypothesis as general as this one. At
last, case (ii) is interesting by its generality and case (iii) deals with a situation opposite,
in some sense, to (ii).
Let Hj for j = 0, . . . , N be a finite set of number fields and let K be the normal closure
of their composition field. Let G be the Galois group of K over Q and let Hj be the Galois
group of K over Hj . Then
L(s, indG
Hj idHj ) = ζHj (s)
so that the set of Dedekind zeta functions ζHj (s) is actually a set of Artin L-functions
generated by a finite set of representations of G. The argument used at the beginning of
previous section shows that for such functions hypothesis (iii) appearing in Theorems 1
and 2 is satisfied.
Corollary 2. Let Hj for j = 0, . . . , N be number fields and let fj be the arithmetical
function whose Dirichlet series is ζHj (s). In each of the following cases the functions
f0 , . . . , fN are globally inequivalent and Theorems 1 and 2 hold.
(i) N = 1 and ζH0 6= ζH1 , i.e., only two distinct Dedekind zeta functions are considered.
(ii) The degrees [Hj : Q] are pairwise distinct.
10
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
√
(iii) Hj = Q[ q aj ] where q is a fixed odd prime, aj ∈ N and the normal closures Hj are
distinct.
(iv) The fields Hj are distinct and normal over Q.
Remark 6. Note that in (iii) it is not assumed that aj is q-power free.
Proof. (i) Evident by (i) of Corollary 1.
(ii) The degree of indG
Hj idHj is [G : Hj ] = [Hj : Q], hence this claim follows by (iii) of
Corollary 1.
√
√
(iii) The normal closure of the fields Hj is K := Q[ζq , q a0 , . . . , q aN ], where ζq is a
primitive q-th root of unity. Every σ ∈ G = Gal(K, Q) is characterized by the integers a
and wj such that
√
√
σ(ζq ) = ζqa , σ( q aj ) = ζqwj q aj , for every j
and such integers are unique when they are taken from F∗q := (Z/qZ)∗ and Fq := Z/qZ,
respectively. This fact suggests to identify σ with the couple (a, w) where a ∈ F∗q and w
belongs to the N + 1-dimensional vector space V over Fq . Under this identification the
composition law of G becomes
σ1 = (a1 , w1 ), σ2 = (a2 , w2 ),
=⇒
σ1 σ2 = (a1 a2 , a1 w2 + w1 ),
therefore G is identified with a subgroup of the group GN which is the group whose
elements are
GN := {(a, w)| a ∈ Fq , w ∈ V },
with product
(a1 , w1 )(a2 , w2 ) := (a1 a2 , a1 w2 + w1 ).
We note that G contains (a, 0) for every a ∈ F∗q since these elements represent the auto√
√
morphisms of K fixing Q[ q a0 , . . . , q aN ]. This fact and the identity
(c, w1 )(ac−1 , 0)(d, w2 )(ba−1 d−1 , 0) = (b, w1 + aw2 ),
prove that G is actually isomorphic to a subgroup of GN of type
G = {(a, w)| a ∈ Fq , w ∈ W },
where W is a suitable subspace of V . Under this isomorphism, every Galois group Hj :=
Gal(K, Hj ) is identified with
Gvj := {(a, w)| a ∈ Fq , w ⊥ v j , w ∈ V },
for a suitable vector v j ∈ V . We assumed that the normal closures Hj are pairwise distinct
so that the vectors v j are not proportional and the groups Gvj are pairwise distinct, too.
Let χj be the character of the representation πj of G which is induced by the trivial
representation of Gvj . A direct check shows that the values of χj are
(
q if w ⊥ v j
χj ((1, w)) =
and χj ((a, ∗)) = 1 for a 6= 1.
0 otherwise,
Q
Therefore, denoting by χ̂j := l6=j χl the character of the representation π̂j := ⊗l6=j πl , we
get
χ̂j ((1, v i )) 6= 0
⇐⇒
i = j,
GENERAL LINEAR INDEPENDENCE OF A CLASS OF MULTIPLICATIVE FUNCTIONS
11
proving that the characters χ̂j are distinct. The claim follows by Corollary 1, case (i),
since L(s, χj ) = ζHj (s).
(iv) Corollary 1 is not sufficiently strong to prove this claim: a different and direct
approach is necessary. Actually, it is sufficient to prove that
k
k
(f0 (p ) − f1 (p ))
(13)
N
Y
fl (pk ) 6= 0
l=2
for infinitely many primes p, for some k depending on p, since the same argument will
work for different choices of the pair of indexes i, j in (1). We recall that for number field
H
X
ζH (s) :=
(N a)−s for <s > 1,
a∈OH
where OH is the ring of algebraic integers of H and a is an ideal. Therefore,
∞
X
ζH (s) =
f (n)n−s
n=1
with f (n) := ]{a ∈ OH : N a = n}. Let p be a non-ramified prime, let
p OH = p1 · · · pg
be its decomposition in OH and let r be the dimension of OH /pj over Z/pZ so that
N pj = pr . When H is normal over Q, r does not depend on pj so that for large k we have
X
(14)
1
f (pk ) = ]{n1 , . . . , ng ∈ N : N (pn1 1 · · · png g ) = pk } =
n1 ,...,ng ∈N
n1 +n2 +···+ng =k/r
=
(k/r)g−1
(δr|k
(g−1)!
+ o(1)) =
(k/r)[H:Q]/r−1
(δr|k
([H:Q]/r−1)!
+ o(1)),
where δr|k = 1 if r divides k, 0 otherwise. Now we can prove the claim.
Let us suppose [H0 : Q] 6= [H1 : Q]. By Chebotarev’s density theorem there are
infinitely many primes p splitting totally in K (the normal closure of the composition
field of H0 , . . . , HN ). Such primes split totally in every Hj as well, as a consequence
fj (pk ) 6= 0 for every k and for every j (since r = 1 in (14)). From (14) we have
f0 (pk ) ∼
k[H0 :Q]−1
([H0 :Q]−1)!
and f1 (pk ) ∼
k[H1 :Q]−1
,
([H1 :Q]−1)!
therefore f0 (pk ) 6= f1 (pk ) for every sufficiently large value of k, thus (13) holds in this
case.
Let us suppose [H0 : Q] = [H1 : Q]. Since H0 6= H1 , by Bauer’s theorem (see [3],
Theorem 9-1-3) there are infinitely many primes p splitting totally in H0 which do not
split totally in H1 . Take p in such set of primes so that the dimension r of p is 1 for H0
and greater than 1 for H1 . Then, from (14) we have
f0 (pk ) ∼
k[H0 :Q]−1
([H0 :Q]−1)!
and f1 (pk ) =
(k/r)[H0 :Q]/r−1
(δr|k
([H0 :Q]/r−1)!
+ o(1)) k [H0 :Q]/2−1
Q
therefore f0 (pk ) 6= f1 (pk ), whenever k is sufficiently large. Moreover, let d := N
j=2 [Hj :
rj
Q]. Let qj be a prime ideal of Hj over p; then N (qj ) = p for some rj . By assumption
d/r
Hj is normal, hence rj |d and N (qj j ) = pd . This proves that fj (pk ) is not zero for every
j = 2, . . . , N whenever d|k and the proof of (13) is now complete.
12
G. MOLTENI ARCH. MATH. 83, 27-40 (2004).
Acknowledgments. I wish to thank the anonymous referee whose accurate reading,
suggestions and comments have considerably improved the presentation of these results.
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Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy
E-mail address: giuseppe.molteni@mat.unimi.it