Table Of ContentQUANTUM LIMITS OF EISENSTEIN SERIES AND
SCATTERING STATES
1
1
YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
0
2
v Abstract. Weidentifythequantumlimitsofscatteringstatesforthemodu-
o lar surface. This isobtained through the studyof quantum measures ofnon-
N holomorphicEisensteinseriesawayfromthecriticalline. Weprovidearange
ofstabilityforthequantum uniqueergodicitytheoremofLuoandSarnak.
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2
]
T 1. Introduction
N
Animportantproblemofquantumchaosistodescribethelimiting behaviourof
. eigenfunctions. OnacompactnegativelycurvedRiemannianmanifoldX Shnirelman,
h
t Colinde Verdi`ere,andZelditch[18,2,22]haveprovedthatfora‘generic’family of
a eigenfunctions φ oftheLaplaciantheassociatedmeasuresdµ (z)= φ (z)2dµ(z)
m j j j
{ } | |
converge weakly to the standard volume element dµ(z) of X, which we write as
[
(1.1) dµj(z) dµ(z) as j .
1 → →∞
v Zelditch[23]extendedtheresulttofinitevolumehyperbolicsurfaces. Lindenstrauss
5 andSoundararajan[11,19] haveprovedthatfor X =Γ H2 whereΓ PSL (Z) is
2
1 \ ⊂
of a certain arithmetic type (1.1) holds if φ runs through the set of Hecke–Maaß
6 j
cusp forms. Earlier Luo and Sarnak [12] investigated the question of quantum
6
. chaos for Eisenstein series E(z,1/2+it), i.e. generalized eigenfunctions on X =
1
PSL (Z) H2. Since this series is not square integrable a certain normalization is
1 2 \
needed. The actual statement in [12] is the following: Let A and B be compact
1
1 Jordan measurable subsets of X. Then
v: (1.2) lim A|E(z,1/2+it)|2dµ(z) = µ(A).
Xi t→∞RB|E(z,1/2+it)|2dµ(z) µ(B)
r In fact, see [12], this followRs from the result
a
6
(1.3) E(z,1/2+it)2dµ(z) µ(A)logt, t .
| | ∼ π · →∞
ZA
A general cofinite subgroup likely has few embedded eigenvalues, possibly finite,
so that (1.1) may be irrelevant. So far the quantum unique ergodicity of Eisen-
stein series is unproven for a general cofinite subgroup. A good substitute for the
embedded eigenvalues are the scattering poles (resonances). A natural question is
to study the quantum limits of these scattering states. We address this question
Date:November30,2011.
2000 Mathematics Subject Classification. Primary11F72;Secondary 58G25,35P25.
Key words and phrases. QuantumLimits,Eisensteinseries,Scattering poles.
ThefirstauthorwassupportedbyNSFgrantDMS-0401318,thesecondauthorwassupported
byagrantfromDAAD(GermanAcademicExchangeService),andthethirdauthorwassupported
byagrantfromTheDanishNaturalScienceResearchCouncil.
1
2 YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
for Γ=PSL (Z). As these states are not in L2(Γ H2), some normalization is also
2
\
needed. Consider a simple pole ρ of the scattering matrix. By the explicit calcu-
lation of the scattering matrix, see (2.3), such a pole is equal to half a zero of the
Riemann zeta function. The Eisenstein series has a pole at this point also and the
residue has Fourier expansion
(1.4) resE(z,s)=(resφ(s))y1−ρ+ c √yK (2πmy)e2πimx.
m ρ−1/2
s=ρ s=ρ
m6=0
X
These scattering states are formal eigenfunctions of the Laplace operator. We
choose to normalize them as follows: Set
(1.5) u (z)=(resφ(s))−1 resE(z,s),
ρ
s=ρ s=ρ
so that the scattering functions have the simplest possible growth behaviour at
infinity, namely y1−ρ.
We let γ be a sequence of zeroes of the Riemann zeta function with 1/2
n
{ } ≤
(γ )whichsatisfieslim γ =γ <1. Automatically (γ )<1andtheRiemann
n n n ∞ n
ℜ ℜ
hypothesis is equivalent to γ =γ =1/2,but we shall not assume it. The points
n ∞
ρ =γ /2 are poles of the scattering matrix φ(s).
n n
Theorem 1.1. Let A be a compact Jordan measurable subset of X. Then
u (z)2dµ(z) E(z,2 γ )dµ(z)
| ρn | → − ∞
ZA ZA
as n . This means that the quantum limit of the measures u (z)2dµ(z) is
→ ∞ | ρn |
the invariant, absolutely continuous measure E(z,2 γ )dµ(z).
