fourier coefficients of harmonic weak maass ...masri/partition-7-14.pdfis a heegner point on the...
TRANSCRIPT
FOURIER COEFFICIENTS OF HARMONIC WEAK MAASS FORMSAND THE PARTITION FUNCTION
RIAD MASRI
Abstract. In a recent paper, Bruinier and Ono proved that certain harmonic weak Maassforms have the property that the Fourier coefficients of their holomorphic parts are algebraictraces of weak Maass forms evaluated on Heegner points. As a special case they obtained aremarkable finite algebraic formula for the Hardy-Ramanujan partition function p(n), whichcounts the number of partitions of a positive integer n. We establish an asymptotic formulawith a power saving error term for the Fourier coefficients in the Bruinier-Ono formula. Asa consequence, we obtain a new asymptotic formula for p(n). One interesting feature ofthis formula is that the main term contains essentially 3 · h(−24n + 1) fewer terms thanthe truncated main term in Rademacher’s exact formula for p(n), where h(−24n+ 1) is theclass number of the imaginary quadratic field Q(
√−24n+ 1).
1. Introduction and statement of results
During the last ten years there have been remarkable advances in the study of q-series,modular forms, and L–functions through their connection to harmonic weak Maass forms(see for example the excellent survey articles of Ono [O] and Zagier [Za]). Roughly speaking,a harmonic weak Maass form of weight k ∈ 1
2Z is a real-analytic function on the complex
upper half-plane H which transforms like a modular form with respect to some congruencesubgroup of SL2(Z), vanishes under the weight k hyperbolic Laplacian, and has at mostlinear exponential growth in the cusps of the group. A detailed discussion of harmonic weakMaass forms can be found in the fundamental paper of Bruinier and Funke [BF] (see alsoSection 3).
A harmonic weak Maass form has a Fourier expansion consisting of a non-holomorphicand holomorphic part. There is a great amount of interest in understanding the arithmeticmeaning of these Fourier coefficients. For example, Bruinier and Ono [BO] related the Fouriercoefficients of the non-holomorphic and holomorphic parts of weight 1/2 harmonic weakMaass forms to central values and central derivatives of modular L-functions, respectively.Bruinier [B2] has since related the Fourier coefficients of the holomorphic parts of weight1/2 harmonic weak Maass forms to periods of algebraic differentials of the third kind onmodular and elliptic curves.
In a recent paper, Bruinier and Ono [BO2] constructed a theta lift from the space ofharmonic weak Maass forms of weight -2 to the space of vector-valued harmonic weak Maassforms of weight -1/2. They used this lift to prove that certain vector-valued harmonicweak Maass forms of weight -1/2 have the property that the Fourier coefficients of theirholomorphic parts are algebraic traces of fixed, weight 0 weak Maass forms evaluated onHeegner points. As a special case they established a finite algebraic formula for the Hardy-Ramanujan partition function p(n), which counts the number of partitions of a positiveinteger n. In this paper we will establish an asymptotic formula with a power saving errorterm for the Fourier coefficients in the Bruinier-Ono formula. As a consequence, we will
1
2 RIAD MASRI
obtain a new asymptotic formula for p(n) (see Theorem 1.6). One interesting feature ofthis formula is that the main term contains essentially 3 · h(−24n + 1) fewer terms thanthe truncated main term in Rademacher’s exact formula for p(n), where h(−24n+ 1) is theclass number of the imaginary quadratic field Q(
√−24n+ 1) (see the discussion following
Theorem 1.6).In order to state our results we briefly review the Bruinier-Ono formula. Bruinier and Ono
[BO2, Section 3.1] defined a theta lift
Λ : H−2(N)→ H− 12,ρ
from the space H−2(N) of harmonic weak Maass forms of weight −2 and level N to the spaceH− 1
2,ρ of vector-valued harmonic weak Maass forms of weight −1/2 and level N with Weil
representation ρ (see Sections 3 and 4 for the precise definitions). The image of f ∈ H−2(N)under the theta lift Λ has a Fourier expansion
Λ(f, w) = Λ−(f, w) + Λ+(f, w), w = u+ iv ∈ Hwith non-holomorphic part
Λ−(f, w) :=∑
h mod 2N
∑D∈ZD<0
c−Λf (D, h)Γ(1− k, π|D|v/N)qD/4Neh, q := e(w) = e2πiw
and holomorphic part
Λ+(f, w) :=∑
h mod 2N
N∞∑D=1
c+Λf
(−D, h)q−D/4Neh +∑
h mod 2N
∞∑D=0
c+Λf
(D, h)qD/4Neh
where N∞ ≥ 0 is an integer, Γ(a, t) is the incomplete Gamma function, and {eh} is thestandard basis for the group algebra C[Z/2NZ]. Note that c±Λf (D, h) = 0 unless D ≡ h2
mod 4N .Assume now that N is squarefree and
(H) −D < −4 is an odd fundamental discriminant coprime to N such that every primedivisor of N splits in K = Q(
√−D).
Fix a solution h ∈ Z/2NZ of h2 ≡ −D mod 4N , and for a primitive integral ideal A of Kwrite
A = Za+ Zb+√−D
2, a = NK/Q(A), b ∈ Z
with b ≡ h mod 2N and b2 ≡ −D mod 4Na. Then
τ(h)A =
b+√−D
2Na
is a Heegner point on the modular curve X0(N). It is known that τ(h)A depends only on the
ideal class of A and on h mod 2N , so we denote it by τ(h)[A] . For details concerning these
facts, see [GZ, part II, Section 1].Let CL(K) be the ideal class group of K and h(−D) = #CL(K) be the class number.
By Minkowski’s theorem we may choose a primitive integral ideal A in each ideal class[A] ∈ CL(K) such that
NK/Q(A) ≤ 2
π
√D.
3
Having fixed such a choice A for each ideal class, we define
OD,N,h := {τ (h)[A] : [A] ∈ CL(K)}.
Let Γτ be the image of the stabilizer of τ = τ(h)[A] in PSL2(Z).
The Maass level-raising operator
R−2 := 2i∂
∂z− 2Im(z)−1, z ∈ H
maps weak Maass forms of weight -2 to weak Maass forms of weight 0. Define the operator
∂ :=1
4πR−2.
Bruinier and Ono [BO2, Theorem 3.6] established a formula for the Fourier coefficientsof holomorphic parts of weight -1/2 vector-valued harmonic weak Maass forms in the imageΛ(H−2(N)). With the setup in this paper, their formula can be stated as follows.
Theorem 1.1 (Bruinier-Ono). For each f ∈ H−2(N), the (D, h)-th Fourier coefficient ofthe holomorphic part of Λ(f, w) ∈ H− 1
2,ρ is given by
c+Λf
(D, h) = −4N
D
∑τ∈OD,N,h
∂f(τ)
#Γτ. (1.1)
Let M !−2(N) ⊂ H−2(N) be the space of weakly holomorphic modular forms of weight -2
and level N , and M!− 1
2,ρ⊂ H− 1
2,ρ be the space of vector-valued weakly holomorphic modular
forms of weight -1/2 and level N with Weil representation ρ (see Section 3). Each formf ∈M !
−2(N) has a Fourier expansion
f(z) =N∞∑n=1
cf (−n)q−n +∞∑n=0
cf (n)qn.
Bruinier and Ono [BO2, Theorem 3.5] proved that the theta lift Λ restricts to a map
Λ : M !−2(N)→M!
− 12,ρ.
We will establish the following asymptotic formula with a power saving error term for theFourier coefficients of holomorphic parts of weight -1/2 vector-valued weakly holomorphicmodular forms in the image Λ(M !
−2(N)).
Theorem 1.2. For each f ∈ M !−2(N), the (D, h)-th Fourier coefficient of the holomorphic
part of Λ(f, w) ∈M!− 1
2,ρ
satisfies
c+Λf
(D, h) =1
D
∑τ∈OD,N,h
Im(τ)> 4Nπ
+D−1
176
Mf,N(τ) + Cf,Nh(−D)
D+Oε,N(D−
89176
+ε)
as D →∞ through all D satisfying the hypothesis (H). Here
Mf,N(z) := −4NN∞∑n=1
cf (−n)n
(1− 1
2πnIm(z)
)e(−nz)
4 RIAD MASRI
and
Cf,N := −Nπ
∫reg
R−2f(z)dµ(z)
is a Borcherds-type regularized integral where dµ(z) is the normalized hyperbolic measure onΓ0(N)\H (see Section 13).
