Table Of ContentPARTITIONS OF LARGE UNBALANCED BIPARTITES
JULIEN BUREAUX
Abstract. Wecomputetheasymptoticbehaviourofthenumberofpartitionsoflarge
√
vectors (n ,n ) of Z2 in the critical regime n (cid:16) n and in the subcritical regime
√1 2 + 1 2
n = o( n ). This work completes the results established in the fifties by Auluck,
1 2
Nanda, and Wright.
4
1
0 1. Introduction
2
How many ways are there to decompose a finite-dimensional vector whose compo-
t
c
nents are non-negative integers as a sum of non-zero vectors of the same kind, up to
O
permutation of the summands? The celebrated theory of integer partitions deals with
5 the one-dimensional version of this problem. We refer the reader to the monograph of
1
Andrews [1] for an account on the subject. In a famous paper, Hardy and Ramanujan
[2] discovered the asymptotic behaviour of the number of partitions of a large integer n:
]
O
( r )
1 2n
C (1) pN(n) ∼ √ exp π .
. 4n 3 3
h
t In the two-dimensional setting, the number of partitions of a large vector has been
a
m studied by many authors, by various approaches. In the early fifties, the physicist
Fermi [3] introduced thermodynamical models characterized by the conservation of two
[
parameters instead of just one (corresponding to integer partitions). This led Auluck [4]
2
to search for an asymptotic expression of the number of partitions of an integer vector
v
9 (n ,n ). He established formulæ in two very different regimes. His first formula holds
1 2
6 when n is fixed and n tends to infinity:
1 2
1
√
8 1 6n !n1 1 ( r2n )
1. (2) pZ2+(n1,n2) ∼ n1! π 2 4n2√3 exp π 32 .
0
4 His second one concerns the case where both components n and n tend to infinity with
1 2
1
the same order of magnitude. The corresponding formula is much more involved but it
:
v can be simplified in the special case n = n = n. For some explicit constants a,b,c, one
1 2
i
X has
a n 2 1o
ar pZ2+(n,n) ∼ n5356 exp bn3 +cn3 .
In the late fifties, Nanda [5] managed to extend the domain of validity of Auluck’s first
1/4
formulatotheweakerconditionn = o(n ). Shortlyafter, Wright[6]wasabletoprove
1 2
2010 Mathematics Subject Classification. 05A16, 05A17, 11P82, 60F05, 82B05.
Key words and phrases. bipartite partitions, statistical mechanics, canonical ensemble, Gibbs mea-
sure, partition function, Mellin transform, multivariate local limit theorem.
1
PARTITIONS OF LARGE UNBALANCED BIPARTITES 2
that Auluck’s second formula can be extended to the more general regime
1 logn logn
1 1
< liminf ≤ limsup < 2.
2 logn logn
2 2
Note finally that Robertson [7, 8] proved analogous formulae in higher dimensions.
√
Our article covers the case n = O( n ), which completes these previous results. In
1 2 √
particular, we deal with the case where n and n have the same order of magnitude,
1 2
which appears as a critical regime.
The papers of Auluck, Nanda and Wright all rely on generating function techniques.
At the exception of Nanda’s work, which is directly based on integer partition estimates,
the main idea is to extract the asymptotic behaviour of the coefficients from the gen-
erating function with a Tauberian theorem or a saddle-point analysis. The extension
proven by Wright was made possible by a more precise approximation of the generating
function.
logn2 4 2 1
N. ?
Wright [6]
1
2
? 1
4
Nanda [5]
logn
1
Figure 1. Phase diagram of the previously studied asymptotic regions.
Our work covers the unknown grey region, including the thick critical
lines 1 and 2, as well as Nanda’s region.
2
The method we use in this article differs from the previous ones by relying heavily
on a probabilistic embedding of the problem which is inspired by the Boltzmann model
in statistical mechanics. The first ingredient of our proof is a precise estimate of the
associated logarithmic partition function, based on an contour-integral representation
of this function and Cauchy’s residue theorem. The second ingredient is a bivariate
local limit theorem, which follows from a general framework developed at the end of the
paper.
