Random characters under the $L$-measure, I : Dirichlet characters

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by  Yacine Barhoumi-Andréani

RANDOM CHARACTERS UNDER THE L-MEASURE, I :
DIRICHLET CHARACTERS
YACINE BARHOUMI-ANDRÉANI
Abstract. We define the L-measure on the set of Dirichlet characters as an analogue of
7
1 the Plancherel measure, once considered as a measure on the irreducible characters of the
0 symmetric group.
2 We compare the two measures and study the limit in distribution of characters eval-

n uations when the size of the underlying group grows. These evaluations are proven to
a converge in law to imaginary exponentials of a Cauchy distribution in the same way as
J
the rescaled windings of the complex Brownian motion. This contrasts with the case of

6
thesymmetricgroupwheretherenormalisedcharactersconvergeinlawtoGaussiansafter
1
rescaling (Kerov Central Limit Theorem).

]
R
P
.
h Contents
t
a
m 1. Introduction 2
[ 1.1. Asymptotic representation theory of (Z/qZ)× 2

1.2. Motivations 4
1
v 1.3. Natural analogues of the Plancherel measure on other groups 5
8
1.4. Plan 7
5
2 1.5. Notations 7
4 2. From S(cid:98) to G(cid:98) 8
0 n q
2.1. Reminders on the irreducible characters of S 8
. n
1
2.2. Moments of characters under the Plancherel measure 9
0
7 2.3. Reminders on Dirichlet characters 12
1 2.4. Moments of characters under the L-measure 13
:
v 3. Fluctuations when q → +∞ 16
i
X 3.1. Evaluations 16
r 3.2. Joint evaluations 18
a
3.3. Speed of convergence in the Wasserstein metric 19
3.4. The case s = ∞ : the uniform measure 21
4. Another type of L-measure 24
5. Conclusion and perspectives 25
Acknowledgements 26
References 26
Date: January 17, 2017.
2010 Mathematics Subject Classification. 60B15, 11K99, 11N64, 60F05.
1
2 YACINE BARHOUMI-ANDRÉANI
1. Introduction
1.1. Asymptotic representation theory of (Z/qZ)×. For q ∈ N∗, set
G := (Z/qZ)× (1)
q
The goal of this article is to study the properties of characters of G when q → +∞ and
q
when the characters are selected at random according to the L-measure that we introduce
in definition 1.1.
This article focuses on the case of Dirichlet characters modulo q that are multiplicative
group morphisms χ : G → C× extended to Z/qZ by setting χ(n) = 0 if n ∈ (Z/qZ) \
q
(Z/qZ)× and finally extended by periodicity to Z by setting χ(n) := χ(n mod q) (see e.g.
[15, § 5]). These maps were used by Dirichlet to prove his celebrated theorem on the
infinitude of primes in arithmetic progressions.
Dirichlet characters have the following properties :
(1) χ is periodic modulo q : χ(n+q) = χ(n) for all m,n ∈ N,
(2) χ is completely multiplicative : χ(mn) = χ(m)χ(n) for all m,n ∈ N,
(3) χ(n) (cid:54)= 0 if and only if gcd(n,q) = 1.
Note that these properties imply that χ(1) = 1, as χ(1) = χ(1×1) = χ(1)2 and χ(1) (cid:54)= 0
since gcd(1,q) = 1.
We define G(cid:98) to be the set of Dirichlet characters modulo q. Apart from the value 0,
q
these characters take values in the ϕ(q)-roots of unity (cid:8)e2iπk/ϕ(q),k (cid:62) 0(cid:9), where ϕ is Euler’s
totient function. There are exactly ϕ(q) such characters.
The L-function attached to a Dirichlet character χ is the following function defined for
all s ∈ {Re > 1} by (see e.g. [28])
(cid:88) χ(n)
L (χ) := (2)
s ns
n(cid:62)1
We consider these functions as linear forms in the character χ, hence the choice of
notation compared with the usual one that writes L(s,χ).