∞
−
Remark 1.2. We know that a positive proportion of the zeros ρ of ζ(s) lie on
the critical line and are simple. In fact, this proportion is at least 40.58%, see
[1]. Under the Riemann hypothesis and the conjectured simplicity of the Riemann
zeros, there is only one quantum limit, the one described in the theorem above:
E(z,3/2)dµ(z).
Theorem 1.1 follows rather easily by studying the quantum limits of Eisenstein
series off the critical line. We present two such theorems. The first addresses the
stability of (1.2) if, instead of real spectral value 1/4+t2, we move in the complex
plane. To be precise let
dµ (z)= E(z,s(t))2dµ(z),
s(t)
| |
where
s(t)=σ +it,
t
σ > 1/2. We investigate what happens in the limit as t , assuming that
t
→ ∞
σ σ 1/2. We find qualitative differences depending on whether σ = 1/2
t ∞ ∞
→ ≥
or not. If σ =1/2 the situation is very similar to that of [12].
∞
Theorem 1.3. Assume σ =1/2. Let A, B be compact Jordan mesurable subsets
∞
of X. Then
µ (A) µ(A)
s(t)
µ (B) → µ(B)
s(t)
as t . In fact we have
→∞
3
(1.6) µ (A) µ(A) .
s(t) ∼ π(2σ 1)
t
−
LIMITS OF EISENSTEIN SERIES 3
We see that the rate of increase depends on the rate that σ tends to σ =1/2.
t ∞
Moreover, Theorem 1.3 implies that the quantum unique ergodicity of Eisenstein
seriesholdsinaquitebigregioninthecomplexplane(physicalplane): forspectral
value λ tending to infinity, the results holds as long as (λ) = o( (λ)) in the
ℑ ℜ
region (λ) 0. To see this we write λ = s(1 s) with s = σ+it and σ 1/2.
ℜ ≥ − p ≥
Then (λ)=o( (λ)) implies (λ)=σ(1 σ)+t2 . We easily deduce that
ℑ ℜ ℜ − →∞
σ isbounded. Then(1 2σ)t=o( σ(1 σ)+t2)givesσ 1/2. Sowecanapply
p − − →
Theorem 1.3.
p
Surprisingly the situation is qualitatively different when σ > 1/2. We prove
∞
the following theorem:
Theorem 1.4. Assume σ > 1/2. Let A be a compact Jordan measurable subset
∞
of X. Then
µ (A) E(z,2σ )dµ(z)
s(t) ∞
→
ZA
as t .
→∞
This proves that, when σ > 1/2, the measures dµ do not become equidis-
∞ s(t)
tributed. In fact it suggests in this case to consider different measures
2
E(z,s(t))
(1.7) ν (z)= dµ(z).
s(t)
(cid:12) E(z,2σ∞)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We note that, since 2σ∞ > 1, we(cid:12)phave E(z,2σ(cid:12)∞) > 0. The downside of this
(cid:12) (cid:12)
definitionisthatthe functionE(z,s(t))/ E(z,2σ )isnotaneigenfunctionofthe
∞
Laplacianin contrastto E(z,s(t)). The upside is that the corresponding measures
p
become equidistributed:
Corollary 1.5. Assume σ >1/2. Let A be a compact Jordan measurable subset
∞
of X. Then
ν (A) µ(A), t .
s(t)
→ →∞
The result of Theorem 1.4 looks similar to Theorem 1 in [5], where the authors
consider the equidistribution of Eisenstein series for convex co-compact subgroups
ΓofIso(Hn+1)withHausdorffdimensionofthelimitsetδ satisfyingδ <n/2. In
Γ Γ
both theorems the Eisenstein series E(z,2σ ) is well-defined and E(z,2σ )dµ(z)
∞ ∞
is the quantum limit.
Similar results to Theorem 1.1 and Theorem 1.4 for more general surfaces with
cuspidal ends have recently been announced by Dyatlov [3].