Remark 1.3. We briefly explain how Theorem 1.2 can be generalized to give an asymptoticformula for the Fourier coefficients of holomorphic parts of weight -1/2 vector-valued har-monic weak Maass forms in the image Λ(H−2(N)). A key step in the proof of Theorem 1.2is to express the image ∂(f) of a weakly holomorphic modular form f ∈M !
−2(N) under theoperator ∂ = R−2/4π as a finite linear combination of certain weight 0 Maass-Poincare series(see Section 5). By (1.1) it follows that the asymptotic distribution of c+
Λf(D, h) is deter-
mined by the asymptotic distribution of traces of these Maass-Poincare series. Using [BO2,Corollary 3.4] and [BO2, Proposition 2.2], one can obtain a similar (but more complicated)expression for the image ∂(f) of any harmonic weak Maass form f ∈ H−2(N). Theorem 1.2can then be generalized by using this expression and modifying the proof accordingly. Wefocus here on the case f ∈ M !
−2(N) since this is what is needed for our application to thepartition function.
Remark 1.4. Let `|N with (`,N/`) = 1, and let f 7→ f |−2WN` be the Atkin-Lehner involu-
tion of M !−2(N) defined in [BO2, Section 4.3]. Then assuming that f |−2W
N` has coefficients
in Z for every such `, Bruinier and Ono [BO2, Theorem 4.5] proved that 6D · ∂f(τ) is analgebraic integer in the ring class field for the order OD ⊂ Q(
√−D).
Remark 1.5. The Fourier coefficients c+Λf
(D, h) are likely related to periods of differentials
of the third kind in a manner similar to [B2].
We now discuss our application to the partition function p(n). Define the weakly holo-morphic modular form fp ∈M !
−2(6) by
fp(z) :=1
2
E2(z)− 2E2(2z)− 3E2(3z) + 6E2(6z)
η(z)2η(2z)2η(3z)2η(6z)2= q−1 − 10− 29q − · · · (1.2)
where
η(z) := q1/24
∞∏n=1
(1− qn)
is the Dedekind eta function and
E2(z) := 1− 24∞∑n=1
(∑d|n
d)qn
is the usual weight 2 Eisenstein series. Bruinier and Ono [BO2, Section 3.2] proved that
p(n) = − 1
24c+
Λfp(24n− 1, 1). (1.3)
By combining (1.3) with (1.1), they obtained the following formula for p(n) (see [BO2,Theorem 1.1])
p(n) =1
24n− 1
∑τ∈Q24n−1,6,1
∂fp(τ).
5
Bruinier and Ono [BO2, Theorem 4.2] also proved that fp(z) satisfies the assumption on theAtkin-Lehner involutions in Remark 1.4, and hence 6(24n− 1)∂fp(τ) is an algebraic integerin the ring class field of discriminant −24n + 1. Note that Larson and Rolen [LR] latershowed this result holds without the 6, answering a question of Bruinier and Ono [BO2,Section 5, (1)].
By combining Theorem 1.2 with (1.3), we will obtain the following new asymptotic formulafor p(n).
Theorem 1.6. Let n be a positive integer with 24n− 1 squarefree. Then
p(n) =1
24n− 1
∑τ∈O24n−1,6,1
Im(τ)> 24π
+(24n−1)−1
176
(1− 1
2πIm(τ)
)e(−τ) + C∗fp,6
h(−24n+ 1)
24n− 1+Oε(n
− 89176
+ε)
as n→∞ where
C∗fp,6 :=1
4π
∫reg
R−2fp(z)dµ(z).
Remark 1.7. If n is a positive integer with 24n − 1 squarefree, then −24n + 1 is an oddfundamental discriminant which satisfies the hypothesis (H) for N = 6.
The asymptotic distribution of p(n) has been studied extensively since the early part ofthe 20th century. Hardy and Ramanujan [HR] invented the “circle method” and used it toestablish the well-known asymptotic
p(n) ∼ 1
4√
3nexp
(2π
√n
6
).
Rademacher [R] later used a refinement of the circle method to establish the exact formula
p(n) =1
2π√
2
∞∑c=1
√cAc(n)
d
dn
exp
(πλn√
2/3
c
)λn
.
Here λn :=√n− 1/24 and Ac(n) is the exponential sum
Ac(n) :=∑
b mod c(b,c)=1
ωb,ce
(−nbc
),
where ωb,c is a certain root of unity. By truncating the series at N := b√n/6c and estimating
the remainder, Rademacher [R2] obtained the asymptotic formula
p(n) = MT(n) +O(n−38 ) (1.4)
with main term
MT(n) :=1
2π√
2
N∑c=1
√cAc(n)
d
dn
exp
(πλn√
2/3
c
)λn
.
Lehmer [L] improved the error term in (1.4) to O(n−12
+ε). Using an arithmetic reformulationof Rademacher’s exact formula due to Bringmann and Ono [BrO], the author and Folsom
6 RIAD MASRI
[FM] established an asymptotic formula for p(n) with an error term which is O(n−δ) forsome absolute δ > 1/2.
We now compare the main term in Theorem 1.6 with the main term MT(n) in Rademacher’sasymptotic formula (1.4), which reveals some interesting new features in the asymptotic dis-tribution of p(n). Suppose that n 6= 6`2 for any ` ∈ Z. Then using an analysis withexponential sums (see [FM, Proposition 6.1]), one can show that
MT(n) =1
24n− 1
∑τ∈Λ24n−1,6
Im(τ)>1
χ12(τ)
(1− 1
2πIm(τ)
)e(−τ) (1.5)
where Λ24n−1,6 is the set of Heegner points of discriminant −24n+ 1 on X0(6) and χ12(τ) =( 12−b) where −b is the middle coefficient of the quadratic form Q(X, Y ) = 6aX2 − bXY +
cY 2 corresponding to the Heegner point τ . Let H denote the Hilbert class field of K =Q(√−24n+ 1). The set of Heegner points decomposes as (see e.g. [GKZ, p. 507] and [GZ,
pp. 235-236])
Λ24n−1,6 =⋃
h mod 12h2≡−24n+1 mod 24
O24n−1,6,h (1.6)
where each set O24n−1,6,h is a simple, transitive Gal(H/K)-orbit. Now, recall that the mainterm in Theorem 1.6 contains no character twist and is summed over one Galois orbit{τ ∈ O24n−1,6,1} subject to Im(τ) > 24/π + (24n − 1)−1/176. On the other hand, from (1.5)we see that MT(n) is summed over all Heegner points {τ ∈ Λ24n−1,6} subject to Im(τ) > 1.The number of elements in each Galois orbit is h(−24n+ 1) since Gal(H/K) ∼= CL(K), andthere are 4 orbits corresponding to the solutions h = 1, 5, 7 and 11 of the congruence in (1.6).Hence the main term in Theorem 1.6 contains essentially 3 · h(−24n + 1) fewer terms thanthe main term MT(n). It would be very interesting to know whether the shorter main termin Theorem 1.6, or the form of the main term itself, offers any computational advantages.
Acknowledgments. I would like to thank Jan Bruinier, Sheng-Chi Liu and Matt Youngfor some helpful discussions/correspondence regarding this work. I would also like to thankthe referees for many suggestions which improved the exposition of the paper. The authorwas partially supported by the NSF grant DMS-1162535 during the preparation of this work.
2. Outline of the proof of Theorem 1.2
In this section we give a brief outline of the proof of Theorem 1.2. We want to establishan asymptotic formula as D →∞ for the trace∑
τ∈OD,N,h
∂f(τ)
#Γτ
appearing in the Bruinier-Ono formula (1.1). The image of a weakly holomorphic modularform f ∈ M !