Local limit theorems happen to play a crucial role in the treatment of questions from
statistical mechanics, where they provide a rigorous justification of the equivalence of
ensembles principle. Twenty years ago, Fristedt [9] introduced similar ideas to study
the structure of uniformly drawn random partitions of large integers. A few years later,
Báez-Duarte [10] applied a local limit theorem technique to derive a short proof of
the Hardy-Ramanujan formula (1). The first implementation of these ideas in a two-
dimensional context seems to be due to Sina˘ı [11], although the setting differs from ours.
PARTITIONS OF LARGE UNBALANCED BIPARTITES 3
A general presentation of these techniques was discussed by Vershik [12]. In a recent
paper, Bogachev and Zarbaliev [13] presented among other results a detailed proof of
Sina˘ı’s approach. Let us mention that the strong anisotropy which is inherent in the
problem that we address makes the implementation of this program more delicate.
2. Notations and statement of the results
Let Z = {0,1,2,3,...} and N = Z \{0} denote respectively the set of non-negative
+ +
numbers and the set of positive integers.
We will use the standard Landau notations a = o(b ) or a = O(b ) for sequences
n n n n
(cid:12) (cid:12) (cid:12) (cid:12)
(a ) and (b ) satisfying respectively limsup(cid:12)an(cid:12) = 0 or limsup(cid:12)an(cid:12) < +∞. Also, we
n n (cid:12)bn(cid:12) (cid:12)bn(cid:12)
will write a (cid:16) b if a and b have the same order of magnitude as n tends to infinity,
n n n n
that is to say if both a = O(b ) and b = O(a ) hold.
n n n n
Definition. Let X be a subset of Z2. For every n ∈ Z2, a partition of n with parts in
+ +
X is a finite unordered family of elements of X whose sum is n. It can be represented
by a multiplicity function ω: X → Z such that P ω(x)·x = n. For x ∈ X, we say
+ x∈X
that ω(x) is the multiplicity of the part x in the partition. The partitions of n with parts
in X constitute the set
( )
Ω (n) := ω ∈ ZX : X ω(x)·x = n .
X +
x∈X
Finally, we write p (n) := |Ω (n)| for the number of partitions of n with parts in X.
X X
Following the works of Wright and Robertson [6, 7, 8], we will focus on two particular
sets of parts in this article, namely:
• X = N2 which corresponds to partitions in which no part has a zero component,
• X = Z2 \{0} which corresponds to the case of general partitions, in which parts
+
may have a zero component. We still have to exclude the zero part in order to
ensure that every vector has only finitely many partitions.
Thefollowingtheoremstatesthemainresultsofthepaper. Itdescribestheasymptotic
behaviour of p (n) in the case of partitions without zero components as well as in the
X
case of general partitions, outside Wright’s region. First, we need to introduce the
following auxiliary functions of α > 0:
X 1 e−αr Φ0(α) π2 Φ0(α)
Φ(α) = , Θ(α) = − , Φ(α) = Φ(α)+ , Θ(α) = − ,
r21−e−αr pΦ(α) 6 q
r≥1 Φ(α)
Ψ(α) = X 1 e−αr , ∆(α) = 2Φ(α)Φ00(α)−Φ0(α)2, ∆(α) = 2Φ(α)Φ00(α)−Φ0(α)2.
r1−e−αr
r≥1
Consider two sequences (n (k)) and (n (k)) of positive integers. In the sequel, the
1 k 2 k
limits and asymptotic comparisons are to be understood as k approaches infinity. The
index k will remain implicit.
Theorem 2.1. Assume that both n and n tend to infinity under the conditions n =
√ 1 2 1
O( n ) and log(n ) = o(n ).
