Journal de Théorie des Nombres de Bordeaux

Reza Akhtar

Paul Larson

Small-sum pairs in abelian groups

Alexey Zykin

Alexey ZYKIN
Asymptotic properties of Dedekind zeta functions in families of number
fields
Tome 22, no 3 (2010), p. 771-778.
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Journal de Théorie des Nombres
de Bordeaux 22 (2010), 771-778
Asymptotic properties of Dedekind zeta functions
in families of number fields
par Alexey ZYKIN
Résumé. Le but de cet article est de démontrer une formule
qui exprime le comportement asymptotique de la fonction zêta de
Dedekind dans des familles de corps globaux pour Re s > 1/2 en
supposant que l’Hypothèse de Riemann Généralisée est vérifiée.
On peut voir ce résultat comme une généralisation du théorème
de Brauer-Siegel. Comme corollaire, on obtient une formule limite
pour les constants d’Euler-Kronecker dans des familles de corps
globaux.
Abstract. The main goal of this paper is to prove a formula
that expresses the limit behaviour of Dedekind zeta functions for
Re s > 1/2 in families of number fields, assuming that the Gener-
alized Riemann Hypothesis holds. This result can be viewed as a
generalization of the Brauer–Siegel theorem. As an application we
obtain a limit formula for Euler–Kronecker constants in families
of number fields.
1. Introduction
Our starting point is the classical Brauer–Siegel theorem for number
fields first proven by Siegel in the case of quadratic fields and then by
Brauer (see [1]) in a more general situation. This theorem states that if
K runs through a sequence of number fields normal over Q such that
nK/ log |DK | → 0, then log(hKRK)/ log
√
|DK | → 1. Here DK , hK , RK
and nK are respectively the discriminant, the class number, the regulator
and the degree of the field K.
In [11] this theorem was generalized by Tsfasman and Vlăduţ to the case
when the condition nK/ log |DK | → 0 no longer holds. To formulate this
result we will need to introduce some notation.
Manuscrit reçu le 20 décembre 2009.
The author was partially supported by the EADS foundation chair in mathematics, by the
laboratory of Mathematical Studies of HSE (project T3-62.0), by the grants RFBR 09-01-12170-
ofi_m, RFBR 08-07-92495-CNRSa, RFBR 10-01-93110-CNRSa, by the grant of the Ministry of
Education and Science of Russia No. 2010-1.3.1-111-017-029. The individual research project N
10-01-0054 "Asymptotic properties of zeta functions" was carried out with the support of the
program "Scientific fund of HSE".
772 Alexey Zykin
For a finite extension K/Q, let Φq(K) be the number of prime ideals
of the ring of integers OK with norm q, i.e. Φq(K) = |{p | Norm p = q}|.
Furthermore, denote by ΦR(K) and ΦC(K) the number of real and complex
places of K respectively. Let gK = log
√
|DK | be the genus of the field K
(in analogy with the function field case). An extension K/Q is called almost
normal if there exists a tower of extensions K = Kn ⊇ Kn−1 · · · ⊇ K1 ⊇
K0 = Q such that Ki/Ki−1 is normal for all i.
Consider a family of pairwise non-isomorphic number fields {Ki}.
Definition 1. If the limits φα = lim
i→∞
Φα(Ki)
gKi
, α ∈ {R,C, 2, 3, 4, 5, 7, 9, . . .}
exist for each α then the family {Ki} is called asymptotically exact. It is
asymptotically good if there exists φα 6= 0 and asymptotically bad oth-
erwise. The numbers φα are called the Tsfasman–Vlăduţ invariants of the
family {Ki}.
It is not difficult to check (see [11, Lemma 2.7]) that the condition
nK/ log |DK | → 0 from the Brauer–Siegel theorem is equivalent to the fact
that the corresponding family is asymptotically bad. One can prove that
any family contains an asymptotically exact subfamily and that an infinite
tower of number fields is always asymptotically exact (see [11, Lemma 2.2
and Lemma 2.4]).
Now we can formulate the Tsfasman–Vlăduţ theorem proven in [11, The-
orem 7.3] in the asymptotically good case and in [12, Theorem 1] in the
asymptotically bad one.
Theorem 1. For an asymptotically exact family {Ki} we have
(1.1) lim
i→∞
log(hKiRKi)
gKi
= 1 +
∑
q
φq log
q
q − 1
− φR log 2− φC log 2π,
provided either all Ki are almost normal over Q or the Generalized Rie-
mann Hypothesis (GRH) holds for zeta functions of the fields Ki.
To generalize this theorem still further we will have to use the concept
of limit zeta functions from [11].