−2(N) under the operator ∂ = R−2/4π is a weight 0 weak Maass form onΓ0(N). As a consequence, the function ∂(f) can be expressed as a finite linear combinationof certain Maass-Poincare series {FN,n}N∞n=1, so it suffices to study the trace of FN,n. Wewill compute the Fourier expansion of FN,n in the cusps of Γ0(N) and find that it has apart with linear exponential growth and a part with polynomial growth. Now, the Galoisorbit of Heegner points OD,N,h becomes (quantitatively) equidistributed with respect to the
7
normalized hyperbolic measure on Γ0(N)\H as D → ∞. However, we cannot directly usethis fact to obtain an asymptotic formula for the trace of FN,n because the “test function”FN,n grows very rapidly in the cusps and hence is not admissible. We will overcome thisdifficulty using two different regularizations. We first prove an equidistribution theorem forGalois orbits of Heegner points in which the test functions are allowed to grow polynomiallyin the cusps (see Theorem 6.3). We then construct for each η > 0 a certain smooth Poincareseries Pn,η which regularizes the linear exponential growth of FN,n in the cusps of Γ0(N).This is inspired by a construction of Duke [D] to regularize the pole at ∞ of the j-function.Upon substituting the regularized function FN,n−Pn,η into the equidistribution theorem, wewill obtain a “smooth” asymptotic formula for the trace of FN,η. Finally, using a Borcherds-type integration along with a delicate analysis to un-smooth the main term and bound (as afunction of η) the Sobolev norm of FN,n −Pn,η appearing in the error term, we will obtainthe desired asymptotic formula for the trace of FN,n (see Theorem 9.1). Note that our proofhas some elements in common with [FM2], though considerable new difficulties arise becauseof the presence of level in all of our arguments (which is crucial for our application to thepartition function).
3. harmonic weak Maass forms
In this section we review some facts concerning harmonic weak Maass forms. For moredetails, see [BF]. Let k ∈ 1
2Z and z = x + iy ∈ H. The weight k hyperbolic Laplacian is
defined by
∆k := −y2(∂2x + ∂2
y
)+ iky (∂x + i∂y) .
Let N be a positive integer. A weak Maass form of weight k on Γ0(N) is a smooth functionf : H→ C satisfying:
(1) f |kM = f for all M ∈ Γ0(N).(2) There is a complex number λ such that ∆kf = λf .(3) There is a constant C > 0 such that f(z) = O(eCy) as y → ∞. An analogous
condition is required at all cusps.
Note. The slash operator |k is defined as in Shimura’s theory of half-integral weight forms.A weak Maass form is harmonic if ∆kf = 0. Every harmonic weak Maass form has a
Fourier expansion
f(z) = f−(z) + f+(z)
with non-holomorphic part
f−(z) :=∑n<0
c−f (n)Γ(1− k, 4π |n| y)qn
and holomorphic part
f+(z) :=N∞∑n=1
c+f (−n)q−n +
∞∑n=0
c+f (n)qn,
where N∞ ≥ 0 is an integer and Γ(a, t) is the incomplete Gamma function. A harmonicweak Maass form with trivial non-holomorphic part is called a weakly holomorphic modularform. Let
M !k(N) ⊂ Hk(N)
8 RIAD MASRI
denote the spaces of weakly holomorphic modular forms and harmonic weak Maass forms,respectively.
We also require the notion of a vector-valued weak Maass form. Let w = u + iv ∈ H,and let Mp2(R) be the metaplectic two-fold cover of SL2(R) realized as the group of pairs(M,φ(w)) where M ∈ SL2(R) and φ : H→ C is a holomorphic function with φ(w)2 = cw+d.The multiplication is defined by
(M,φ(w))(M ′, φ′(w)) = (MM ′, φ(M ′w)φ′(w)).
Let Γ = Mp2(Z) be the inverse image of SL2(Z) under the covering map. The group Γ isgenerated by T = (( 1 1
0 1 ), 1) and S = (( 0 −11 0 ),
√w). Given h ∈ Z/2NZ let eh be the corre-
sponding standard basis vector for the group algebra C[Z/2NZ]. The Weil representation ρ
is the unitary representation of Γ on C[Z/2NZ] defined in terms of the generators T and S
of Γ by
ρ(T )(eh) = e
(h2
4N
)eh,
ρ(S)(eh) =1√2iN
∑h′ mod 2N
e
(−hh
′
2N
)eh′ .
A vector-valued weak Maass form of weight k (with respect to Γ and ρ) is a smoothfunction g : H→ C[Z/2NZ] satisfying:
(1) g(Mw) = φ(w)2kρ(M,φ)g(w) for all (M,φ) ∈ Γ.(2) There is a complex number λ such that ∆kg = λg.(3) There is a constant C > 0 such that g(w) = O(eCv) as v → ∞. An analogous
condition is required at all cusps.
A vector-valued weak Maass form is harmonic if ∆kg = 0. Every vector-valued harmonicweak Maass form has a Fourier expansion
g(w) = g−(w) + g+(w)
with non-holomorphic part
g−(w) :=∑
h mod 2N
∑D<0
c−g (D, h)Γ(1− k, π|D|v/N)qD/4Neh
and holomorphic part
g+(w) :=∑
h mod 2N
N∞∑D=1
c+g (−D, h)q−D/4Neh +
∑h mod 2N
∞∑D=0
c+g (D, h)qD/4Neh.
Note that c±g (D, h) = 0 unless D ≡ h2 mod 4N . Let
M!k,ρ ⊂ Hk,ρ
denote the spaces of vector-valued weakly holomorphic modular forms and vector-valuedharmonic weak Maass forms, respectively.
9
4. The Bruinier-Ono theta lift
Let k ∈ 12Z and z ∈ H. The Maass level raising and lowering differential operators are
defined by
Rk,z := 2i∂
∂z+ kIm(z)−1,
Lk,z := −2iIm(z)2 ∂
∂z.
Bruinier and Ono (see Sections 2.3 and 3.1, and Corollary 3.4 of [BO2]) defined a theta lift
Λ : H−2(N)→ H− 12,ρ
by
Λ(f, w) := L3/2,w
∫Γ0(N)\H
(R−2,zf(z))Θ(w, z, φKM),
where Θ(w, z, φKM) is a C[Z/2NZ]-valued theta function constructed from the Kudla-MillsonSchwartz function φKM (see [KM] and [BO2, Section 2.3]). As a function of z, the thetakernel Θ(w, z, φKM) is a Γ0(N)-invariant harmonic (1, 1)-form on H. When restricting Λ toweakly holomorphic modular forms, one has (see [BO2, Theorem 3.5])
Λ : M !−2(N)→M!
− 12,ρ.
5. Differential operators and weak Maass forms
Define the operator
∂−2 :=1
2πi
∂
∂z+
1
2πy.
Then ∂−2 maps weak Maass forms of weight -2 on Γ0(N) to weak Maass forms of weight 0on Γ0(N). It turns out that the image ∂−2(f) of a harmonic weak Maass form f ∈ H−2(N)under the operator ∂−2 can be expressed as a finite linear combination of certain Maass-Poincare series. Let m ∈ Z+, and define the Maass-Poincare series (see e.g. [B, Section1.3])
FN,m(z) := πm∑
γ∈Γ∞\Γ0(N)
Im(mγz)1/2I3/2(2πmIm(γz))e(−mRe(γz)), z = x+ iy ∈ H
where I3/2 is the I-Bessel function of order 3/2. It is known that FN,m is a weak Maass formof weight 0 on Γ0(N). Now, suppose that f ∈M !
−2(N) has the Fourier expansion
f(z) =N∞∑n=1
cf (−n)q−n +∞∑n=0
cf (n)e(nz).
Then by [MP, Theorem 1.2],
∂−2(f) = −2N∞∑n=1
cf (−n)FN,n.
10 RIAD MASRI
In particular, the identity
1
4πR−2 = −∂−2
implies that
1
4πR−2(f) = 2
N∞∑n=1
cf (−n)FN,n. (5.1)
As explained in Remark 1.3, a similar (but more complicated) expression exists for anyharmonic weak Maass form f ∈ H−2(N), but we will not need this here.
6. quantitative equidistribution
Let f1, f2 : H→ C be Γ0(N)-invariant functions and F(N) be a fundamental domain forΓ0(N). Define the Petersson inner product
〈f1, f2〉2 :=
∫F(N)
f1(z)f2(z)dµ(z)
where
dµ(z) :=1
vol(F(N))
dxdy
y2
is the normalized hyperbolic measure on F(N). Let
∆ := −y2(∂2x + ∂2
y)
be the weight 0 hyperbolic Laplacian and ∆A be the composition of ∆ with itself A-timeswhere A ∈ Z≥0. For a cusp a of Γ0(N), let σa ∈ SL2(R) be a scaling matrix such thatσa(∞) = a (see [I, p. 47]).