2 2 1
PARTITIONS OF LARGE UNBALANCED BIPARTITES 4
n
1
(ii) If α > 0 is the unique solution of Θ(α ) = √ , then
n n
n
2
1 Φ(αn)e−12Ψ(αn) (cid:26)(cid:18) q (cid:19)√ (cid:27)
p (n ,n ) ∼ exp α Θ(α )+2 Φ(α ) n .
N2 1 2 p n n n 2
2π n ∆(α )
2 n
n
1
(iiii) If α > 0 is the unique solution of Θ(α ) = √ , then
n n
n
2
1 Φ(αn)!54 e12Ψ(αn) (cid:26)(cid:18) q (cid:19)√ (cid:27)
p (n ,n ) ∼ exp α Θ(α )+2 Φ(α ) n .
Z2+\{0} 1 2 (2π)32 n2 q∆(α ) n n n 2
n
We will present a complete proof of (i) and will state along the proof the additional
arguments which are needed for (ii).
Although the formulæ in Theorem 2.1 involve an implicit function α of (n ,n ),
n 1 2
remark that we can actually derive explicit expansions in terms of (n ,n ) when n is
√ 1 2 1
negligible compared to n , which condition is equivalent to α → +∞. Notice indeed
2 n
thattheauxiliaryfunctionsΦ,Φ,Θ,Θ,∆,∆, andΨadmitsimpleasymptoticexpansions
in terms of the arithmetic function σ (m) = P d2 as α → +∞. This follows from the
2 d|m
Lambert series [14, Section 4.71] elementary formulæ
(3) Φ(α) = X σ2(m)e−αm, −Φ0(α) = X σ2(m)e−αm.
m2 m
m≥1 m≥1
Asymptotic expansions of Θ−1,Θ−1 can be computed effectively from there by an iter-
ative method or by using the Lagrange reversion formula.
LetusshowhowthesesimplesideasallowsustoextendthepreviousresultsbyNanda
and Robertson. For example, the following application of case (i) provides additional
corrective terms in the expansion given by Robertson [7, Theorem 2] which was stated
1/3
for the special case K = 1, that is to say n = o(n ).
1 2
Corollary 2.2. There exists a sequence (c ) of rational numbers such that for all K ∈ N,
k
if n and n tend to +∞ such that n2K+1 = o(nK) and log(n ) = o(n ),
1 2 1 2 2 1
p (n ,n ) ∼ n1 nn21 exp(KX−1c n21k+1).
N2 1 2 n (n !)2 k nk
2 1 k=1 2
Notice that the proof of this formula, which is given below, can be directly translated
into an effective algorithm. For instance, we give here the first terms of the sequence
(c ) which have been computed with the help of the Sage mathematical software [15]:
k
5 805 6731 133046081 170097821
c = , c = − , c = , c = − , c = , ...
1 2 3 4 5
4 288 576 2073600 414720
In the same way, an application of case (ii) of our theorem leads to an extension of
1/4
formula (2) which was stated under the condition n = o(n ), or equivalently K = 1,
1 2
in the work of Nanda [5].
PARTITIONS OF LARGE UNBALANCED BIPARTITES 5
Corollary 2.3. There exists a sequence (c ) of real numbers such that for all K ∈ N, if
k
n and n tend to +∞ such that nK+1 = o(nK/2) and log(n ) = o(n ),
1 2 1 2 2 1
√
p (n ,n ) ∼ 1 6n2!n1 1√ exp(πr2n2)exp(KX−1c nk1+1)
Z2+ 1 2 n1! π 4n2 3 3 k=1 k nk2/2
p
Aneffectivecomputationofthefirsttermsofthesequence(c )gives, witha = ζ(2),
k
5a 1 5 145a2 1385a 5 1
c = − , c = − , c = 6a3− + + , ...