Remark 1.6. The crucial ingredients in [12] are
(i) Asubconvexestimateforthe L-seriesofaMaaßcuspformonitscriticalline,
e.g. L(φ ,1/2+it) (1+ t)1/3+ǫ, see [14].
j
≪ | |
(ii) A subconvex estimate for the Riemann zeta function on its critical line, e.g.
ζ(1/2+it) (1+ t)1/6+ǫ.
≪ | |
(iii) Estimates for ζ(1+it) and (ζ′/ζ)(1+it).
ForTheorem1.3aboveweusesubconvexboundsonL-functionsandζ(s). When
σ > 1/2, i.e. in Theorem 1.4 only convexity bounds are used. While we use
∞
estimates on ζ(1+it) and 1/ζ(1+it) in both cases, the estimate for (ζ′/ζ)(1+it)
is requiredonly for the theorem of Luo–Sarnak. Our results clarify the mechanism
for quantum unique ergodicity of Eisenstein series.
4 YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
Remark 1.7. Equation (1.2) was extended by Jakobson [9] to the unit tangent
bundle of X. Koyama [10] extended the result to Eisenstein series for PSL (Z[i]),
2
and Truelsen [21] to Eisenstein series for PSL ( ), with the integers of a
2 K K
O O
totally real field K of finite degree over Q with narrow class number one.
In bothcasesbounds ofthe type (i), (ii), and(iii) aboveare used. Inthe case of
K =Q(i) the subconvex estimate analogous to (i) was established by Petridis and
Sarnak[15],andthegeneralcasewasestablishedbyMichelandVenkatesh[13]. As
a substitute of (ii) and (iii) one uses estimates for the Dedekind zeta function ζ .
K
TheanalogousquestionforholomorphicHeckecuspformofweightkhasrecently
been resolved by Holowinsky and Soundararajan [6]. Let f be a sequence of L2-
k
normalized holomorphic Hecke cusp forms for the group SL (Z) of weight k and
2
letφ (z):=yk/2f (z). Thenthe measures φ (z)2dµ(z)convergeweaklytodµ(z),
k k k
| |
as previously conjectured by Rudnick and Sarnak. We note that in this case φ is
k
eigenfunction of the weight k Laplacian with eigenvalue k/2(1 k/2).
−
2. Proofs
The non-holomorphic Eisenstein series E(z,s), (z,s) H2 C is defined for
∈ ×
(s)>1 by
ℜ
(2.1) E(z,s)= (γz)s.
ℑ
γ∈XΓ∞\Γ
Here Γ = PSL (Z) and Γ is the cyclic subgroup generated by z z +1. The
2 ∞
7→
Eisenstein series E(z,s) admits a Fourier expansion of the cusp i , see e.g. [8,
∞
(3.25)]
E(z,s)= a (y,s)e2πinx
n
n∈Z
X
2y1/2
(2.2) =ys+φ(s)y1−s+ ns−1/2σ (n)K (2π n y)e2πinx.
ξ(2s) | | 1−2s | | s−1/2 | |
n6=0
X
Here ξ(s) = π−s/2Γ(s/2)ζ(s) is the completed Riemann zeta function satisfying
the functional equation ξ(s)=ξ(1 s), σ (n) is the sum of the c’th powers of the
c
−
divisors of n, and K (y) is the K-Bessel function. The scattering matrix is
s
ξ(2 2s)
(2.3) φ(s)= − .
ξ(2s)
We notice that the corresponding expression in [12] is missing a factor of 2 in the
non-zero terms. This is irrelevant for their purpose but becomes crucial for ours.
The spectral decomposition of L2(Γ H2) allows us to consider separately Maaß
\
cusp forms and incomplete Eisenstein series.
2.1. Maaß cusp forms. Since there is a basisof the cuspidaleigenspacesconsist-
ing of Hecke–Maaß cusp forms, we restrict our attention to those.
Lemma 2.1. Let φ be a Hecke–Maaß cusp form. Then
j
(2.4) φ E(z,σ +it)2dµ(z) 0,
j t
ZΓ\H2 | | →
as t .
