FOURIER COEFFICIENTS OF HARMONIC WEAK MAASS FORMS AND THE PARTITION FUNCTION

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1 FOURIER COEFFICIENTS OF HARMONIC WEAK MAASS FORMS AND THE PARTITION FUNCTION RIAD MASRI Abstrat. In a reent paper, Bruinier Ono proved that ertain harmoni weak Maass forms have the property that the Fourier oeffiients of their holomorphi parts are algebrai traes of weak Maass forms evaluated on Heegner points. As a speial ase they obtained a remarkable finite algebrai formula for the Hardy-Ramanujan partition funtion pn, whih ounts the number of partitions of a positive integer n. We establish an asymptoti formula with a power saving error term for the Fourier oeffiients in the Bruinier-Ono formula. As a onsequene, we obtain a new asymptoti formula for pn. One interesting feature of this formula is that the main term ontains essentially 3 h 4n + fewer terms than the trunated main term in Rademaher s exat formula for pn, h 4n + is the lass number of the imaginary quadrati field Q 4n +.. Introdution statement of results During the last ten years there have been remarkable advanes in the study of q-series, modular forms, L funtions through their onnetion to harmoni weak Maass forms see for example the exellent survey artiles of Ono [O] Zagier [Za]. Roughly speaking, a harmoni weak Maass form of weight k Z is a real-analyti funtion on the omplex upper half-plane H whih transforms like a modular form with respet to some ongruene subgroup of SL Z, vanishes under the weight k hyperboli Laplaian, has at most linear exponential growth in the usps of the group. A detailed disussion of harmoni weak Maass forms an be found in the fundamental paper of Bruinier Funke [BF] see also Setion 3. A harmoni weak Maass form has a Fourier expansion onsisting of a non-holomorphi holomorphi part. There is a great amount of interest in understing the arithmeti meaning of these Fourier oeffiients. For example, Bruinier Ono [BO] related the Fourier oeffiients of the non-holomorphi holomorphi parts of weight / harmoni weak Maass forms to entral values entral derivatives of modular L-funtions, respetively. Bruinier [B] has sine related the Fourier oeffiients of the holomorphi parts of weight / harmoni weak Maass forms to periods of algebrai differentials of the third kind on modular ellipti urves. In a reent paper, Bruinier Ono [BO] onstruted a theta lift from the spae of harmoni weak Maass forms of weight - to the spae of vetor-valued harmoni weak Maass forms of weight -/. They used this lift to prove that ertain vetor-valued harmoni weak Maass forms of weight -/ have the property that the Fourier oeffiients of their holomorphi parts are algebrai traes of fixed, weight 0 weak Maass forms evaluated on Heegner points. As a speial ase they established a finite algebrai formula for the Hardy- Ramanujan partition funtion pn, whih ounts the number of partitions of a positive integer n. In this paper we will establish an asymptoti formula with a power saving error term for the Fourier oeffiients in the Bruinier-Ono formula. As a onsequene, we will

2 RIAD MASRI obtain a new asymptoti formula for pn see Theorem.6. One interesting feature of this formula is that the main term ontains essentially 3 h 4n + fewer terms than the trunated main term in Rademaher s exat formula for pn, h 4n + is the lass number of the imaginary quadrati field Q 4n + see the disussion following Theorem.6. In order to state our results we briefly review the Bruinier-Ono formula. Bruinier Ono [BO, Setion 3.] defined a theta lift Λ : H N H,ρ from the spae H N of harmoni weak Maass forms of weight level N to the spae H,ρ of vetor-valued harmoni weak Maass forms of weight / level N with Weil representation ρ see Setions 3 4 for the preise definitions. The image of f H N under the theta lift Λ has a Fourier expansion Λf, w = Λ f, w + Λ + f, w, w = u + iv H with non-holomorphi part Λ f, w := Λ f D, hγ k, π D v/nq D/4N e h, holomorphi part h Λ + f, w := mod N h D Z D<0 N mod N D= + Λ f D, hq D/4N e h + h mod N D=0 q := ew = e πiw + Λ f D, hq D/4N e h N 0 is an integer, Γa, t is the inomplete Gamma funtion, {e h } is the stard basis for the group algebra C[Z/NZ]. Note that ± Λ f D, h = 0 unless D h mod 4N. Assume now that N is squarefree H D < 4 is an odd fundamental disriminant oprime to N suh that every prime divisor of N splits in K = Q D. Fix a solution h Z/NZ of h D mod 4N, for a primitive integral ideal A of K write A = Za + Z b + D, a = N K/Q A, b Z with b h mod N b D mod 4Na. Then τ h A = b + D Na is a Heegner point on the modular urve X 0 N. It is known that τ h A depends only on the ideal lass of A on h mod N, so we denote it by τ h [A]. For details onerning these fats, see [GZ, part II, Setion ]. Let CLK be the ideal lass group of K h D = #CLK be the lass number. By Minkowski s theorem we may hoose a primitive integral ideal A in eah ideal lass [A] CLK suh that N K/Q A π D.