1 4 4a 2 8 72 3 576 32a 192a3
√
Proof of Corollary 2.2. Asnotedabove,theconditionn = o( n )impliesthatα tends
1 2 n
p
to +∞ (see the proof of Proposition 5.2 for details). Since the both of Φ(α) and ∆(α)
areequivalenttoe−α asα tendsto+∞, andΨ(α)tendsto0, thenon-exponentialfactor
in the formula for p (n ,n ) of Theorem 2.1 reduces to 1 . Also, the identities (3)
N2 1 2 2πn2
for Φ(α) and −Φ0(α) show that we can now work with the formal power series
f(z) = X∞ σ2(m) zm, g(z) = 1 (cid:18)−z d f(z)(cid:19)2.
m2 f(z) dz
m=1
√
Namely, the equation Θ(α ) = n / n corresponds formally to g(z ) = n2/n with
n 1 2 n 1 2
zn := e−αn. Since g(z) has no constant term, we obtain by reversion an infinite asymp-
totic expansion (we use here the Poincaré notation ∼ to denote an asymptotic series):
e−αn ∼ n21 +X∞ a n21k.
n k nk
2 k=2 2
for some sequence of rationnals (a ) expressible with the help of the Lagrange inversion
k
formula. From this point, it is now easy to derive the existence of two expansions (where
(b ) and (b0) are two sequences of rational numbers)
k k
α ∼ logn −2logn +X∞ b n21k, qΦ(α ) ∼ √n1 "1+X∞ b0 n21k#,
n 2 1 k nk n n k nk
k=1 2 2 k=2 2
which, together with Theorem 2.1 and Stirling’s formula, prove the result. (cid:3)
In both Corollary 2.2 and Corollary 2.3, notice that the boundary of the domain of
validity for the expansions is actually asymptotic, as K grows larger, to the critical
√
regime n (cid:16) n which is represented by the thick line in Figure 1. In this critical
1 2
case, Theorem 2.1 still applies but does not lead to any much simpler expression since,
√
when n / n converges quickly enough to some positive constant, all terms depending
1 2
on α tend to constant coefficients. Still, the theorem provides the existence and some
n
expressions of the exponential rate functions
1 √ 1 √
h(t) := lim √ logp (bt nc,n), h(t) = lim √ logp (bt nc,n),
n→+∞ n N2 n→+∞ n Z2+\{0}
which are defined for all t > 0. Figure 2 shows the graphs of these functions.
PARTITIONS OF LARGE UNBALANCED BIPARTITES 6
7
6
5 h(t)
4 h(t)
3
π 2
3
q
2
1
t
2 4 6 8 10
Figure 2. Comparison of the rate functions h and h corresponding re-
spectively to partitions without zero components and to partitions where
zero components are allowed.
3. The probabilistic model
In this section, we introduce a family of Gibbs probability measures on the set Ω :=
X
S Ω (n) of all partitions, given some fixed set of parts X. The idea is that, while
n∈Z2 X
+
the uniform distribution on the set Ω (n) of partitions on n is hard to describe, it is
X
mucheasiertodefineadistributiononthelargerspaceΩ , thatwillgivetheexactsame
X
weight to every partition of n. Let α,β ∈ (0,+∞) be two shape parameters to be chosen
later and write λ = (α,β). To each choice of λ, we associate a probability measure P
λ
on the discrete space Ω such that for every ω ∈ Ω ,
X X
exp{−P ω(x)hλ,xi}
P (ω) := x∈X .
λ P exp{−P ω(x)hλ,xi}
ω∈ΩX x∈X
For each partition ω ∈ Ω , let us introduce the key quantity N(ω) := P ω(x)·x,
X x∈X
which we will see as a random variable with values in Z2. By definition, we have
+
ω ∈ Ω (N(ω)) for every partition ω. Furthermore, the probability P (ω) becomes
X λ
P (ω) = 1 e−hλ,N(ω)i, where Z := X e−hλ,N(ω)i.
