II. MARCO TEÓRICO
2.2. BASES TEÓRICAS CIENTÍFICAS
2.2.1. LA INTELIGENCIA EMOCIONAL:
We have reduced our problem to an analytic one: we want to understand the asymptotic behavior of κW,m,ts,p(c, X) as X → ∞. We shall prove now, that, for
certain counting function W, it does not depend on the class c.
As we shall immediately see, the proof of this fact will be a direct consequence of the results contained in [1, Sections 4-8]. So, without repeating arguments which are already in [1, Sections 4-8], we shall explain the main steps which lead to the analytic investigation of the behavior of κW,m,ts,p(c, X) as X → ∞, refering to [1]
when the proofs given by Agboola apply to our case too.
Given a cosetc of (KΛ)×·Um0(Λ)in J(KΛ), let us consider the function of z ∈C
Dc,m,ts,p(z) := X a∈co(Fp∩c) (a,m)=1 [Λ :dW(a)]−z. Using Cl\0
m(Λ), the group of characters of Cl
0 m(Λ), we can write Dc,m,ts,p(z) = 1 |Cl0m(Λ)| X χ∈Cl\0m(Λ) ˜ χ(c)Dts,p(z, χ),
whereDts,p(z, χ) :=Pa∈Fpχ(a)[Λ :dW(a)]−z and χ is considered as a map on the
set of integrals ideals ofΛ, by establishingχ(a) = 0ifa1 6=OK (wherea= (at)t∈T)
Since, in our case, we are considering m divisible by a sufficiently high power of
|G| and of p, we understand that, by the description ofFp given above, we have
{a∈ co(Fp∩c),(a,m) = 1}={a∈ co(F ∩c), (a,m) = 1}.
Thus, with our choice of m, the series Dc,m,ts,p(z) previously defined, is equal to
the series Dc,m(z) := X a∈co(F∩c) (a,m)=1 [Λ :dW(a)]−z
considered by Agboola in his paper (see [1, Definition 4.1]). This is the reason why the analytic part remains exactly the same of [1]: we are considering the same series studied by Agboola with just an extra condition on the primem.
Proposition 5.2.8. The series Dc,m,ts,p(z) is convergent in some right hand half-
plane.
Proof. From our discussion above, this result follows from the analogous one
proved in [1, Section 4] for the series Dc,m(z). See in particular [1, Proposition
4.5] where, through the use of Euler product expansions, Agboola showed that
Dc,m(z) converges in some right hand half-plane.
From its definition we know thatDc,m,ts,p(z)is a Dirichlet series
P∞
n=0ann−z, where
the an are non-negative coefficients, and it is moreover clear that
κW,m,ts,p(c, X) =
X
n≤X
an.
So we have reduced our problem to study the asymptotic behavior of the sum of the first X coefficients of a convergent Dirichlet series. A classical theorem which helps at this point is the Délange–Ikehara Tauberian Theorem, which is recalled below.
Theorem 5.2.9 (Délange–Ikehara Tauberian Theorem). Let P∞ n=1ann
−z be a Dirichlet series with non-negative coefficients which is convergent on the half-plane
Re(z)> a >0. Assume that in its domain of convergence
∞
X
n=1
where w >0andg(z) andh(z)are holomorphic functions on the closed half-plane
on the right of a, such that g(a)6= 0. Then
X n≤X an ∼ g(a) a·Γ(w) ·X a·(log(X))w−1,
asX → ∞. At the denominatorΓ(w)is the value of the classical Gamma function
at w.
In order to apply Theorem 5.2.9 to our case, first of all we need to find the most right pole β(c,m) of Dc,m,ts,p(z) (which corresponds to Dc,m(z) in [1]), its order
δ(c,m) and the “residue” atβ(c,m) given by
τ(c,m) := lim
z→β(c,m)(z−β(c,m)) δ(c,m)D
c,m,ts,p(z).