Proposition 6.1. Let g : H→ C be a C∞, Γ0(N)-invariant function, and suppose that foreach cusp a we have
∆Ag(σaz) = O(e−Cy), A = 0, 1, 2, . . . (6.1)
for some constant C > 0 (depending on a and A). Then∑τ∈OD,N,h
g(τ) = h(−D)
∫F(N)
g(z)dµ(z) +Oε,N(||∆2g||2D12− 1
16+ε).
Proof. Let {um} be an orthonormal basis of Hecke-Maass cusp forms of weight 0 for Γ0(N)with ∆-eigenvalues λm = 1
4+ t2m for m ∈ Z+. For each cusp a define the real-analytic
Eisenstein series
Ea(z, s) :=∑
γ∈Γa\Γ0(N)
Im(σ−1a γz)s, z ∈ H, Re(s) > 1.
Because g satisfies the bound (6.1) we have the spectral expansion
g(z) = 〈g, 1〉2 +∞∑m=1
〈g, um〉2um(z) +∑a
1
4π
∫R〈g, Ea(·, 1
2+ it)〉2Ea(z,
12
+ it)dt,
11
which converges pointwise absolutely and uniformly on compact subsets of F(N). Summingover the spectral expansion yields∑τ∈OD,N,h
g(τ) = h(−D)〈g, 1〉2 +∞∑m=1
〈g, um〉2Wm(D) +∑a
1
4π
∫R〈g, Ea(·, 1
2+ it)〉2Wa(D, t)dt,
(6.2)
where the hyperbolic Weyl sums are defined by
Wm(D) :=∑
τ∈OD,N,h
um(τ) and Wa(D, t) :=∑
τ∈OD,N,h
Ea(τ,12
+ it).
To estimate the contribution of the discrete spectrum, it suffices to consider only L2-normalized Hecke-Maass newforms for Γ0(N) (see e.g. the proof of [HM, Theorem 6]). Bya formula of Waldspurger/Zhang (see e.g. [W, W2] and [Z]), for such a newform um one has∣∣∣∣∣∣
∑τ∈OD,N,h
um(τ)
∣∣∣∣∣∣2
= cD√D |am(1)|2 Λ(um,
12)Λ(um × χ−D, 1
2), (6.3)
where cD is a positive constant which takes only finitely many different values, am(1) is thefirst Fourier coefficient of um, and
Λ(Π, s) := L∞(Π, s)L(Π, s)
is the completed L–function. Note that the term |am(1)|2 appears on the right hand sidebecause in the Waldspurger/Zhang formula the Hecke-Maass newform is arithmetically nor-malized.
By a subconvexity bound of Jutila and Motohashi [JM] we have
L(um,12)�ε,N λ
16
+εm ,
and by a hybrid subconvexity bound of Blomer and Harcos [BH, Theorem 2] we have
L(um × χ−D, 12)�ε,N λ
74
+εm D
12− 1
8+ε.
Then using the following estimate of Hoffstein and Lockhart [HL]
|am(1)|2 �ε,N λεmeπ|t|,
and the fact that the contribution from the infinite parts of the L–functions in (6.3) is� e−π|tm| by Stirling’s formula, we obtain the estimate
Wm(D)�ε,N λ2324
+εm D
12− 1
16+ε. (6.4)
Following the argument in [HM, section 6.4], one can reduce the estimate of Wa(D, t) toan analogous estimate for ∑
τ∈OD,1,1
E(τ, 12
+ it)
where E(z, s) is the real-analytic Eisenstein series for SL2(Z). One has the identity (see e.g.[GZ, p. 248]) ∑
τ∈OD,1,1
E(τ, s) = 2−sDs/2ζ(2s)−1ζ(s)L(χ−D, s).
12 RIAD MASRI
Then using a standard lower bound for ζ(2s), the subconvexity bound
ζ(12
+ it)� (14
+ t2)112
+ε,
and the following hybrid subconvexity bound of Heath-Brown [HB]
L(χ−D,12
+ it)�ε (14
+ t2)332
+εD14− 1
16+ε,
we obtain
Wa(D, t)�ε (14
+ t2)1796
+εD12− 1
16+ε. (6.5)
Using the bound (6.1), a repeated application of Stokes’ theorem (see e.g. [I, Lemma1.18]) yields the identities
〈g, um〉2 = λ−Am 〈∆Ag, um〉2and
〈g, Ea(·, 12
+ it)〉2 = (14
+ t2)−A〈∆Ag, Ea(·, 12
+ it)〉2.
By Parseval’s formula (see [IK, (15.17)]),
∞∑m=1
|〈∆Ag, um〉2|2 +∑a
1
4π
∫R|〈∆Ag, Ea(·, 1
2+ it)〉2|2dt = ||∆Ag||22.
Then by the Cauchy-Schwarz inequality we have∞∑m=1
|〈g, um〉2|λ2324
+εm � ||∆2g||2 (6.6)
and ∫R|〈g, Ea(·, 1
2+ it)〉2|(1
4+ t2)
1796
+εdt� ||∆2g||2. (6.7)
Finally, the proposition follows by combining (6.2) with the estimates (6.4)–(6.5) and (6.6)–(6.7). �
Definition 6.2. Let g : H→ C be a C∞, Γ0(N)-invariant function and α be a real number.We say that g has cuspidal growth of power α if for each cusp a of Γ0(N) there exists aconstant ca ∈ C such that
∆A(g(σaz)− cayα) = O(e−Cy), A = 0, 1, 2, . . .
for some constant C > 0 (depending on a and A).
We now combine Proposition 6.1 with a suitable regularization to establish the following
Theorem 6.3. Suppose that F has cuspidal growth of power α < 5/8. Then∑τ∈OD,N,h
F (τ) = h(−D)
∫F(N)
F (z)dµ(z)
+Oε,N(||∆2FT0 ||2D12− 1
16+ε) +Oε,N(D
12−c(α)+ε) +ON(D
12− (1−α)
2+ε),
13
where FT0 is a regularized version of F for a certain constant T0 > 1 independent of D (see(6.9)) and
c(α) :=
{116, α ≤ 1
2116− (α
2− 1
4), 1
2< α < 5
8.
Proof. For T > 1, define
ψT (t) := tαχ(t/T )
where χ : R+ → [0, 1] is a C∞ function such that
χ(t) =
{0, t < 1
1, t > 2.
Define the incomplete Eisenstein series
Eb(ψT |z) :=∑
γ∈Γb\Γ0(N)
ψT(Im(σ−1
b γz))
and let
ηT (z) :=∑b
cbEb(ψT |z).
Then by [I, (3.17)] and (8.2) we have
ηT (σaz) =
0, 1 < y < T
cayαχ(y/T ), T ≤ y ≤ 2T
cayα, y > 2T.
It follows that for y > 2T the regularized function
FT (z) := F (z)− ηT (z)
satisfies the bound (6.1).Now, let
T = TD := 1 + max
{√D
2N,4N
π
}.
Then by Lemma 6.4 we have ηT (τ) = 0 for all τ ∈ OD,N,h, and thus∑τ∈OD,N,h
F (τ) =∑
τ∈OD,N,h
FT (τ). (6.8)
Let T0 be a constant independent of D with T > T0 > 1 and decompose the regularizedfunction as
FT (z) = FT0(z) + ηT (z) (6.9)
where
ηT (z) := ηT0(z)− ηT (z).
14 RIAD MASRI
Since FT0 satisfies the bound (6.1), it follows from Proposition 6.1 that∑τ∈OD,N,h
FT0(τ) = h(−D)
∫F(N)
FT0(z)dµ(z) +Oε,N(||∆2FT0||2D12− 1
16+ε).
Then by (6.8)–(6.9), to complete the proof it suffices to show that∑τ∈OD,N,h
ηT (τ) = h(−D)
∫F(N)
ηT0(z)dµ(z) +Oε,N(D12−c(α)+ε) +ON(D
12− (1−α)
2+ε).
By [I, Theorem 11.3 and (7.12)–(7.13)] we have
ηT (z) = 〈ηT , 1〉2 +1
2πi
∑b
cb
∫R
(ψT0(
12
+ it)− ψT (12
+ it))Eb(z,
12
+ it)dt,
where
ψ(t) :=
∫ ∞0
ψ(t)t−(s+1)dt.
It follows that∑τ∈OD,N,h
ηT (τ) = h(−D)
∫F(N)
ηT0(z)dµ(z)− h(−D)
∫F(N)
ηT (z)dµ(z) + E(D,T ),
where
E(D,T ) :=1
2πi
∑b
cb
∫R
(ψT0(
12
+ it)− ψT (12
+ it))Wb(D, t)dt.