λ λ
Z
λ
ω∈ΩX
The normalization constant Z is usually referred to as the partition function of the
λ
system in the statistical mechanics literature.
In accordance with the previous discussion, remark that the conditional distribution
ofP onΩ (n)yieldstheuniformmeasuresincetheprobabilityofapartitionω onlyde-
λ X
pendsonN(ω). Moreover,sinceallpartitionsofΩ (n)aregivenequalweight 1 e−hλ,ni,
X Z
λ
PARTITIONS OF LARGE UNBALANCED BIPARTITES 7
the total weight of Ω (n) is equal to 1 p (n)e−hλ,ni and therefore
X Z X
λ
Z
(4) p (n) = λ P (N = n).
X e−hλ,ni λ
Probabilistic intuition dictates our strategy: calibrate the parameter λ as a function
of n so that the distribution of the random vector N concentrates around n under the
probabilitymeasureP . Thisway,wewillbeabletoensureapolynomialdecreaseforthe
λ
quantity P (N = n). A natural choice to enforce this behaviour is to take λ = (α ,β )
λ n n n
such that E (N) is close enough to n. This is achieved by choosing the couple (α ,β )
λ n n
defined by the equations (7) of Section 5. Looking back to p (n), we see that we need
X
to estimate precisely the partition function Z as well as P (N = n). The former will
λ λ
be done by a careful approximation of logZ in section 4 and the latter will be deduced
λ
from estimates of the first and second derivatives of logZ together with a Gaussian
λ
local limit theorem statement proven in section 5.
Before we turn to more technical discussions, let us remark that under the probability
measure P , the random variables ω(x) for x ∈ X are mutually independent and that
λ
their distribution is geometric. More precisely, we have for all k ∈ Z ,
+
P (ω(x) = k) = e−khλ,xi(1−e−hλ,xi).
λ
Finally, a fruitful consequence of the independence in this model is the fact that the
partition function Z can be written as an infinite product:
λ
Y 1
(5) Z = .
λ 1−e−hλ,xi
x∈X
Let us mention that (5) is the bipartite partition analogue of the famous Euler product
formula for the usual partition generating function.
4. Approximation of the logarithmic partition function
Because of the product formula (5), the logarithm of the partition function Z can be
λ
expressed as the sum of an absolutely convergent series:
logZ = − X log(1−e−hλ,xi) = X X∞ e−rhλ,xi.
λ
r
x∈X x∈Xr=1
Let us recall that we consider the case X = N2 of partitions whose parts have non-zero
components. The logarithmic partition function thus writes
(6) logZ = X X X e−αx1re−βx2r.
λ
r
x1≥1x2≥1r≥1
Letζ andΓdenoterespectivelytheRiemannzetafunctionandtheEulergammafunction
[14]. Also consider for every α > 0 and s ∈ C the Dirichlet series D (s) defined by
α
XX e−αkr X 1 e−αr
D (s) := = .
α rs rs1−e−αr
k≥1r≥1 r≥1
PARTITIONS OF LARGE UNBALANCED BIPARTITES 8
Recalling that the Cahen-Mellin inversion formula [14] yields for every c > 0 and t > 0,
1 Z c+i∞
e−t = Γ(s)t−sds,
2iπ
c−i∞
we can rewrite the identity (6) for every c > 1 as
logZ = 1 X∞ X∞ X∞ Z c+i∞ e−rαx1Γ(s)(rβx )−sds
λ 2
2iπ r
c−i∞
x1=1x2=1r=1
1 Z c+i∞ ds
= Γ(s)ζ(s)D (s+1) ,
2iπ α βs
c−i∞
the exchange in the order of summation being justified by the Fubini theorem. We
proved that the logarithmic partition function admits an integral representation of the
form
1 Z c+i∞ ds
logZ = M (s) ,
λ 2iπ α βs
c−i∞
where M (s) := Γ(s)ζ(s)D (s+1). We will now see how to recover from the residues
α α
of the meromorphic function M the asymptotic behaviour of logZ and its derivatives
α λ
when β tends towards 0.