Definition 2. The limit zeta function of an asymptotically exact family of
number fields {Ki} is defined as
ζ{Ki}(s) =
∏
q
(1− q−s)−φq .
Theorem C from [11] gives us the convergence of the above infinite prod-
uct for Re s ≥ 1. Let κK = Res
s=1
ζK(s) be the residue of the Dedekind zeta
function of the field K at s = 1. Using the residue formula (see [7, Chapter
Properties of Dedekind zeta functions 773
VIII, Theorem 5])
κK =
2ΦR(K)(2π)ΦC(K)hKRK
wK
√
|DK |
(here wK is the number of roots of unity in K) and the estimate wK =
O(n2K) (see [7, p. 322]) one can see that the question about the behaviour
of the ratio from the Brauer–Siegel theorem is immediately reduced to the
corresponding question for κK .
The formula (1.1) can be rewritten as lim
i→∞
log κKi
gKi
= log ζ{Ki}(1). Further-
more, Tsfasman and Vlăduţ prove in [11, Proposition 4.2] that for Re s > 1
the equality lim
i→∞
log ζKi (s)
gKi
= log ζ{Ki}(s) holds.
Our main goal is to investigate the question of the validity of the above
equality for Re s < 1. We work in the number field case, for the function
field case see [13], where the same problem was treated in a much broader
context.
The case s = 1 is in a sense equivalent to the Brauer–Siegel theorem
so current techniques does not allow to treat it in full generality without
the assumption of GRH. From now on we will assume that GRH holds for
Dedekind zeta functions of the fields under consideration. Assuming GRH,
Tsfasman and Vlăduţ proved ([11, Corollary from Theorem A]) that the
infinite product for ζ{Ki}(s) is absolutely convergent for Re s ≥
1
2 . We can
now formulate our main results.
Theorem 2. Assuming GRH, for an asymptotically exact family of number
fields {Ki} for Re s > 12 we have
lim
i→∞
log((s− 1)ζKi(s))
gKi
= log ζ{Ki}(s).
The convergence is uniform on compact subsets of the half-plane {s | Re s >
1
2}.
The result for s = 12 is considerably weaker and we can only prove the
following upper bound:
Theorem 3. Let ρKi be the first non-zero coefficient in the Taylor series
expansion of ζKi(s) at s = 12 , i. e.
ζKi(s) = ρKi
(
s− 1
2
)rKi
+ o
((
s− 1
2
)rKi)
.
Then, assuming GRH, for any asymptotically exact family of number fields
{Ki} the following inequality holds:
(1.2) lim sup
i→∞
log |ρKi |
gKi
≤ log ζ{Ki}
(1
2
)
.
774 Alexey Zykin
The question whether the equality holds in theorem 3 is rather delicate.
It is related to the so called low-lying zeroes of zeta functions, that is the
zeroes of ζK(s) having small imaginary part compared to gK . We doubt that
the equality lim
i→∞
log |ρKi |
gKi
= log ζ{Ki}(
1
2) holds for any asymptotically exact
family {Ki} since the behaviour of low-lying zeroes is known to be rather
random. Nevertheless, it might hold for "most" families (whatever it might
mean). A more thorough discussion of this question in a slightly different
situation (low-lying zeroes of L-functions of modular forms on SL2(R)) can
be found in [4].
To illustrate how hard the problem may be, let us remark that Iwaniec
and Sarnak studied a similar question for the central values of L-functions
of Dirichlet characters [5] and modular forms [6]. They manage to prove
that there exists a positive proportion of Dirichlet characters (modular
forms) for which the logarithms of the central values of the corresponding
L-functions divided by the logarithms of the analytic conductors tend to
zero. The techniques of the evaluation of mollified moments used in these
papers are rather involved. We also note that, to our knowledge, there has
been no investigation of low-lying zeroes of L-functions of growing degree.
It seems that the analogous problem in the function field has neither been
very well studied.
Let us formulate a corollary of the theorem 2. We will need the following
definition:
Definition 3. The Euler–Kronecker constant of a number field K is defined
as γK = c0(K)c−1(K) , where ζK(s) = c−1(K)(s− 1)
−1 + c0(K) +O(s− 1).
Ihara made an extensive study of the Euler-Kronecker constant in [2].
In particular, he obtained an asymptotic formula for the behaviour of γ in
families of curves over finite fields. As a corollary of theorem 2, we prove
the following analogue of Ihara’s result in the number field case:
Corollary 1. Assuming GRH, for any asymptotically exact family of num-
ber fields {Ki} we have
lim
i→∞
γKi
gKi
= −
∑
q
φq
log q
q − 1
.