By [KMY, Lemma 5.6], for all B > 0 we have∫R|ψT0(1
2+ it)− ψT (1
2+ it)|(1 + |t|)Bdt� C(α, T ), (6.10)
where
C(α, T ) :=
{log(T ), α ≤ 1
2
Tα−12 , α > 1
2.
Since T < (1 + 4Nπ
)√D, we combine (6.5) and (6.10) to obtain
E(D,T ) =
{Oε,N(D
12− 1
16+ε), α ≤ 1
2
Oε,N(D12−( 1
16−(α
2− 1
4))+ε), 1
2< α < 5
8.
Finally, a straightforward estimate yields
h(−D)
∫F(N)
ηT (z)dµ(z) = O(h(−D)Tα−1) = Oε,N(D12− (1−α)
2+ε),
where we used
h(−D)�ε D12
+ε.
�
15
Lemma 6.4. For γ ∈ Γa\Γ0(N) and τ ∈ OD,N,h we have
Im(σ−1a γτ) ≤ max
{Im(τ),
4N
π
}≤ max
{√D
2N,4N
π
}.
Proof. Recall that we chose OD,N,h so that a Heegner point τ ∈ OD,N,h has the form
τ = τ(h)A =
b+√−D
2Na
with
a ≤ 2
π
√D. (6.11)
Write σ−1a γ =
(a′ b′
c′ d′
)∈ SL2(R). Then we have
Im(σ−1a γτ) =
Im(τ)
|c′τ + d′|2=
1
|c′τ + d′|2
√D
2Na.
If c′ = 0, then d′ = 1 (see [I, (2.15)-(2.17)]) so that
Im(σ−1a γτ) ≤
√D
2N
(recall that a = NK/Q(A) ≥ 1). On the other hand, if c′ 6= 0 then by (8.2) we have (c′)2 ≥ 1so that
|c′τ + d′|2 =
(c′b
2Na+ d′
)2
+
(c′√D
2Na
)2
≥ D
4N2a2.
Then using (6.11) we obtain
Im(σ−1a γτ) ≤ 4N2a2
D
√D
2Na=
2Na√D≤ 4N
π.
�
7. Fourier expansion of FN,m(z)
Write
FN,m(z) =∑
γ∈Γ∞\Γ0(N)
pm(γz)
where
pm(z) := ψm(Im(z))e(−mz)
and
ψm(t) := πm√mtI3/2(2πmt)e(mit), t ∈ R+.
Then by [I, p. 60] the Fourier expansion of FN,m in the cusp b is given by
FN,m(σbz) = δ∞,bψm(y)e(−mz) +∑n∈Z
e(nx)∑c∈R+
S∞,b(−m,n; c)Am(n, c, y),
16 RIAD MASRI
where S∞,b(−m,n; c) is the Kloosterman sum
S∞,b(−m,n; c) :=∑
a ∗c d
∈B\σ−1∞ Γ0(N)σb/B
e
(−md+ na
c
)
with group of integral translations
B :=
{(1 b
1
): b ∈ Z
}and
Am(n, c, y) :=
∫Rψm
(y
c2(t2 + y2)
)e
(m
c2(t+ iy)− nt
)dt.
In Lemma 7.1 we will evaluate the integral Am(n, c, y). Then inserting this evaluation intothe Fourier expansion yields
FN,m(σbz) = δ∞,bπm3/2√yI3/2(2πmy)e(−mx) + Cb(m)y−1
+∑n∈Zn6=0
Cb(m,n)√yK3/2(2π |n| y)e(nx)
where
Cb(m) :=2π3m3
3
∑c∈R+
S∞,b(−m, 0; c)
c4
and
Cb(m,n) :=
2πm3/2
∑c∈R+
S∞,b(−m,n; c)
cJ3
(4π√m |n|c
), n < 0
2πm3/2∑c∈R+
S∞,b(−m,n; c)
cI3
(4π√mn
c
), n > 0.
Finally, using the identities
√tI3/2(t) =
1√2π
(et(
1− 1
t
)+ e−t
(1 +
1
t
))and
√tK3/2(t) =
√π
2e−t(
1 +1
t
),
we obtain
FN,m(σbz) = Cb(m)y−1 + δ∞,bm
2
(1− 1
2πmy
)e(−mz) + Eb(m,x, y) (7.1)
17
where
Eb(m,x, y) := δ∞,bm
2e−2πmy
(1 +
1
2πmy
)e(−mx)
+1
2
∑n∈Zn 6=0
Cb(m,n) |n|−1/2 e−2π|n|y(
1 +1
2π |n| y
)e(nx).
Lemma 7.1. We have
Am(n, c, y) =
2πm3/2
c
√yK3/2(2π|n|y)J3
(4π√m |n|c
), n < 0
2π3m3
3c4y−1, n = 0
2πm3/2
c
√yK3/2(2πny)I3
(4π√mn
c
), n > 0.
Proof. Using the identity
M0,3/2(2u) = 27/2Γ(5/2)√uI3/2(u)
where M0,3/2 is the usual M -Whittaker function of order (0, 3/2), we find that
ψm(t) = C1mM0,3/2(4πmt)e(mit)
where
C1 :=π√
2π27/2Γ(5/2)=
1
12.
Therefore
Am(n, c, y) =m
12Im(n, c, y),
where
Im(n, c, y) :=
∫RM0,3/2
(4πmy
c2(t2 + y2)
)e
(mt
c2(t2 + y2)− nt
)dt.
By a simple change of variables, one can show that Im(n, c, y) equals the integral I in [B,p. 33] with the choices k = 0 and s = 2. Then using the evaluation of the integral I giventhere, we have
Im(n, c, y) =
C2
√m/ |n|c
W0,3/2(4π |n| y)J3
(4π√m |n|c
), n < 0
C3m2
c4y−1, n = 0
C2
√m/n
cW0,3/2(4πny)I3
(4π√mn
c
), n > 0,
where
C2 :=2πΓ(4)
Γ(2)= 12π, C3 :=
4π3Γ(4)
3Γ(2)2= 8π3,
18 RIAD MASRI
K3 and J3 are the usual K and J-Bessel functions of order 3, respectively, and W0,3/2 is theusual W -Whittaker function of order (0, 3/2). Using the identity
W0,3/2(2u) =√
2u/πK3/2(u)
where K3/2 is the usual K-Bessel function of order 3/2, we have
W0,3/2(4π |n| y) = 2√|n| yK3/2(2π |n| y).
The result now follows after simplification. �
8. Poincare series
Let λ : R→ [0, 1] be a C∞ function such that
λ(t) =
{0, t ≤ 0
1, t ≥ 1.
Let η > 0 and define
Pm,η(z) :=∑
γ∈Γ∞\Γ0(N)
ψm,η(Im(γz))e(−mγz),
where
ψm,η(t) := λ
(t− 4N
π
η
)m
2
(1− 1
2πmt
).
Then define the regularized function
FN,m,η(z) := FN,m(z)−Pm,η(z).
Proposition 8.1. For y > 4Nπ
+ η we have
FN,m,η(σbz) = Cb(m)y−1 + Eb(m,x, y).
In particular, the regularized function FN,m,η has cuspidal growth of power α = −1.
Proof. We have the Fourier expansion
Pm,η(σbz) = δ∞,bψm,η(y)e(−mz) +∑n∈Z
e(nx)∑c∈R+
S∞,b(−m,n; c)Am,η(n, c, y), (8.1)
where
Am,η(n, c, y) :=
∫Rψm,η
(y
c2(t2 + y2)
)e
(m
c2(t+ iy)− nt
)dt.
The function ψm,η : R→ [0,m] is C∞ and satisfies
ψm,η(t) =
0, t ≤ 4Nπ,
m
2
(1− 1
2πmt
), t ≥ 4N
π+ η.
Moreover, since
min
{c ∈ R+ :
(∗ ∗c ∗
)∈ σ−1
a Γ0(N)σb
}≥ 1 (8.2)
19
for all cusps a, b of Γ0(N) (see [I, eqs. (2.28)–(2.31)]), we have
y
c2(t2 + y2)≤ 4N
π
for y ≥ 4Nπ
. It follows from (8.1) that
Pm,η(σbz) =
δ∞,bψm,η(y)e(−mz), y ≥ 4Nπ
δ∞,bm
2
(1− 1
2πmy
)e(−mz), y ≥ 4N
π+ η.