→∞
LIMITS OF EISENSTEIN SERIES 5
Proof. The Maaß cusp form φ has a Fourier expansion
j
φ (z)=y1/2 λ(n)K (2πny)e(nx),
j itj
n6=0
X
with λ(1)=1. We assume that it is even, since, if it is odd, φ ,µ =0. Being
j s(t)
h i
a Hecke eigenform, φ has an L-series with Euler product
j
∞
λ(n)
L(φ ,s)= = (1 λ(p)p−s+p−2s)−1.
j ns −
n=1 p
X Y
We want to understand the behavior as t of
→∞
(2.5) J (t)= φ (z) E(z,s(t))2dµ(z).
j j
ZΓ\H2 | |
We calculate
(2.6) I (s)= φ (z)E(z,s(t))E(z,s)dµ(z),
j j
ZΓ\H2
and set s = s(t) to recover J (t). For fixed s, I (s) is a holomorphic function of
j j
w=s(t). In [12] this function is identified for w =1/2+it, so we use the principle
of analytic continuation to deduce that
Γ s±itj±(s(t)−1/2)
R(s) 2
I (s)= ,
j ξ(2s(t))Q (cid:16) 2πsΓ(s) (cid:17)
with
L(φ ,s s(t)+1/2)L(φ ,s+s(t) 1/2)
j j
R(s)= − − .
ζ(2s)
We plug s=s(t) to get
Γ s(t)±itj±(s(t)−1/2)
J (t)= 2−1πs(t)−s(t)L(φ ,1/2 2it)L(φ ,2σ 1/2) 2 .
j j − j t− Q Γ(cid:16)(s(t))ζ(2s(t))2 (cid:17)
| |
We apply Stirling’s formula [7, 5.112] in the form
(2.7) Γ(σ+it) =√2π tσ−1/2e−π2|t|(1+O(t−1))
| | | | | |
uniformly for σ M. Using this we find that the quotient of Gamma factors is
| | ≤
t1/2−2σt.
j
≪ | |
If σ is bounded away from 1/2, the function ζ(2s(t))−2 is bounded and the
t
| |
convexity estimate L(φ ,1/2+it) t1/2 suffices to guarantee that limJ (t)=0.
j j
≪
If σ is not bounded away from 1/2 and we need non-trivial estimates for
t
ζ(2s(t))−1 and L(φ ,1/2+it) to reach the same conclusion. Such estimates are
j
certainly available: the estimate
(2.8) log−1 t ζ(2s(t)) log t
| |≪| |≪ | |
is classical in the theory of the Riemann zeta function (see [20, 3.6.5 and 3.11.8]),
and the subconvexity estimate
(2.9) L(φ ,1/2+it)=O (t1/3+ǫ)
j j,ǫ
| |
wasprovedbyMeurman[14]. We notethatany subconvexityestimateL(φ ,1/2+
j
it)=O(t1/2−ǫ) suffices to show that limJ (t)=0. (cid:3)
j
| |
6 YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
2.2. Incomplete Eisenstein series. We now concentrate on the contribution of
theincompleteEisensteinseries. Leth(y) C∞(R+)beafunctionwhichdecreases
∈
rapidly at 0 and . This means that h(y) = O (yN) for 0 < y 1 and h(y) =
N
∞ ≤
O(y−N) for y 1 for all N N. Its Mellin transform is
≫ ∈
∞ dy
(2.10) H(s)= h(y)y−s
y
Z0
and the Mellin inversion formula gives
1 a+i∞
(2.11) h(y)= H(s)ysds
2πi
Za−i∞
for any a R. The function H(s) is entire and H(σ+it) is in the Schwartz space
∈
in the t variable for any σ R. We consider the incomplete Eisenstein series
∈
1 a+i∞
(2.12) F (z)= h( (γz))= H(s)E(z,s)ds.
h
ℑ 2πi
γ∈XΓ∞\Γ Za−i∞
Lemma 2.2. Let F be an incomplete Eisenstein series as above. Then
h
F (z) E(z,σ +it)2dµ(z)
h t
ZΓ\H2 | | ∼
F (z)E(z,2σ )dµ(z), if σ =1/2,
Γ\H2 h ∞ ∞ 6
(RΓ\H2Fh(z)dµ(z)(4ξ(2)(σt−1/2))−1, if σ∞ =1/2,
as t . R
→∞
Proof. We choose a such that a>2σ for all t. The function F (z) is smooth and
t h
rapidly decreasing in the cusp. Then unfolding and using Parseval we get
F (z)dµ (z)= F (z)E(z,s(t))2dµ(z)
h s(t) h
ZΓ\H2 ZΓ\H2 | |
∞ 1 dxdy
= h(y)E(z,s(t))2
| | y2
Z0 Z0
∞ dy
(2.13) = h(y) a (y,s(t))2 .