Proposition 4.1. For every non-negative integers m,p,q, there exists a decreasing func-
tion Cp,q(α) of α > 0 with a positive limit as α → ∞, such that the remainder function
m
R (α,β) defined by
m
R (α,β) := logZ − Dα(2) −Xm (−1)kζ(−k)Dα(1−k)βk
m (α,β) β k!
k=0
satisfies |∂αp∂βqRm(α,β)| ≤ Cmp,q(α)e−αβm−q+12 for all α,β > 0.
Proof. Let us recall that the Riemann function ζ(s) is meromorphic on C and that it
has a unique pole at s = 1, at which the residue is 1. The Euler gamma function Γ(s)
is also meromorphic on C and has poles at every integer k ≤ 0.
Letmbeapositiveintegerandγ = −m−1. WearegoingtoapplyCauchy’sresidue
m 2
theorem to the meromorphic function M with the rectangular contour C defined by
α
the segments [2−iT,2+iT], [2+iT,γ +iT], [γ +iT,γ −iT] and [γ −iT,2−iT],
m m m m
where T > 0 is some positive real we will let go to infinity. Computation of the residues
of M in this stripe yields
α
1 Z M (s)ds = Dα(2) +Xm (−1)kζ(−k)Dα(1−k)βk.
2iπ α βs β k!
C+
k=0
In order to prove that the contributions of the horizontal segments in the left-hand side
integral vanish for T → ∞, we use the three following facts:
(iii) From the complex version of Stirling’s formula, we know that |Γ(σ + iτ)|
decreasesexponentiallyfastwhen|τ|tendsto+∞,uniformlyineverybounded
stripe [14, p. 151].
(iiiiii) Also |ζ(σ + iτ)| is polynomially bounded in |τ| as |τ| → +∞, uniformly in
every bounded stripe [16, p. 95].
PARTITIONS OF LARGE UNBALANCED BIPARTITES 9
(iiiiiiiii) Finally, note that for all σ ∈ R,
0
X 1 e−αr
sup |D (σ+iτ)| ≤ D (σ ) = < ∞.
σ≥σ0 α α 0 r≥1 rσ0 1−e−αr
From these observations, we see that the integrals along horizontal lines vanish as T →
∞:
Z γm+iT ds Z 2−iT ds
lim M (s) = lim M (s) = 0.
T→+∞ 2+iT α βs T→+∞ γm−iT α βs
Furthermore, M (s)β−s is integrable on the vertical line (γ −i∞,γ +i∞), so that in
α m m
the limit T → ∞, we obtain
logZ = Dα(2) +Xm (−1)kζ(−k)Dα(1−k)βk + 1 Z γm+i∞M (s)ds.
λ β k! 2iπ α βs
k=0 γm−i∞
Thus, the remainder function R (α,β) is actually equal to the integral term in the
m
right-hand side. We need to control its derivatives. Note that the derivatives
M (s) Γ(s+q)
∂p∂q α = (−1)q ζ(s)∂pD (s+1)
α β βs βs+q α α
are well defined and integrable on the vertical line (γ −i∞,γ +i∞) thanks again to
m m
the facts (i) and (ii), as well as the analogue of (iii) for the ∂p derivative of D . It is
α α
easy to check that the result follows with
|∂pD (γ +1)| Z ∞
Cp,q(α)e−α = α α m |ζ(γ +iτ)||Γ(γ +q+iτ)|dτ. (cid:3)
m 2π m m
−∞
Remark. In the case X = Z2 \{0}, the logarithmic partition function becomes logZ +
+ λ
Ψ(α)+Ψ(β), where Ψ(·) = D(1) is the function defined in Section 2. The two addi-
·
tional terms correspond respectively to horizontal and vertical one-dimensional integer
partitions. In that case, the expansion as β → 0+ involves the expansion of Ψ(β) (which
was the basis of [2]) and can be written informally as
Φ(α)+ζ(2) 1 Ψ(α) 1 (cid:18)D (0) 1 (cid:19)
α
+ logβ + − log(2π)+ − β+...