(8.3)
The proposition now follows from the Fourier expansion (7.1). �
9. traces of weak maass forms
Define the trace
TrD(FN,m) :=∑
τ∈OD,N,h
FN,m(τ).
Theorem 9.1. We have
TrD(FN,m) =m
2
∑τ∈OD,N,h
Im(τ)> 4Nπ
+D−1
176
(1− 1
2πmIm(τ)
)e(−mτ) + h(−D)cN,m +Oε,N(D
12− 1
176+ε)
as D →∞ through all D satisfying the hypothesis (H). Here
cN,m :=
∫reg
FN,m(z)dµ(z)
is a Borcherds-type regularized integral (see Section 13).
Proof. Assume that 0 < η < 9981000
(we will eventually choose η to be very small as a functionof D). By Proposition 8.1, the regularized function FN,m,η has cuspidal growth of powerα = −1. Moreover, this growth is uniform in η for y > 4N
π+ 998
1000, hence the same choice of
constant T0 := 4Nπ
+ 9991000
and corresponding function ηT0 given by
ηT0(σbz) =
0, 1 < y < 4N
π+ 999
1000
Cb(m)y−1χ(y/2), 4Nπ
+ 9991000≤ y ≤ 2(4N
π+ 999
1000)
Cb(m)y−1, y > 2(4Nπ
+ 9991000
)
(9.1)
can be used to regularize each function FN,m,η as in the proof of Theorem 6.3. Upon substi-tuting FN,m,η into Theorem 6.3, we obtain the asymptotic formula
TrD(FN,m) = TrD(Pm,η) + h(−D)
∫F(N)
FN,m,η(z)dµ(z) (9.2)
+Oε,N(||∆2FN,m,η,T0||2D12− 1
16+ε) +Oε,N(D
12− 1
16+ε) +ON(D−
12
+ε),
where
FN,m,η,T0(z) := FN,m,η(z)− ηT0(z).
20 RIAD MASRI
By Lemma 9.2 we have
TrD(Pm,η) =∑
Im(τ)> 4Nπ
ψm,η(Im(τ))e(−mτ).
Split the sum on the right hand side into the ranges Im(τ) ≤ 4Nπ
+ η and Im(τ) > 4Nπ
+ η,and define
RN,m,η(D) := TrD(FN,m)− m
2
∑Im(τ)> 4N
π+η
(1− 1
2πmIm(τ)
)e(−mτ).
Then (9.2) can be written as
RN,m,η(D) = h(−D)
∫F(N)
FN,m,η(z)dµ(z)
+Oε,N(||∆2FN,m,η,T0||2D12− 1
16+ε) +Oε,N(D
12− 1
16+ε) +ON(D−
12
+ε)
+∑
4Nπ<Im(τ)≤ 4N
π+η
ψm,η(Im(τ))e(−mτ).
In Lemma 13.2 we will show that∫F(N)
FN,m,η(z)dµ(z) =
∫reg
FN,m(z)dµ(z) =: cN,m
where the right hand side is a Borcherds-type regularized integral (note that the right handside is independent of η).
A straightforward estimate yields∑4Nπ<Im(τ)≤ 4N
π+η
ψm,η(Im(τ))e(−mτ)� m · e2πm( 4Nπ
+ 9981000
)#ΛN,h,η(D),
where
ΛN,h,η(D) := {τ ∈ OD,N,h :4N
π< Im(τ) ≤ 4N
π+ η}.
By Lemma 11.1 we have the estimate
#ΛN,h,η(D) = ON(ηh(−D)) +Oε,N(η−1D12− 1
16+ε).
Moreover, by Lemma 12.1 we have the estimate
||∆2FN,m,η,T0||2 = ON,m(η−10).
Combining the preceding estimates yields
RN,m,η(D) = h(−D)cN,m +Oε,N(η−10D12− 1
16+ε) +ON(ηh(−D)).
If we let η = D−b for some b > 0, then
RN,m,η(D) = h(−D)cN,m +Oε,N(D12−( 1
16−10b)+ε) +Oε,N(D
12−b+ε)
where we used
h(−D)�ε D12
+ε.
21
The exponent is optimized when 116− 10b = b, or b = 1/176, thus1
RN,m,η(D) = h(−D)cN,m +Oε,N(D12− 1
176+ε).
�
Lemma 9.2. We have
TrD(Pm,η) =∑
Im(τ)> 4Nπ
ψm,η(Im(τ))e(−mτ).
Proof. Write
TrD(Pm,η) =∑
Im(τ)≤ 4Nπ
Pm,η(τ) +∑
Im(τ)> 4Nπ
Pm,η(τ) =: I + II.
Let γ ∈ Γ∞\Γ0(N) and fix a Heegner point τ ∈ OD,N,h with Im(τ) ≤ 4N/π. Then by Lemma6.4, Im(γτ) ≤ 4N/π. Since ψm,η(t) = 0 for t ≤ 4N
π, it follows that
Pm,η(τ) =∑
γ∈Γ∞\Γ0(N)
ψm,η(Im(γτ))e(−mγτ) = 0,
and thus I = 0. On the other hand, if we fix a Heegner point τ ∈ OD,N,h with Im(τ) > 4N/π,then by (8.3) we have
Pm,η(τ) = ψm,η(Im(τ))e(−mτ),
and thus
II =∑
Im(τ)> 4Nπ
ψm,η(Im(τ))e(−mτ).
�
10. proof of Theorems 1.2 and 1.6
Proof of Theorem 1.2. By combining the Bruinier-Ono period formula (1.1), the iden-tity (5.1), and Theorem 9.1 we obtain
c+Λf
(D, h) = −4N
DTrD(
1
4πR−2f)
= −8N
D
N∞∑n=1
cf (−n)TrD(FN,n)
=1
D
∑Im(τ)> 4N
π+D−
1176
Mf,N(τ) + Cf,Nh(−D)
D+Oε,N(D−
89176
+ε),
where
Mf,N(z) := −4NN∞∑n=1
cf (−n)n
(1− 1
2πnIm(z)
)e(−nz)
1Recall that we assumed η ≤ 9981000 , which is equivalent to D ≥
(1000998
)176 ≈ 1.4224 for the choice η =
D−1/176. Since D ≥ 4 by assumption, the latter inequality is satisfied.
22 RIAD MASRI
and
Cf,N := −8NN∞∑n=1
cf (−n)cN,n = −8N
∫reg
N∞∑n=1
cf (−n)FN,n(z)dµ(z) = −Nπ
∫reg
R−2f(z)dµ(z),
where for the last equality we again used (5.1). �
Proof of Theorem 1.6. Recall the Bruinier-Ono formula (1.3) for the partition function,
p(n) = − 1
24c+
Λfp(24n− 1, 1),
where for the weakly holomorphic modular form fp ∈ M !−2(6) defined by (1.2) we have
N∞ = 1 and cfp(−1) = 1. Then by Theorem 1.2 we obtain
c+Λfp
(24n− 1, 1) =1
24n− 1
∑τ∈O24n−1,6,1
Im(τ)> 24π
+(24n−1)−1
176
Mfp,6(τ) + Cfp,6h(−24n+ 1)
24n− 1+Oε(n
− 89176
+ε),
where
Mfp,6(z) = −24
(1− 1
2πIm(z)
)e(−z)
and
Cfp,6 = − 6
π
∫reg
R−2fp(z)dµ(z).
The theorem now follows after multiplying by −1/24. �
11. Proof of Lemma 11.1
In this section we establish the following estimate by modifying the argument in [D, p.248-249].
Lemma 11.1. For each number 0 < η ≤ 1 we have
#ΛN,h,η(D) = ON(ηh(−D)) +Oε(η−1D
12− 1
16+ε).
Proof. Let 0 < η ≤ 1 and φη : R→ [0, 1] be a C∞ function such that
(1) φη is supported on (4Nπ− η, 4N
π+ 2η).
(2) φη = 1 on [4Nπ, 4Nπ
+ η].
(3) φη satisfies the bound
φ(A)η � η−A, A = 0, 1, 2. (11.1)
Define
gη(z) :=∑
γ∈Γ∞\Γ0(N)
φη(Im(γz)).