Then, using Theorem 5.2.9, we shall get
κW,m,ts,p(c, X)∼
τ(c,m)
β(c,m)·Γ(δ(c,m)) ·X
β(c,m)·
(log(X))δ(c,m)−1. (5.11) Hence, from this, if we are able to show the independence of β(c,m), τ(c,m)
and δ(c,m) from c, we will deduce the independence of κW,m,ts,p(c, X) from c, as
X → ∞.
LetI(Λ)denote the group of fractional ideals ofΛ. The valuesβ(c,m),τ(c,m)and
δ(c,m) can be found, as already done in [1, Sections 6-7], through a comparison of Dc,m,ts,p(z) with the Dirichlet L-series associated to Λ
LΛ(z, χ) := X a∈I(Λ) a⊆Λ χ(a)[Λ :dW(a)]−z. In particular, if we define bαW(χ) := lim z→ 1 αW z− 1 αW δ(c,m) Dts,p(z, χ),
from [1, Section 7] (be careful that in Proposition 7.1, the assumption p - m is superfluous), we get: β(c,m) = 1 αW δ(c,m) = t∈ T \ {1} s.t. W(t) =αW τ(c,m) = 1 |Cl0m(Λ)| X χ∈Cl\0m(Λ) ˜ χ(c)bαW(χ).
Finally, we have the following result.
Proposition 5.2.10. Let κW,m,ts,p(c, X) be defined as in (5.9). Then
κW,m,ts,p(c, X) is asymptotically independent from c⇐⇒bαW(χ) = 0, ∀χ6= 1.
Moreover, in this case we have
lim z→ 1 αW z− 1 αW dαW(1) Dc,m,ts,p(z) = bαW(1)/|Cl 0 m(Λ))|.
Proof. For a proof of it see [1, Lemma 7.5 - Proposition 7.6].
As explained in [1], the easiest case when bαW(χ) = 0, ∀χ 6= 1, is when W is
chosen to be constant on T\{1}. While in the non-constant case we can lose the asymptotic independence as explained in [1, Proposition 7.8].
An example, as we have seen, of constant W is given by Wram. In this case we
haveαWram = 1 and so
κWram,m,ts,p(c, X)∼
bαW
ram(1)
Γ(|T \ {1}|)· |Cl0m(Λ))| ·X·(log(X))
|T\{1}|−1,
as X → ∞, and hence, from (5.10),
NWram,m,ts,p(c, X)∼ K · |Ker(qm)| · bαW ram(1) Γ(|T \ {1}|)· |Cl0m(Λ))| ·X·(log(X)) |T\{1}|−1, as X → ∞.
Thus the analog of [1, Theorem B] now clearly follows (this is a more precise version of Theorem 0.0.9 in the Introduction).
Theorem 5.2.11. Letcbe a given class in Rts,p(OK[G]). The asymptotic behavior
of NWram,m,ts,p(c, X), as X → ∞, does not depend on c. Moreover Pr
0
Wram,m,ts,p(c)
exists and does not depend on c.
Proof. Clear from the previous asymptotic approximation and from the definition
of Pr0Wram,m,ts,p(c).
Remark 5.2.12. Analogously to [1, Section 9, Proposition 9.5], one can prove that
in our definition of probability, if instead of considering G-Galois K-algebras we
restrict our attention to just field extensions, the result of Theorem 5.2.11 remains the same.
Corollary 5.2.13. The limit Pr((1,1),p, c), defined in the introduction of this
chapter (see (5.2)), exists and does not depend on the given realizable class c ∈
Rts,p(OK[G]).
Proof. Since DWram(L/K) coincides with D(L/K) (see at the beginning of the
chapter), taking m := (|G| · p)mO
K (where m is the sufficiently high natural
number given by Theorem 5.1.10 and Proposition 2.2.9), and thanks to Corollary 5.2.13, we see that Pr0Wram,m,ts,p(c)coincides with Pr((1,1),p, c).