Definition 1.1. We define the L-measure on G(cid:98) by
q
|L (χ)|2 (cid:88)
P (χ) := s , Z := |L (η)|2, s ∈ {Re > 1} (3)
s,q s,q s
Z
s,q
η∈G(cid:98)q
This measure can be written in the following way using the infamous Euler formula (12) :
P (χ) := 1 exp(−2H (χ)), H (χ) := (cid:88)log(cid:12)(cid:12)1−p−sχ(p)(cid:12)(cid:12) (4)
s,q s s
Z
s,q
p∈P
where P is the set of prime numbers.
We can thus interpret (3) in the setting of statistical mechanics as a system of particles
on a circle (whose positions are given by the angles of the character evaluated in prime
numbers) submitted to a particular logarithmic confinement potential at inverse tempera-
ture 2.
RANDOM DIRICHLET CHARACTERS 3
Let χ ≡ χ(q) denote the canonical evaluation on G(cid:98) in the Dynkin formalism, namely
q
χ (η) = η(k), ∀η ∈ G(cid:98) (5)
k q
We will be interested in the behaviour of the random variables χ for a fixed integer k.
k
The main theorem of this paper states
Theorem 1.2 (Convergence in law of the evaluations). Let k ∈ 2,q be a fixed integer
and s ∈ (1,+∞). Then, under P , the following convergence in la(cid:74)w is(cid:75)satisfied
s,q
L
χ −−−−→ eislog(k)C
k
q→+∞
where C is a standard Cauchy-distributed random variable of density x (cid:55)→ 1 1 .
π1+x2
This theorem is proven in section 3.1. An immediate striking comparison with the
winding number of the complex Brownian motion can be made : write Z := 1+W +iW(cid:48) =
t t t
R eiΘt with (W,W(cid:48)) two independent real Brownian motions. Here, (Θ ) is a continuous
t t t
determination of the argument of Z around 0. Then, one has (see [27] or [24, X-4.1], [19,
ch. 7])
Θ L
√t −−−−→ C (6)
log( t) t→+∞
Spitzer proved this last convergence using an explicit computation of the Fourier trans-
form of Θ (see [24, X-4.1] for another proof). We will prove this convergence in the same
t
vein, with a direct computation of the Laplace transform of χ .
k
The similarity stops nevertheless here : the windings of (Z ) around several points
t t
z ,...,z ofthecomplexplanedonotconvergeinlawafterrescalingtoindependentCauchy
1 k
random variables. A precise description of the limiting joint distribution is given in [24,
ch. XIII cor. 3.9] and involves a mixing by a random variable that creates a dependency
in the limiting angles. In the case of evaluations of random Dirichlet characters, one has
the following
Theorem 1.3 (Fluctuationsofjointevaluations). Let s ∈ (1,+∞). Let (cid:96) (cid:62) 1 and (p )
j 1(cid:54)j(cid:54)(cid:96)
be fixed prime numbers. Then, under P , we have the following convergence in distribution
s,q
L (cid:0) (cid:1)
(χ ,...,χ ) −−−−→ eislog(p1)C1,...,eislog(p(cid:96))C(cid:96)
p1 p(cid:96) q→+∞
where (C ,...,C ) are independent Cauchy-distributed random variables.
1 (cid:96)
It is enough to consider the evaluations in prime numbers (χ ) by multiplicativity
p p∈P
of characters. More general evaluations will be dependent, provided that the numbers in
which they are evaluated are not coprime.
4 YACINE BARHOUMI-ANDRÉANI
1.2. Motivations. Let n ∈ N. The Plancherel measure on the set Y of Young diagrams
n
with n boxes is defined by (see e.g. [17, 21])
d2
P(cid:96) (λ) := λ
n
n!