3 3 Having fixed suh a hoie A for eah ideal lass, we define O D,N,h := {τ h [A] Let Γ τ be the image of the stabilizer of τ = τ h [A] The Maass level-raising operator : [A] CLK}. in PSL Z. R := i z Imz, z H maps weak Maass forms of weight - to weak Maass forms of weight 0. Define the operator := 4π R. Bruinier Ono [BO, Theorem 3.6] established a formula for the Fourier oeffiients of holomorphi parts of weight -/ vetor-valued harmoni weak Maass forms in the image ΛH N. With the setup in this paper, their formula an be stated as follows. Theorem. Bruinier-Ono. For eah f H N, the D, h-th Fourier oeffiient of the holomorphi part of Λf, w H,ρ is given by + Λ f D, h = 4N fτ.. D #Γ τ O τ D,N,h Let M N! H N be the spae of weakly holomorphi modular forms of weight - level N, M! H,ρ,ρ be the spae of vetor-valued weakly holomorphi modular forms of weight -/ level N with Weil representation ρ see Setion 3. Eah form f M N! has a Fourier expansion fz = N n= f nq n + f nq n. Bruinier Ono [BO, Theorem 3.5] proved that the theta lift Λ restrits to a map n=0 Λ : M! N M!,ρ. We will establish the following asymptoti formula with a power saving error term for the Fourier oeffiients of holomorphi parts of weight -/ vetor-valued weakly holomorphi modular forms in the image ΛM! N. Theorem.. For eah f M N,! the D, h-th Fourier oeffiient of the holomorphi part of Λf, w M! satisfies,ρ + Λ f D, h = D τ O D,N,h Imτ> 4N π +D 76 M f,n τ + C f,n h D D as D through all D satisfying the hypothesis H. Here N M f,n z := 4N f nn n= πnimz + O ɛ,nd e nz ɛ

4 4 RIAD MASRI C f,n := N π reg R fzdµz is a Borherds-type regularized integral dµz is the normalized hyperboli measure on Γ 0 N\H see Setion 3. Remark.3. We briefly explain how Theorem. an be generalized to give an asymptoti formula for the Fourier oeffiients of holomorphi parts of weight -/ vetor-valued harmoni weak Maass forms in the image ΛH N. A key step in the proof of Theorem. is to express the image f of a weakly holomorphi modular form f M! N under the operator = R /4π as a finite linear ombination of ertain weight 0 Maass-Poinaré series see Setion 5. By. it follows that the asymptoti distribution of + Λ f D, h is determined by the asymptoti distribution of traes of these Maass-Poinaré series. Using [BO, Corollary 3.4] [BO, Proposition.], one an obtain a similar but more ompliated expression for the image f of any harmoni weak Maass form f H N. Theorem. an then be generalized by using this expression modifying the proof aordingly. We fous here on the ase f M! N sine this is what is needed for our appliation to the partition funtion. Remark.4. Let l N with l, N/l =, let f f Wl N be the Atkin-Lehner involution of M N! defined in [BO, Setion 4.3]. Then assuming that f Wl N has oeffiients in Z for every suh l, Bruinier Ono [BO, Theorem 4.5] proved that 6D fτ is an algebrai integer in the ring lass field for the order O D Q D. Remark.5. The Fourier oeffiients + Λ f D, h are likely related to periods of differentials of the third kind in a manner similar to [B]. We now disuss our appliation to the partition funtion pn. Define the weakly holomorphi modular form f p M! 6 by f p z := E z E z 3E 3z + 6E 6z = q 0 9q. ηz ηz η3z η6z is the Dedekind eta funtion ηz := q /4 E z := 4 n= q n n= d n is the usual weight Eisenstein series. Bruinier Ono [BO, Setion 3.] proved that dq n pn = 4 + Λ fp 4n,..3 By ombining.3 with., they obtained the following formula for pn see [BO, Theorem.] pn = f p τ. 4n τ Q 4n,6,