Z0 n∈Z| n | ! y2
X
2.3. Contribution of the constant term. By (2.2) we have
a (y,s(t))2 =y2σt +2 (φ(s(t))y1−2it)+ φ(s(t))2y2−2σt.
0
| | ℜ | |
We analyze the three terms separately. The first term is
∞ dy
h(y)y2σt−1 =H(1 2σ ),
t
y −
Z0
which converges to H(1 2σ ) when t . Next
∞
− →∞
∞ dy
φ(s(t)) h(y)y−2it =φ(s(t))H(2it).
y
Z0
The function H(2it) decays rapidly and φ(s(t)) is bounded, see [16, (8.6)]. By
analyzing the same expression with φ(s(t))y1−2it instead of φ(s(t))y1+2it we find
that the term in (2.13) involving (φ(s(t))y1−2it) tends to zero.
ℜ
LIMITS OF EISENSTEIN SERIES 7
The last expression coming from the constant term is
∞ dy
φ(s(t))2 h(y)y1−2σt = φ(s(t))2H(2σ 1).
t
| | y | | −
Z0
Certainly H(2σ 1) H(2σ 1), as t and φ(s(t)) is bounded.
t ∞
− → − →∞ | |
Using the explicit expression for φ(s) in (2.3) we have better control of the
behavior of φ(s(t)) when σ =1/2. We have
∞
6
ξ(2 2s(t)) ζ(2(1 σ ) 2it) Γ(1 σ it)
φ(s(t)) = | − | =π2σt−1 − t − − t− .
| | ξ(2s(t)) ζ(2σ +2it) Γ(σ +it)
| | (cid:12) t (cid:12)(cid:12) t (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
Using the convexity bound ζ(σ+it)=O(cid:12)(cid:12)(t(1−σ)/2+ǫ) we g(cid:12)(cid:12)e(cid:12)(cid:12)t (cid:12)(cid:12)
| |
ζ(2(1 σ )+it)=O(tσt−1/2+ǫ).
t
− | |
By (2.8)
1
=O(log t).
ζ(2σ +2it) | |
t
The quotient of Γ-factors is asymptotic to t1−2σt by (2.7). We therefore conclude
| |
that, when σ =1/2, we have
∞
6
(2.14) φ(s(t)) 0
| |→
as t .
→∞
To summarize we have proved that the contribution of the constant term in
(2.13) converges to H(1 2σ ) if σ =1/2 and is O(1) if σ =1/2.
∞ ∞ ∞
− 6
2.4. Contribution of the non-constant terms. By (2.2) and (2.11) the contri-
bution equals
A(t)= ∞ 1 H(s)ysds ∞ 8y n2σt−1 σ (n) 2 K (2πny) 2 dy
Z0 2πiZℜ(s)=a nX=1|ξ(2s(t))|2 (cid:12) 1−2s(t) (cid:12) (cid:12) s(t)−1/2 (cid:12) y2
= ∞ 1 H(s) ∞ ys 8 n2σt−1(cid:12)σ (n)(cid:12)2(cid:12)K (y) 2ds(cid:12)dy
Z0 2πiZℜ(s)=a nX=1(2πn)s|ξ(2s(t))|2 (cid:12) 1−2s(t) (cid:12) (cid:12) s(t)−1/2 (cid:12) y
= 1 H(s) 1 8 ∞ σ1−2s(t)(n) 2(cid:12) ∞ys K(cid:12) (cid:12) (y) 2 dyd(cid:12)s.