Here, d denotes the dimension of the irreducible S -module indexed by the Young
λ n
diagram λ ∈ Y and S is the symmetric group ; this quantity has an explicit expression
n n
in terms of λ, but it can also be written in terms of characters of S as its irreducible
n
charactersareinbijectionwithYoungdiagrams(see[18, I-7]). Forλ ∈ Y , letχλ : S → Z
n n
be the associated irreducible character. A classical formula states (see [18, ch. I-7 (7.6),
ch. I-6 ex. 2 (a)])
d = χλ(id )
λ n
whereid ∈ S istheidentitypermutation. ThislastformulaallowstowritethePlancherel
n n
measure as a measure on characters by setting
χ(id )2
n
P(cid:96) (χ) := (7)
n
n!
Character evaluations form a natural set of observables for this measure. Using the
Dynkin formalism, we define the canonical evaluation χ on S(cid:98) by
n
χ (η) := η(σ), ∀η ∈ S(cid:98) , ∀σ ∈ S
σ n n
A natural question concerns the behaviour of χ under P(cid:96) for a certain σ ∈ S and
σ n n
n → +∞; suchtypesofquestionsgavebirthtothefieldofasymptotic representation theory
[17]. In the case where σ is a k-cycle c for a fixed integer k, one has the fondamental
k
result due to Kerov [14] independently discovered by Hora [13] :
Theorem 1.4 (Gaussian fluctuations of character evaluations (Kerov CLT)). Let k (cid:62) 2
be a fixed integer and c a k-cycle. Then, under P(cid:96) , one has the following convergence in
k n
law
1 χ L √
ck −−−−→ kG G ∼ N (0,1)
nk/2χ k k
idn n→+∞
Moreover, different evaluations (n−k/2χ /χ ) for a fixed integer r converge to
ck idn 2(cid:54)k(cid:54)r
independent Gaussian random variables.
This result concerns the Gaussian fluctuations of the renormalised character, namely
χ/χ(id ). It has been extended in various ways, by computing the speed of convergence
n
in the total variation distance (see [11]), by letting k depend on n in the cycle c (for
√ k
k = O( n), see[26]andreferencescited), bychangingthecyclec withaproductofcycles,
k
in which case the limiting distribution changes radically (see [13, 14]) or by changing the
measure (see e.g. [10, 16, 8, 1, 20, 5] and references cited).
This article is in the lineage of these results ; it aims at giving another direction of
generalisation of this last analysis by replacing the group S with the group (Z/qZ)×.
n
The analogue of the Plancherel measure on (Z/qZ)× will be the uniform measure, which is
RANDOM DIRICHLET CHARACTERS 5
a limiting case of L-measure that we will study separately in section 3.4. The case of the L-
measure corresponds to the aforementioned other types of measures, particular cases of the
Schur measure that we describe in the next section. In this framework, theorems 1.2 and
1.3 are clear analogues of the Kerov CLT, the equivalence betwen disjoint cycles and prime
numbersbeingmadeinaccordancetotheusualanalogiesbetweenpermutationsandprimes
(see e.g. [12] or [2, p. 22]). These theorems have nevertheless some disparities : random
Dirichlet characters need not be renormalised, and the limiting random variables involve
the Cauchy distribution (the stable law of index 1) and not the Gaussian one (the stable
law of index 2).
1.3. Natural analogues of the Plancherel measure on other groups. Thefirstpoint
to investigate when trying to generalise theorem 1.4 to other groups concerns the natural
analogue of the Plancherel measure (7). Given its form, the Plancherel measure on G(cid:98) is
q
χ(1)2 1
P(χ) := = (8)
(cid:80)
χ(1)2 ϕ(q)
η∈G(cid:98)q
as χ(1) = 1. This is thus the uniform measure on G(cid:98) . This measure corresponds to a
q
limiting type of L-measure, where we have set s → +∞ in (3) (see section 3.4). For this
degenerate type of L-measure, the limiting behaviour of the characters evaluations is given
by the following theorem proven in section 3.4 :
Theorem 1.5 (Limit in law of the evaluations under P ). The following convergence in
∞,q
law is satisfied for all fixed integer k ∈ 2,q
(cid:74) (cid:75)
L
χ −−−−→ ei2πU, U ∼ U([0,1])
k
q→+∞
Moreover, (χ ) converges in law to a vector of independent random variables.