β 2 2 2 12 24
5. Calibration of the shape parameters
In this section, we find appropriate values for the parameters λ = (α,β) as functions
of n for which E (N) is asymptotically close to n. Since the distribution of the random
λ
vector N under P is given by a Gibbs measure,
λ
(cid:20)∂ logZ (cid:21)
E (N) = −grad(logZ ) = − α λ .
λ λ ∂ logZ
β λ
Let us recall that by definition, the function Φ introduced in Section 2 is
X 1 e−αr
∀α > 0, Φ(α) := ,
r21−e−αr
r≥1
which is exactly D (2) for the Dirichlet series D (s) introduced in Section 4. The
α α
approximation given in Proposition 4.1 applied with (m,p,q) = (1,1,0) and (m,p,q) =
PARTITIONS OF LARGE UNBALANCED BIPARTITES 10
(1,0,1) yields the existence of two decreasing functions C and C with finite limits as
1 2
α → +∞ such that
(cid:12) Φ0(α)(cid:12) (cid:12) Φ(α)(cid:12)
(cid:12)(cid:12)(cid:12)∂αlogZλ− β (cid:12)(cid:12)(cid:12) ≤ e−αC1(α) and (cid:12)(cid:12)(cid:12)∂βlogZλ+ β2 (cid:12)(cid:12)(cid:12) ≤ e−αC2(α).
This justifies the choice made in the next proposition to define the shape parameters α
n
and β through the implicit equations (7). We start with a lemma.
n
Lemma 5.1. The function Φ is logarithmically convex on (0;+∞).
Proof. The function logΦ being smooth, its convexity is equivalent to the inequality
d2 Φ(α)Φ00(α)−Φ0(α)2
(logΦ(α)) = ≥ 0,
dα2 Φ(α)2
which follows immediately from the Cauchy-Schwarz inequality since
Φ(α) = XX e−αkr, Φ0(α) = −XX ke−αkr and Φ00(α) = XXk2e−αkr.
r2 r
k≥1r≥1 k≥1r≥1 k≥1r≥1
(cid:3)
Proposition 5.2. For all n = (n ,n ) ∈ N2, there exists a unique couple (α ,β ) ∈
1 2 n n
(0;+∞)2 such that
−Φ0(α ) Φ(α )
n n
(7) = n and = n .
β 1 β2 2
n n
Proof. Eliminating β in the implicit equations (7), we need only prove the existence of
n
a unique α > 0 such that
n
1 Φ0(α ) 1 n
n 1
= − √ .
p
2 Φ(α ) 2 n
n 2
We recognize the derivative of the function pΦ(α) = exp(cid:16)1 logΦ(α)(cid:17) which is strictly
√ 2
convex by Lemma 5.1, so that Φ0/ Φ is continuously increasing. In addition, it is easy
to check that
Φ0(α) −ζ(3)/α2 Φ0(α) −e−α
lim = lim = −∞ and lim = lim √ = 0.
p p p
α→0+ Φ(α) α→0+ ζ(3)/α α→+∞ Φ(α) α→+∞ e−α
This concludes the proof. (cid:3)
Recall that the assumptions of Theorem 2.1 are n → +∞ and n → +∞ with the
√ 1 2
conditionsn = O( n )andlog(n ) = o(n ). Consider thecouple of parameters α and
1 2 2 1 n
β defined by the equations (7). A consequence of the proof above is that, under these
n
assumptions, the sequence α is bounded away from 0 (but not from +∞). In addition,
n
e−αn e−αn n1
(8) (cid:16) n , (cid:16) n , and β (cid:16) .
β 1 β2 2 n n
n n 2