2πiZℜ(s)=a (2π)s|ξ(2s(t))|2 nX=1(cid:12)(cid:12) ns−(2σt−1)(cid:12)(cid:12) Z0 (cid:12) s(t)−1/2 (cid:12) y
(cid:12) (cid:12)
Wenowuse[4,6.576(4)]tocalculatetheintegralinvolvingtheK-Besselfunctions,
and the Ramanujan identity
∞
σ (n)σ (n) ζ(s)ζ(s a)ζ(s b)ζ(s a b)
a b
= − − − −
ns ζ(2s a b)
n=1 − −
X
to see that
1 1 ξ(s 2σ +1)ξ(s 2it)ξ(s+2it)ξ(s+2σ 1)
t t
A(t)= H(s) − − − ds
2πiZℜ(s)=a |ξ(2s(t))|2 ξ(2s)
1
= B(s)ds,
|ξ(2s(t))|22πiZℜ(s)=a
8 YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
whereB(s)equalH(s)timestheξ-factors. Sinceξ(s)haspolesats=0,1thepoles
of B(s) in the region (s) 1/2 are at 1 2it, 2σ , 2 2σ , (2σ 1) and 2it.
t t t
ℜ ≥ ± − ± − ±
We now move the line of integration to (s) = 1/2. By considering the Stirling
ℜ
asymptotics for the Γ-factors, convexity bounds for the zeta functions, Eq. (2.8),
and the rapid decay of H(s) we see that B(s) decays rapidly in vertical strips and
this allows to move the line of integration. We find that
1
A(t)= res B(s)+ res B(s)
|ξ(2s(t))|2 (cid:18)s=1±2it s=2σt
1
+δ res B(s)+(1 δ ) res B(s)+ B(s)ds ,
t t
·s=2−2σt − ·s=2σt−1 2πiZℜ(s)=1/2 !
with δ =1 if σ <3/4 and 0 otherwise. We analyze these five terms.
t t
(i) Using Stirling, convexity bounds on the zeta functions. (2.8) and the rapid
decay of H(1 2it) the term
±
1 ξ(1 4it)ξ(1 2it 2σ +1)ξ(1 2it+2σ 1)
t t
res B(s)=H(1 2it) ± ± − ± −
ξ(2s(t))2 s=1±2it ± ξ(2s(t))2ξ(2 4it)
| | | | ±
tends to zero as t .
→∞
(ii) We now consider the second term:
1 ξ(4σ 1)
t
res B(s)=H(2σ ) − ,
ξ(2s(t))2 s=2σt t ξ(4σt)
| |
which convergesto H(2σ )ξ(4σ∞−1) when t and σ =1/2. When σ 1/2
∞ ξ(4σ∞) →∞ ∞ 6 t →
it behaves asymptotically like
1
H(1) .
4ξ(2)(σ 1/2)
t
−
(iii) We then move on to the third term:
1 ξ(1 4σ )ξ(2 2σ 2it)ξ(2 2σ +2it)
t t t
res B(s)=H(2 2σ ) − − − −
ξ(2s(t))2 s=2−2σt − t ξ(2s(t))2ξ(4 4σt)
| | | | −
ξ(1 4σ )
=H(2 2σ ) φ(s(t))2 − t .
t
− | | ξ(4 4σ )
t
−
If σ =1/2we canuse (2.14) to conclude that this tends to zero,andif σ =1/2
∞ ∞
6
it is bounded since φ(s(t))2 is bounded.
| |
(iv) The fourth term is
1
res B(s)=H(2σ 1) φ(s(t))2.
ξ(2s(t))2 s=2σt−1 t− | |
| |
When σ =1/2this tends to zero and when σ =1/2it is bounded, by the same
∞ ∞
6
arguments as for the third term.
(v) We now deal with the last term, i.e.
1 1
B(s)ds=
|ξ(2s(t))|22πiZℜ(s)=1/2 2π|ξ(2σt+2it)|2
∞
ξ(1/2+2σ 1+iτ)2ξ(1/2+i(τ 2t))ξ(1/2+i(τ +2t))
t
H(1/2+iτ)| − | − dτ.