p p∈P
More generally, the Plancherel measure and a wide class of measures studied in [10, 16,
8, 1, 20, 5] are particular specialisations of the following Schur measure restricted to S(cid:98) :
n
s (X)s (Y) (cid:88)
M(n) (λ) := λ λ 1 , h [XY] := s (X)s (Y) (9)
X,Y h [XY] {λ∈Yn} n λ λ
n λ∈Yn
Here, s (X) is the Schur function [18, ch. 1-3] evaluated in an alphabet X and the
λ
normalisation constant h [XY] is a specialisation of the complete homogeneous functions
n
[18, ch. 1-2]. The prominent Schur measure on the set of all Young diagrams introduced
by Okounkov in [22] is an independent randomisation1 of (9).
1ByarandomvariableB satisfyingP(B =n)=h [XY]/H[XY]withH[XY]:=(cid:80) h [XY].
X,Y X,Y n n(cid:62)0 n
The Schur measure writes M (λ):=(cid:80) P(B =n)M(n) (λ)=s (X)s (Y)/H[XY].
X,Y n(cid:62)0 X,Y X,Y λ λ
6 YACINE BARHOUMI-ANDRÉANI
The Schur measure writes as a measure on S(cid:98) using the Frobenius characteristic (see
n
[18, I-7])
1 (cid:88)
s (X) = Ch (χλ) := χλ(σ)p (X)
λ X ct(σ)
n!
σ∈Sn
where p (X) are the power functions, a particular type of symmetric functions (see
ct(σ)
[18, ch. 1-2]) indexed by the cycle-type of a permutation. Write σ = σ ...σ where
1 C(σ)
(σ ) are the disjoint cycles of σ and C(σ) is the total number of cycles. Then,
k 1(cid:54)k(cid:54)C(σ)
the power functions can be expressed in terms of a family (p (X)) associated with the
(cid:96) (cid:96)(cid:62)1
lenght of a cycle
C(σ)
(cid:89) (cid:89)
p (X) = p (X) = p (X)m(cid:96)(σ)
ct(σ) |σ | (cid:96)
k
k=1 (cid:96)(cid:62)1
with m (σ) := (cid:80) 1 and |σ | is the size of the cyle σ (the number of its elements).
(cid:96) k(cid:62)1 {|σk|=(cid:96)} k k
We can thus write
1 (cid:88) (cid:89)
Ch (χ) = χ(σ) p (X)m(cid:96)(σ)
X (cid:96)
n!
σ∈Sn k(cid:62)1
in the same way as
(cid:88) (cid:89)
L (χ) = χ(n) p−svp(n)
s
n∈N∗ p∈P
The point of importance is to notice that one can write the measure (9) as a measure
on S(cid:98) by writing, in the same vein as for the Plancherel measure
n
Ch (χ)Ch (χ)
M(n) (χ) = X Y
X,Y h [XY]
n
The Frobenius characteristic Ch is a linear form on S(cid:98) and so is the evaluation χ ;
X n idn
the common points in all the precedent measures is thus (1) their writing as a product of
two linear forms and (2) the coefficients of each linear form that write as a product over
the considered structures (cycles or primes).
If one replaces the characters of S by characters of another finite group G whose
n
elementshaveanaturalnotionofdecompositionintoelementarystructures(primes, cycles,
etc.), a natural analogue of these measures can hence be defined by means of a product of
two real linear forms on G(cid:98) of the form
L (χ)L (χ) (cid:88)
P (χ) := 1 2 , Z(G(cid:98)) := L (χ)L (χ)
L1,L2 Z(G(cid:98)) 1 2
χ∈G(cid:98) (10)
(cid:88) (cid:89)
L (χ) ≡ L (χ) := χ(g) zm(a), i ∈ {1,2}, z :=(z )
i z a a a
g∈G a∈C(g)
where C(g) is a particular set characteristic of a certain decomposition of g ∈ G and the
z ’s are real numbers.
a
RANDOM DIRICHLET CHARACTERS 7
The measure (3) is clearly a particular case of (10), but it has only a one-dimensional
free parameter s ; from this perspective, it is more an analogue of the z-measure [8] or the
q-Plancherel measure [10, 16] than the Schur measure itself. We will come back on this
last point in section 4.