23
Then we have ∑τ∈OD,N,h
gη(τ) =∑
τ∈OD,N,h
∑γ∈Γ∞\Γ0(N)
γ 6=I
φη(Im(γτ)) +∑
τ∈OD,N,h
φη(Im(τ))
≥∑
τ∈OD,N,h
φη(Im(τ))
=∑
τ∈OD,N,hτ∈ΛN,h,η(D)
1 +∑
τ∈OD,N,hτ /∈ΛN,h,η(D)
φη(Im(τ))
≥ #ΛN,h,η(D).
The real-analytic Eisenstein series E∞(z, s) has a meromorphic continuation to C with asimple pole at s = 1 with residue 1/vol(F(N)) (see [I, Theorem 11.3 and Proposition 6.13]).Then by [I, eq. (7.12)] we have
gη(z) =1
vol(F(N))φη(1) +
1
2πi
∫Rφη(
12
+ it)E∞(z, 12
+ it)dt,
where
φη(s) :=
∫ ∞0
φη(u)u−(s+1)du.
Thus ∑τ∈OD,N,h
gη(τ) =1
vol(F(N))φη(1)h(−D) +
1
2πi
∫Rφη(
12
+ it)W∞(D, t)dt.
Now, a straightforward estimate yields
φη(1)� η.
Moreover, by (6.5) we have
W∞(D, t)�ε (14
+ t2)1796
+εD12− 1
16+ε.
It follows that ∑τ∈OD,N,h
gη(τ) = ON(ηh(−D)) +Oε(CηD12− 1
16+ε),
where
Cη :=
∫R|φη(1
2+ it)|(1
4+ t2)
1796
+εdt.
We integrate by parts 2 times and use the bound (11.1) to obtain
φη(12
+ it)� η−1
2∏j=1
|32
+ it− j|−1,
which yields Cη � η−1.�
24 RIAD MASRI
12. Proof of Lemma 12.1
Lemma 12.1. For each number 0 < η < 9981000
we have
||∆AFN,m,η,T0||2 �N,m η−(4A+2), A = 1, 2, . . . .
Proof. Fix Y > 0 and define
P (Y ) := {z = x+ iy : 0 < x < 1, y ≥ Y }and
L(Y ) := {z = x+ iY : 0 < x < 1}.
One can choose a fundamental domain D for Γ0(N) such that
D = D(Y ) ∪⋃a
Da(Y )
where Da(Y ) := σaP (Y ) and
D(Y ) := D\⋃a
Da(Y )
has compact closure and is adjacent to each cuspidal zone Da(Y ) along the horocycle σaL(Y )(see [I, Section 2.2]). Using the SL2(R)-invariance of the measure dµ(z) we find that
||∆AFN,m,η,T0||22 :=
∫D
|∆AFN,m,η,T0(z)|2dµ(z) ≤ I + II,
where
I :=
∫D( 4N
π)
|∆AFN,m,η,T0(z)|2dµ(z)
and
II :=∑b
∫P ( 4N
π)
|∆AFN,m,η,T0(σbz)|2dµ(z).
First we estimate II. By (7.1), (8.3) and (9.1) we have (recall that η < 9981000
)
FN,m,η,T0(σbz) =
FN,m(σbz)− δ∞,b
m
2ψm,η(y)e(−mz), 4N
π≤ y < 4N
π+ η
Cb(m)y−1 + Eb(m,x, y), 4Nπ
+ η ≤ y < 4Nπ
+ 9991000
Cb(m)y−1(1− χ(y/2)) + Eb(m,x, y), 4Nπ
+ 9991000≤ y < 2(4N
π+ 999
1000)
Eb(m,x, y), y ≥ 2(4Nπ
+ 9991000
).
(12.1)
By splitting the y-integral in II into the different ranges considered in (12.1), we obtain
II =∑b
∫ ∞4Nπ
∫ 1
0
|∆AFN,m,η,T0(σbz)|2dµ(z) = III +O(1),
where
III :=∑b
∫ 4Nπ
+η
4Nπ
∫ 1
0
|∆A(FN,m(σbz)− δ∞,bm
2ψm,η(y)e(−mz))|2dµ(z).
25
By linearity of ∆ and the triangle inequality,
III ≤ IV + V +O(1),
where
IV := 2∑b
∫ 4Nπ
+ 9981000
4Nπ
∫ 1
0
|∆AFN,m(σbz)| · |δ∞,bm
2∆A(ψm,η(y)e(−mz))|dµ(z)
and
V :=∑b
∫ 4Nπ
+ 9981000
4Nπ
∫ 1
0
|δ∞,bm
2∆A(ψm,η(y)e(−mz))|2dµ(z).
Using the estimate
max(x,y)∈[0,1]×[ 4N
π, 4Nπ
+ 9981000
]|∆A(ψm,η(y)e(−mz))| � η−A,
we obtain
IV� η−A and V� η−2A.
We conclude that
II� η−2A.
Next we estimate I. Observe that D(4N/π) can be contained in a rectangle
RN := [−BN , BN ]× [CN ,4N
π]
for some BN ≥ 1 and 0 < CN ≤√
3/2. Since ψm,η(t) = 0 for t ≤ 4N/π, by (8.1) we have
FN,m,η,T0(z) = (FN,m(z)− fm,η(x, y))− ηT0(z),
where
fm,η(x, y) :=∑n∈Z
e(nx)∑
1≤c≤C−1N
S(−m,n; c)Am,η(n, c, y).
By linearity of ∆ and three applications of the triangle inequality, we have
I ≤∫RN
|∆A((FN,m(z)− fm,η(x, y))− ηT0(z))|2dµ(z) ≤ VI + VII +O(1),
where
VI := 2
∫RN
(|∆AFN,m(z)|+ |∆AηT0(z)|
)|∆Afm,η(x, y)|dµ(z)
and
VII :=
∫RN
|∆Afm,η(x, y)|2dµ(z).
Using the estimate
max(x,y)∈RN
|∆Afm,η(x, y)| � η−(4A+2), (12.2)
we have
VI� η−(4A+2) and VII� η−2(4A+2).
26 RIAD MASRI
We conclude that
I� η−2(4A+2).
It remains to establish the estimate (12.2). Define
Φm,η,c,y(u) := ψm,η
(y
c2(u2 + y2)
)e
(m
c2(u+ iy)
),
so that
Am,η(n, c, y) =
∫R
Φm,η,c,y(u)e(−nu)du.
Since
Φm,η,c,y(u) = 0 for |u| ≥√
1− C2N ,
integrating by parts (2A+ 2)-times yields
Am,η(n, c, y) =1
(2πin)2A+2
∫ √1−C2N
−√
1−C2N
Φ(2A+2)m,η,c,y (u)e(−nu)du.
In particular, we have
∆Afm,η(x, y) =∑n∈Z
∑1≤c≤C−1
N
S(−m,n; c)∆AIm,η,n,c(x, y),
where
Im,η,n,c(x, y) :=e(nx)
(2πin)2A+2
∫ √1−C2N
−√
1−C2N
Φ(2A+2)m,η,c,y (u)e(−nu)du.
For clarity, we first assume that A = 1. Then
∆Im,η,n,c(x, y) =
− y2 e(nx)
(2πin)2
∫ √1−C2N
−√
1−C2N
Φ(4)m,η,c,y(u)e(−nu)du− y2 e(nx)
(2πin)4
∫ √1−C2N
−√
1−C2N
∂2yΦ
(4)m,η,c,y(u)e(−nu)du.
Using the estimate ([I, (2.37)])
S(−m,n; c)� c2,
and the estimates
maxu∈[−√
1−C2N ,√
1−C2N ]
|Φ(4)m,η,c,y(u)| �m,c,y η
−4
and
maxu∈[−√
1−C2N ,√
1−C2N ]
|∂2yΦ
(4)m,η,c,y(u)| �m,c,y η
−6,
we obtain
max(x,y)∈RN
|∆fm,η(x, y)| � η−6.
The preceding argument generalizes in a straightforward way to A ≥ 1, and the estimate(12.2) follows.
�
27
13. Regularized integrals
First we recall the notion of a regularized integral in the sense of Borcherds [Bo] andHarvey-Moore [HMo]. Let F be the standard fundamental domain for SL2(Z). Then afundamental domain for Γ0(N) is given by
F(N) :=⋃
σ∈Γ0(N)\SL2(Z)
σF .
For a fixed Y > 1, define the truncated domains
FY := {z ∈ F : Im(z) ≤ Y }
and
FY (N) :=⋃
σ∈Γ0(N)\SL2(Z)
σFY .