This result was formulated in [10] without the assumption of the Rie-
mann hypothesis. Unfortunately, the proof given there is flawed. It uses
an unjustified change of limits in the summation over prime powers and
the limit taken over the family {Ki}. Thus, the question about the validity
of this equality without the assumption of GRH is still open. It would be
interesting to have a result of this type at least under a certain normality
condition on our family {Ki}. Even the study of abelian extensions is not
uninteresting in this setting.
Properties of Dedekind zeta functions 775
2. Proofs of the main results
Proof of theorem 2. The statement of the theorem is known for Re s > 1
(see [11, Proposition 4.2]) thus we can freely assume that Re s < 2.
We will use the following well known result [3, Proposition 5.7] which
can be proven using Hadamard’s factorization theorem.
Proposition 1. (1) For −12 ≤ σ ≤ 2, s = σ + it we have
ζ ′K(s)
ζK(s)
+ 1
s
+ 1
s− 1
−
∑
|s−ρ|<1
1
s− ρ
= O(gK),
where ρ runs through all non-trivial zeroes of ζK(s) and the constant
in O is absolute.
(2) The number m(T,K) of zeroes ρ = β + γi of ζK(s) such that |γ −
T | ≤ 1 satisfies m(T,K) < C(gK+nK log(|T |+4)) with an absolute
constant C.
Now, applying this proposition, we see that for fixed T > 0, ε > 0 and
any s ∈ DT,ε = {s ∈ C | | Im s| ≤ T, ε+ 12 ≤ Re s ≤ 2} we have
(2.1) ζ
′
K(s)
ζK(s)
+ 1
s− 1
=
∑
|s−ρ|<ε
1
s− ρ
+OT,ε(gK),
for by Minkowski’s theorem [7, Chapter V, Theorem 4] nK < CgK with an
absolute constant C.
If we assume GRH, the sum over zeroes on the right hand side of (2.1)
disappears. Integrating, we finally get that in DT,ε
log(ζK(s)(s− 1))
gK
= OT,ε(1)
Now, we can use the so called Vitali theorem [9, 5.21]:
Proposition 2. Let fn(s) be a sequence of functions holomorphic in a
domain D. Assume that for some M ∈ R we have |fn(s)| < M for any n
and s ∈ D. Let also fn(s) tend to a limit at a set of points having a limit
point in D. Then the sequence fn(s) tends to a holomorphic function in D
uniformly on any closed disk contained in D.
It suffices to notice that the convergence of log ζKi(s)/gKi to ζ{Ki}(s) is
known for Re s > 1 by [11, proposition 4.2]. So, applying the above theorem
and using the fact that under GRH ζ{Ki}(s) is holomorphic for Re s ≥
1
2
[11, corollary from theorem A] we get the required result. 
Proof of theorem 3. Denote gk = gKk . Let us write
ζKk(s) = ck
(
s− 1
2
)rk
Fk(s),
776 Alexey Zykin
where Fk(s) is an analytic function in the neighbourhood of s = 12 such
that Fk
(
1
2
)
= 1. Let us put s = 12 + θ, where θ > 0 is a small positive real
number. We have
log ζKk(12 + θ)
gk
= log ck
gk
+ rk
log θ
gk
+
logFk(12 + θ)
gk
.
To prove the theorem we will construct a sequence θk such that
(1) 1gk log ζKk
(
1
2 + θk
)
→ log ζ{Kk}
(
1
2
)
;
(2) rkgk log θk → 0;
(3) lim inf 1gk logFk
(
1
2 + θk
)
≥ 0.
For each natural number N we choose θ(N) a decreasing sequence such
that ∣∣∣∣ζ{Kk} (12
)
− ζ{Kk}
(1
2
+ θ(N)
)∣∣∣∣ < 12N .
This is possible since ζ{Kk}(s) is continuous for Re s ≥
1
2 by [11, corollary
from theorem A]. Next, we choose a sequence k′(N) with the property:∣∣∣∣ 1gk log ζKk
(1
2
+ θ
)
− log ζ{Kk}
(1
2
+ θ
)∣∣∣∣ < 12N
for any θ ∈ [θ(N+1), θ(N)] and any k ≥ k′(N). This is possible by theorem
2. Then we choose k′′(N) such that
−rk log θ(N + 1)
gk
≤ θ(N)
N
for any k ≥ k′′(N), which can be done thanks to the following proposition
(c.f. [3, Proposition 5.34]):
Proposition 3. Assume that GRH holds for ζK(s). Then
ord
s= 12
ζK(s) <
C log 3|DK |
log log 3|DK |
the constant C being absolute.
Finally, we choose an increasing sequence k(N) such that k(N) ≥
max(k′(N), k′′(N)) for any N.