1.4. Plan. The plan of this article is the following : after some reminders on characters
of both S and G , we start by comparing random characters under the aforementioned
n q
measures and show that they share a similarity of structure ; their evaluations can be
writtenbyoperatorsofdilationinbothcases, actingondifferentspacesofsquareintegrable
functions. We then study the asymptotic properties of χ under P and prove theorems
k s,q
1.2 and 1.3. We moreover give the speed of this convergence in the Wasserstein distance
(lemma 3.2). We treat in section 3.4 the case of the uniform measure (8) that corresponds
to the limiting case of the L-measure P . Last, we treat in section 4 the case of the
∞,q
analogue of the Schur measure on G(cid:98) . We conclude with several questions of interest,
q
extensions and future developments.
This article is the first of a series that investigates L-measures on sets of more general
characters such as Hecke characters and automorphic representations of GL(n).
1.5. Notations. We write a◦|b for “a divides b” and b|◦a for “b multiple of a”. We will
make a constant use of the multi-index notation xλ := (cid:81) xλk if x = (x ,x ,...) and
k(cid:62)1 k 1 2
(cid:81)
λ = (λ ,λ ,...). Wealsosetλ! := λ !forthemulti-indexfactorial. Last, weset|λ| =
1 2 k(cid:62)1 k
(cid:80) λ for a multi-index. The raising factorial is defined as x↑n := x(x+1)···(x+n−1)
k(cid:62)1 k
for x ∈ C and n (cid:62) 1.
We will note λ (cid:96) n for λ ∈ Y . The Vandermonde determinant is denoted by a (x) :=
n δ
a (x ,...,x ) if δ ≡ δ = (n−1,n−2,...,1,0) and defined by
δ 1 n n
(cid:89)
a (x) := (x −x )
δ i j
1(cid:54)i<j(cid:54)n
(cid:2) (cid:3) (cid:2) (cid:3)
ForaLaurentseriesf innvariables, wewrite xλ (f)thecoefficientofxλ, i.e. xλ (f) =
(cid:82) xλf(x)dm(x) with dm(x) := (cid:81) dxk and U the unit circle if this last integral con-
Un k(cid:62)1 2iπx
k
verges.
The Euler’s totient function is denoted by ϕ and defined for all n (cid:62) 1 by
(cid:18) (cid:19)
(cid:89) 1
ϕ(n) := n 1−
p
p◦|n
Here, the product is on the set of prime numbers P. Every integer n ∈ N∗ decomposes
into a product of primes according to
(cid:89)
n = pvp(n)
p∈P
8 YACINE BARHOUMI-ANDRÉANI
where v (n) is the p-adic valuation of n. For a sequence of complex numbers (a ) such
p p p∈P
that |a | (cid:54) 1, one has the formula, for all s ∈ {Re > 1}
p
(cid:89)(cid:0)1−p−sa (cid:1)−1 = (cid:88) 1 (cid:89)avp(n) (11)
p ns p
p∈P n(cid:62)1 p∈P
In particular, for a character χ ∈ G(cid:98) , using the prime factor decomposition and the
q
morphism property of χ, we get the well known Euler formula
(cid:89)(cid:0)1−p−sχ(p)(cid:1)−1 = (cid:88) 1 (cid:89)χ(p)vp(n) = (cid:88) 1 χ(n) (12)
ns ns
p∈P n(cid:62)1 p∈P n(cid:62)1
Note that as χ(p) = 0 if p◦|q, the product on the LHS of the last formula is in fact on
the set of primes that do not divide q.