× ξ(1+2iτ)
Z
−∞
LIMITS OF EISENSTEIN SERIES 9
WenotethatH(1/2+iτ)isofrapiddecay. Westudyfirsttheexponentialbehaviour
of the integral as a function of t. Stirling asymptotics for the integrand give:
(e−π|τ|/4)2e−π|τ/2−t|/2e−π|τ/2+t|/2(eπ|τ|/2) e−πt,
≤
which cancels with the exponential growth of 1/ ξ(2s(t))2. Using (2.8), the rapid
| |
decay of H(1/2+iτ), and any polynomial bound in τ of ζ(2σ 1/2+iτ), we are
t
−
reduced to estimate in t the integral
log t ∞
| | H˜(τ)(1+τ+2t)−1/4(1+τ 2t)−1/4 ζ(1/2+i(τ 2t))ζ(1/2+i(τ+2t)) dτ,
(t−1/2+σt)2 Z−∞ | | | − | | − |
whereH˜ is somefunctionofrapiddecay. We separatenowthe twocasesσ >1/2
∞
and σ = 1/2. In the first case we use the convexity bound on the ζ function to
∞
estimate the expression as
log t ∞
| | H˜(τ)(1+ τ +2t)ǫ(1+ τ 2t)ǫdτ =o(1),
(t−1/2+σt)2 Z−∞ | | | − |
as σ > 1/2. For the second case we can use any subconvex bound ζ(1/2+it)=
∞
O(t1/4−δ), for instance Weyl’s bound [20, Theorem 5.5]
| |
ζ(1/2+it) t1/6+ǫ.
≪| |
We are reduced to estimate in t the integral
log t ∞
| | H˜(τ)(1+ τ +2t)−1/4+1/4−δ(1+ τ 2t)−1/4+1/4−δdτ,
(t−1/2+σt)2 Z−∞ | | | − |
∞
=t1−2σt+ǫ H˜(τ)(1+ τ +2t)−δ(1+ τ 2t)−δdτ =o(1).
| | | − |
Z−∞
This concludes the evaluation of the non-constant terms in (2.13).
Tosummarizewehaveprovedthatifσ =1/2thefunction F (z)dµ (z)
∞ 6 Γ\H2 h s(t)
converges to
R
ξ(4σ 1)
∞
H(1 2σ )+H(2σ ) −
∞ ∞
− ξ(4σ )
∞
as t , and, if σ =1/2,
∞
→∞
H(1)
F (z)dµ (z) ,
ZΓ\H2 h s(t) ∼ 4ξ(2)((σt−1/2)
as t . This finishes the proof once we notice that
→∞
ξ(4σ 1)
∞
H(1 2σ )+H(2σ ) −
∞ ∞
− ξ(4σ )
∞
∞ dy
= h(y) y(2σ∞−1)+1+φ(2σ )y−2σ∞+1
∞ y2
Z0 (cid:16) (cid:17)
∞ 1 dy
= h(y) E(z,2σ )dx
∞ y2
Z0 (cid:18)Z0 (cid:19)
= F (z)E(z,2σ )dµ(z)
h ∞
ZΓ\H2
and
H(1)= F (z)dµ(z).
h
ZΓ\H2
(cid:3)
10 YIANNISN.PETRIDIS,NICOLERAULF,ANDMORTENS.RISAGER
It is straightforward to verify that Theorem 1.3, and Theorem 1.4 follow from
Lemma 2.1 and Lemma 2.2 using an approximation argument like in the proof of
[12, Proposition 2.3].
Remark 2.3. In [12] the point s = 1 was a double pole, coming from 2σ and
t
2 2σ with σ =1/2, and this explains the logarithm in (1.3).
t t
−
Proof of Theorem 1.1. We have
2
u 2dµ(z)= ( res φ(s))−1 res E(z,s) dµ(z)
| ρn| (cid:12) s=ρn s=ρn (cid:12)
(cid:12) (cid:12) 2
(cid:12) (cid:12)
=(cid:12)( res φ(s))−1 res φ(s)E((cid:12)z,1 s) dµ(z)
(cid:12) s=ρn s=ρn − (cid:12)
=(cid:12)(cid:12)E(z,1 ρn)2dµ(z). (cid:12)(cid:12)
(cid:12)| − | (cid:12)
The result now follows from Theorem 1.4 with σ =1 γ /2. (cid:3)
∞ ∞
−
Proof of Corollary 1.5. Let f be a test function for the convergence in Corollary
1.5. Then we use
f(z)
E(z,2σ )
∞
as test function for Theorem 1.4 to deduce that, as t
→∞
f(z) f(z)
dµ E(z,2σ )dµ(z)= f(z)dµ(z).
ZΓ\H2 E(z,2σ∞) s(t) →ZΓ\H2 E(z,2σ∞) ∞ ZΓ\H2
Finally one uses the approximation argument in [12, Proposition 2.3] to complete
the proof.
(cid:3)
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