Now, if we define N = N(k) by the inequality k(N) ≤ k ≤ k(N + 1) and
let θk = θ(N(k)), then from the conditions imposed on θk we automatically
get (1) and (2). The delicate point is (3). We will use Hadamard’s product
formula [8, p. 137]:
log |DK | = ΦR(K)(log π − ψ(s/2)) + 2ΦC(K)(log(2π)− ψ(s))
− 2
s
− 2
s− 1
+ 2
∑
ρ
′ 1
s− ρ
− 2 ζ
′
K(s)
ζK(s)
,
Properties of Dedekind zeta functions 777
where ψ(s) = Γ′(s)/Γ(s) is the logarithmic derivative of the gamma func-
tion. In the first sum ρ runs over the zeroes of ζK(s) in the critical strip and∑′ means that ρ and ρ̄ are to be grouped together. This can be rewritten
as
1
gk
(
log ζk
(1
2
+ θ
)
− rk log θ
)′
= −1 + ΦR(Kk)
2gk
(
log π − ψ
(1
4
+ θ
2
))
+
+ ΦC(Kk)
gk
(
log 2π − ψ
(1
2
+ θ
))
+ 8θ
(1− 4θ2)gk
+
∑
ρ 6=1/2
′ 1
(1/2 + θ − ρ)gk
.
(the term rk log θ comes from the contribution of zeroes at s = 12). One
notices that all the terms on the right hand side except for −1 and 8θ(1−4θ2)gk
are positive. Thus, we see that 1gk
(
logFk
(
1
2 + θ
))′
≥ C for any small
enough θ, where C is an absolute constant. From this and from the fact
that Fk
(
1
2
)
= 1 we deduce that
1
gk
logFk
(1
2
+ θk
)
≥ Cθk → 0.
This proves (3) as well as the theorem. 
Proof of the corollary 1. It suffices to take the values at s = 1 of the deriva-
tives of both sides of the equality in theorem 2. This is possible since the
convergence is uniform for Re s > 12 . 
Acknowledgements. I would like to thank my advisor Michael Tsfasman
for many fruitful discussions and constant attention to my work. I would
also like to thank Michel Balazard for giving valuable advices and sharing
his ideas with me.
References
[1] R. Brauer, On zeta-functions of algebraic number fields. Amer. J. Math. 69 (1947), Num.
2, 243–250.
[2] Y. Ihara, On the Euler–Kronecker constants of global fields and primes with small norms.
Algebraic geometry and number theory, 407–451, Progr. Math., 253, Birkhaüser Boston,
Boston, MA, 2006.
[3] H. Iwaniec, E. Kowalski, Analytic number theory. American Mathematical Society Collo-
quium Publications, 53. AMS, Providence, RI, 2004.
[4] H. Iwaniec, W. Luo, P. Sarnak, Low lying zeros of families of L-functions. Inst. Hautes
Études Sci. Publ. Math., Num. 91 (2000), 55–131.
[5] H. Iwaniec, P. Sarnak, Dirichlet L-functions at the central point. Number theory in
progress, Vol. 2 (Zakopane-Koscielisko, 1997), 941–952, de Gruyter, Berlin, 1999.
[6] H. Iwaniec, P. Sarnak, The nonvanishing of central values of automorphic L-functions
and Siegel’s zero. Israel J. Math. A 120 (2000), 155–177.
[7] S. Lang, Algebraic number theory. 2nd ed. Graduate Texts in Mathematics 110, Springer-
Verlag, New York, 1994.
[8] H. M. Stark, Some effective cases of the Brauer-Siegel Theorem. Invent. Math. 23(1974),
pp. 135–152.
778 Alexey Zykin
[9] E. C. Titchmarsh, The theory of functions. 2nd ed. London: Oxford University Press. X,
1975.
[10] M. A. Tsfasman, Asymptotic behaviour of the Euler-Kronecker constant. Algebraic geome-
try and number theory, 453–458, Progr. Math., 253, Birkhaüser Boston, Boston, MA, 2006.
[11] M. A. Tsfasman, S. G. Vlăduţ, Infinite global fields and the generalized Brauer–Siegel
Theorem. Moscow Mathematical Journal, Vol. 2(2002), Num. 2, 329–402.
[12] A. Zykin, Brauer–Siegel and Tsfasman–Vlăduţ theorems for almost normal extensions of
global fields. Moscow Mathematical Journal, Vol. 5 (2005), Num 4, 961–968.
[13] A. Zykin, Asymptotic properties of zeta functions over finite fields. Preprint.
Alexey Zykin
State University — Higher School of Economics,
7, Vavilova st.
117312, Moscow, Russia
E-mail: alzykin@gmail.com
URL: http://www.mccme.ru/lifr/pers/zykin/zykin.html


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