We then define the regularized integral∫reg
FN,m(z)dµ(z) := limY→∞
∫FY (N)
FN,m(z)dµ(z).
For each η > 0 define the function
ψm,η,Y (t) :=
{ψm,η(t), t ≤ Y
0, t > Y,
and the associated Poincare series
Pm,η,Y (z) :=∑
γ∈Γ∞\Γ0(N)
ψm,η,Y (Im(γz))e(−mγz).
Lemma 13.1. For z ∈ FY (N) we have Pm,η,Y (z) = Pm,η(z).
Proof. By definition of ψm,η,Y we have
Pm,η(z) = Pm,η,Y (z) +∑
γ∈Γ∞\Γ0(N)Im(γz)>Y
ψm,η(Im(γz))e(−mγz).
Let γ ∈ Γ∞\Γ0(N) and z ∈ FY (N). Then γz = Az′ for some A ∈ SL2(Z) and z′ ∈ FY , thusIm(γz) = Im(Az′) < Im(z′) < Y . It follows that
#{γ ∈ Γ∞\Γ0(N) : Im(γz) > Y } = 0,
hence Pm,η,Y (z) = Pm,η(z). �
Lemma 13.2. We have∫F(N)
FN,m,η(z)dµ(z) = limY→∞
∫FY (N)
FN,m(z)dµ(z).
Proof. Since FN,m,η := FN,m −Pm,η ∈ L1(F(N)),∫F(N)
FN,m,η(z)dµ(z) = limY→∞
∫FY (N)
(FN,m(z)−Pm,η(z)) dµ(z).
28 RIAD MASRI
Now, by Lemma 13.1 we have∫FY (N)
Pm,η(z)dµ(z) =
∫FY (N)
Pm,η,Y (z)dµ(z).
We claim that if z ∈ F(N) \ FY (N), then Pm,η,Y (z) = 0. Let γ ∈ Γ∞\Γ0(N) and z ∈
F(N) \ FY (N). Then γz = Az′ for some A =
(a bc d
)∈ SL2(Z) and z′ = x′ + iy′ ∈ F \ FY ,
i.e., z′ ∈ F with y′ > Y . We have
Im(γz) = Im(Az′) =y′
|cz′ + d|2.
If c = 0 then d = 1, so that Im(γz) = y′ > Y . Since ψm,η,Y (t) = 0 for t > Y , it follows thatPm,η,Y (z) = 0. On the other hand, if c 6= 0 then c2 ≥ 1, so that
Im(γz) =y′
(cx′ + d)2 + c2(y′)2≤ 1
y′<
1
Y< 1
(recall Y > 1). Since ψm,η,Y (t) = ψm,η(t) = 0 for t < 1, it follows that Pm,η,Y (z) = 0, whichcompletes the proof of the claim. By the claim we have∫
FY (N)
Pm,η,Y (z)dµ(z) =
∫F(N)
Pm,η,Y (z)dµ(z).
Moreover, unfolding yields ∫F(N)
Pm,η,Y (z)dµ(z) = 0.
Thus we conclude that∫F(N)
FN,m,η(z)dµ(z) = limY→∞
∫FY (N)
FN,m(z)dµ(z).
�
Remark 13.3. If FN,m = R−2f for a harmonic weak Maass form f ∈ H−2(N), then∫reg
FN,m(z)dµ(z) =∑a
αac+f,a(0),
where αa is the width of the cusp a and c+f,a(0) is the constant term of the holomorphic part
of the Fourier expansion of f in the cusp a of Γ0(N).2 In particular, for the constants Cf,Nand C∗fp,6 in Theorems 1.2 and 1.6, respectively, we have
Cf,N = −Nπ
∑a
αacf,a(0)
and
C∗fp,6 =1
4π
∑a
αacfp,a(0),
where the last sum is over the 4 cusps of Γ0(6).
2We thank Jan Bruinier for a very helpful correspondence regarding this fact.
29
References
[BH] V. Blomer and G. Harcos, Hybrid bounds for twisted L-functions. J. Reine Angew. Math. 621 (2008),53–79.
[Bo] R. Borcherds, Automorphic forms with singularities on Grassmannians. Inv. Math. 132 (1998), 491–562.
[BrO] K. Bringmann and K. Ono, An arithmetic formula for the partition function. Proc. Amer. Math. Soc.135 (2007), 3507–3514.
[B] J. H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors. Lecture Notes inMathematics, 1780. Springer-Verlag, Berlin, 2002. viii+152 pp.
[B2] J. H. Bruinier, Harmonic Maass forms and periods. Math. Ann. 357 (2013), 1363–1387.[BF] J. H. Bruinier and J. Funke, On two geometric theta lifts. Duke Math. Journal 125 (2004), 45–90.[BO] J. H. Bruinier and K. Ono, Heegner divisors, L-functions and harmonic weak Maass forms. Ann. of
Math. 172 (2010), 2135–2181.[BO2] J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic
weak Maass forms. Adv. Math. 246 (2013), 198–219.[D] W. Duke, Modular functions and the uniform distribution of CM points. Math. Ann. 334 (2006), 241–
252.[FM] A. Folsom and R. Masri, Equidistribution of Heegner points and the partition function. Math. Ann.
348 (2010), 289–317.[FM2] A. Folsom and R. Masri, The asymptotic distribution of traces of Maass-Poincare series. Adv. Math.
226 (2011), 3724–3759.[GKZ] B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of L–series. II. Math. Ann. 278
(1987), 497–562.[GZ] B. Gross and D. Zagier, Heegner points and derivatives of L–series. Invent. Math. 84 (1986), 225–320.[HM] G. Harcos and P. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidistribution
of Heegner points. II. Invent. Math. 163 (2006), 581–655.[HR] G. H. Hardy and S. Ramanujan, Une formule asymptotique pour le nombre des partitions de n. Comptes
Rendus 2, (1917), found in Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence,RI, (2000), 239–241.
[HMo] J. Harvey and G. Moore, Algebras, BPS states, and strings. Nuclear Phys. B 463 (1996), no. 2-3,315–368.
[HB] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. II. Quart. J. Math. Oxford Ser. 31(1980), 157–167.
[HL] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero. With an appendix byDorian Goldfeld, Hoffstein and Daniel Lieman. Ann. of Math. 140 (1994), 161–181.
[I] H. Iwaniec, Introduction to the spectral theory of automorphic forms. Biblioteca de la RevistaMatematica Iberoamericana. Revista Matematica Iberoamericana, Madrid, 1995. xiv+247 pp.
[IK] H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society ColloquiumPublications, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp.
[JM] M. Jutila and Y. Motohashi, Uniform bound for Hecke L-functions. Acta Math. 195 (2005), 61–115.[KM] S. Kudla and J. Millson, The theta correspondence and harmonic forms I. Math. Ann. 274 (1986),
353–378.[KMY] B. D. Kim, R. Masri, and T. H. Yang, Nonvanishing of Hecke L–functions and the Bloch-Kato
conjecture. Math. Ann. 349 (2011), 301–343.[LR] E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms. Forum
Mathematicum, DOI: 10.1515/forum-2012-0112, March 2013.[L] D. H. Lehmer, On the remainders and convergence of the series for the partition function. Trans. Amer.
Math. Soc. 46 (1939), 362–373.[MP] A. Miller and A. Pixton, Arithmetic traces of non-holomorphic modular invariants. Int. J. Number
Theory 6 (2010), 69–87.[O] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory. Current devel-
opments in mathematics, 2008, 347–454, Int. Press, Somerville, MA, 2009.
30 RIAD MASRI
[R] H. Rademacher, On the expansion of the partition function in a series. Ann. of Math. 44 (1943),416–422.
[R2] H. Rademacher, Topics in analytic number theory. Edited by E. Grosswald, J. Lehner and M. Newman.Die Grundlehren der mathematischen Wissenschaften, Band 169. Springer-Verlag, New York-Heidelberg,1973. ix+320 pp.
[W] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier . J. Math.Pures Appl. 60 (1981), 375–484.
[W2] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symetrie.Compos. Math. 54 (1985), 173–242.
[Za] D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Bringmann-Ono). Seminaire Bourbaki. Vol. 2007/2008. Asterisque No. 326 (2009), Exp. No. 986, viiviii, 143–164(2010).
[Z] S. Zhang, Gross–Zagier formula for GL2. Asian J. Math. 5 (2001), 183–290.
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX77843-3368
E-mail address: [email protected]