Last, we set 1(n) := 1 and we define the multiplicative convolution ∗ of two se-
{n(cid:62)1}
quences (a ) and (b ) by
n n(cid:62)1 n n(cid:62)1
(cid:88) (cid:88) (cid:88)
a∗b(n) := a b 1 = a b = a b
k (cid:96) {k(cid:96)=n} (cid:96) n/(cid:96) n/(cid:96) (cid:96)
k,(cid:96)(cid:62)1 (cid:96)◦|n (cid:96)◦|n
In particular, we note a∗k(n) := a ∗ ··· ∗ a(n) (k times). We moreover recall that
Dirichlet series have the following morphism property for the multiplicative convolution,
for z ∈ {Re > 1} and provided that the following series converge
(cid:88) a∗b(n) (cid:88) a(n) (cid:88) b(m)
= (13)
nz nz mz
n(cid:62)1 n(cid:62)1 m(cid:62)1
2. From S(cid:98) to G(cid:98)
n q
2.1. Reminders on the irreducible characters of S . Let χλ ∈ S(cid:98) be the irreducible
n n
character associated to λ (cid:96) n. For σ ∈ S , the value of χλ(σ) only depends on the
n
cycle type µ := ct(σ) (cid:96) n. We denote this value by χλ. The canonical evaluation on
µ
the probability space (Y ,P(cid:96) ) thus becomes χ : λ (cid:55)→ χλ . A classical formula for the
n n µ µ
evaluation of χλ is given by [18, ch. I-7 (7.7)]
µ
χλ = (cid:104)s ,p (cid:105) (14)
µ λ µ
Here, p is the power function, and the scalar product can be realised as the classical L2
µ
scalar product for the Haar measure of the unitary group. One has also the following
Lemma 2.1 ([18], (7.8) p. 114). Write(cid:2)xλ(cid:3)(f) for the coefficient of xλ := (cid:81) xλk in the
k(cid:62)1 k
Laurent series f in n variables, i.e. (cid:2)xλ(cid:3)(f) = (cid:82) xλf(x)dm(x) with dm(x) := (cid:81) dxk
Un k(cid:62)1 2iπx
and U the unit circle. Then k
(cid:90)
(cid:2) (cid:3)
χλ = xλ+δ (p a ) = xλ+δp (x)a (x)dm(x)
µ µ δ µ δ
Un
For the reader’s benefit, we remind the proof of this result.
RANDOM DIRICHLET CHARACTERS 9
Proof. Using equation (14) and the fact that (s ) is an orthonormal basis, one has
λ λ
(cid:88) (cid:88)
p = (cid:104)s ,p (cid:105)s = χλs
µ λ µ λ µ λ
λ(cid:96)n λ(cid:96)n
One has moreover (see e.g. [18]) s = a /a , hence p a = (cid:80) χλa and in
λ λ+δ δ µ δ λ(cid:96)n µ λ+δ
particular,
(cid:2) (cid:3) (cid:88) (cid:2) (cid:3)
xλ+δ (p a ) = χν xλ+δ (a )
µ δ µ ν+δ
ν(cid:96)n
It thus remains to prove that(cid:2)xλ+δ(cid:3)(a ) = 1 . But one has
ν+δ {λ=ν}
(cid:90)
(cid:2) (cid:3)
xλ+δ (a ) = xλ+δa (x)dm(x)
ν+δ ν+δ
Un
(cid:90)
(cid:88)
= ε(σ) xλ+δxσ(ν+δ)dm(x)
Un
σ∈Sn
(cid:88)
= ε(σ)1
{λ+δ=σ(ν+δ)}
σ∈Sn
= 1
{λ=ν}
since the only permutation allowing to have λ+δ = σ(ν+δ) is the identity, the partitions
λ+δ and ν +δ having strictly increasing parts. (cid:3)
2.2. Moments of characters under the Plancherel measure.
Lemma 2.2 (Structure of character evaluations). Let dm(z) := (cid:81) dzk . For f,g ∈
k(cid:62)1 2iπz
k
L2(U,m) ≡ L2(m), define the scalar product
(cid:90) (cid:90)
(cid:0) (cid:1)
(cid:104)f,g(cid:105)L2(Un) := f(z)g(z)dm(z) = f e2iπθ1,...,e2iπθn g(e2iπθ1,...,e2iπθn)dθ1...dθn
Un [0,1]n
For all µ (cid:96) n and x ∈ U, set
f (x) := p (x)a (x)
µ µ δ
We define the operator of anti-dilatation by
δ(cid:101)g(u) := g(x u ,x u ,...,x u )
x 1 1 2 2 n n
and the following operator
(cid:90) (cid:90)
L := f (x)δ(cid:101) dm(x) : g ∈ L2(m) (cid:55)→ f (x)g(x(cid:12)·)dm(x) ∈ L2(m)
µ µ x µ
Un Un
Using the fact that f ∈ L2(m) (as f is a polynomial) and the Cauchy-Schwarz inequal-
µ µ
ity, it is easily proven that L g ∈ L2(m) if g ∈ L2(m).
µ
10 YACINE BARHOUMI-ANDRÉANI
Last, set
(cid:88)
R (x) := xλ
n
λ(cid:96)n
h (x) := L R
n 1n n
Then, we have
(cid:104)Lkh ,L(cid:96)h (cid:105)
E(n)(cid:0)χkχ (cid:96)(cid:1) = µ n ν n L2(Un) (15)
P(cid:96) µ ν (cid:104)h ,h (cid:105)
n n L2(Un)
Proof. For k (cid:62) 1 and µ (cid:96) n, one has
E(n)(cid:16)|χ |2k(cid:17) = 1 (cid:88)d2(cid:0)χλ(cid:1)2k = 1 (cid:88)(cid:104)s ,p (cid:105)2(cid:104)s ,p (cid:105)2k
P(cid:96) µ n! λ µ n! λ 1n λ µ
λ(cid:96)n λ(cid:96)n
(cid:90) (cid:90)
1 (cid:88)
= xλ+δp (x)a (x)dm(x) yλ+δp (y)a (y)dm(y)×
1n δ 1n δ
n!
Un Un
λ(cid:96)n
k+1(cid:90) (cid:90)
(cid:89)
xλ+δp (x )a (x )dm(x ) yλ+δp (y )a (y )dm(y )
(cid:96) µ (cid:96) δ (cid:96) (cid:96) (cid:96) µ (cid:96) δ (cid:96) (cid:96)
Un Un
(cid:96)=2
(cid:32) (cid:33)
(cid:90) k+1
1 (cid:88)(cid:89)
= xλ+δyλ+δ p (x )a (x )p (y )a (y )×
n! (cid:96) (cid:96) 1n 1 δ 1 1n 1 δ 1
(Un)2(k+1)
λ(cid:96)n (cid:96)=1
k+1 k+1
(cid:89) (cid:89)
p (x )a (x )p (y )a (y ) dm(x )dm(y )
µ (cid:96) δ (cid:96) µ (cid:96) δ (cid:96) (cid:96) (cid:96)
(cid:96)=2 (cid:96)=1
Denote x := (x ,...,x ) and y := (y ,...,y ). Last, set
1 k+1 1 k+1
(cid:88)(cid:89) (cid:89)
K (x,y) := xλ+δ yλ+δ (16)
n (cid:96) (cid:96)
λ(cid:96)n (cid:96)(cid:62)1 (cid:96)(cid:62)1
In this last formula, the variables can have a different size. For instance, one can define
(cid:32) (cid:33)
(cid:88) (cid:89)
K (x,y) := xλ+δ yλ+δ
n (cid:96)
λ(cid:96)n (cid:96)(cid:62)1
Then, one has
(cid:90)
(cid:16) (cid:17) 1
E(n) |χ |2k = K (x,y) f ⊗f⊗k(x)·f ⊗f⊗k(y) dm⊗k+1(x)dm⊗k+1(y)
P(cid:96) µ n! n 1n µ 1n µ
(Un)2(k+1)
Using the formula
(cid:90)
zλ+δzν+δ dm(z) = 1
{λ=ν}
Un