5 Bruinier Ono [BO, Theorem 4.] also proved that f p z satisfies the assumption on the Atkin-Lehner involutions in Remark.4, hene 64n f p τ is an algebrai integer in the ring lass field of disriminant 4n +. Note that Larson Rolen [LR] later showed this result holds without the 6, answering a question of Bruinier Ono [BO, Setion 5, ]. By ombining Theorem. with.3, we will obtain the following new asymptoti formula for pn. Theorem.6. Let n be a positive integer with 4n squarefree. Then pn = e τ + C h 4n + f 4n πimτ p,6 + O ɛ n ɛ 4n as n τ O 4n,6, Imτ> 4 π +4n 76 Cf p,6 := R f p zdµz. 4π reg Remark.7. If n is a positive integer with 4n squarefree, then 4n + is an odd fundamental disriminant whih satisfies the hypothesis H for N = 6. The asymptoti distribution of pn has been studied extensively sine the early part of the 0th entury. Hardy Ramanujan [HR] invented the irle method used it to establish the well-known asymptoti pn 4 3n exp π n 6 Rademaher [R] later used a refinement of the irle method to establish the exat formula pn = π = A n d exp dn Here λ n := n /4 A n is the exponential sum A n := ω b, e nb b mod b,=. πλ n /3 λ n. ω b, is a ertain root of unity. By trunating the series at N := n/6 estimating the remainder, Rademaher [R] obtained the asymptoti formula with main term MTn := π, pn = MTn + On N = πλ n /3 A n d exp. dn λ n Lehmer [L] improved the error term in.4 to On +ɛ. Using an arithmeti reformulation of Rademaher s exat formula due to Bringmann Ono [BrO], the author Folsom 5

6 6 RIAD MASRI [FM] established an asymptoti formula for pn with an error term whih is On δ for some absolute δ > /. We now ompare the main term in Theorem.6 with the main term MTn in Rademaher s asymptoti formula.4, whih reveals some interesting new features in the asymptoti distribution of pn. Suppose that n 6l for any l Z. Then using an analysis with exponential sums see [FM, Proposition 6.], one an show that MTn = 4n τ Λ 4n,6 Imτ> χ τ πimτ e τ.5 Λ 4n,6 is the set of Heegner points of disriminant 4n + on X 0 6 χ τ = b is the middle oeffiient of the quadrati form QX, Y = b 6aX bxy + Y orresponding to the Heegner point τ. Let H denote the Hilbert lass field of K = Q 4n +. The set of Heegner points deomposes as see e.g. [GKZ, p. 507] [GZ, pp ] Λ 4n,6 = O 4n,6,h.6 h mod h 4n+ mod 4 eah set O 4n,6,h is a simple, transitive GalH/K-orbit. Now, reall that the main term in Theorem.6 ontains no harater twist is summed over one Galois orbit {τ O 4n,6, } subjet to Imτ > 4/π + 4n /76. On the other h, from.5 we see that MTn is summed over all Heegner points {τ Λ 4n,6 } subjet to Imτ >. The number of elements in eah Galois orbit is h 4n + sine GalH/K = CLK, there are 4 orbits orresponding to the solutions h =, 5, 7 of the ongruene in.6. Hene the main term in Theorem.6 ontains essentially 3 h 4n + fewer terms than the main term MTn. It would be very interesting to know whether the shorter main term in Theorem.6, or the form of the main term itself, offers any omputational advantages. Aknowledgments. I would like to thank Jan Bruinier, Sheng-Chi Liu Matt Young for some helpful disussions/orrespondene regarding this work. I would also like to thank the referees for many suggestions whih improved the exposition of the paper. The author was partially supported by the NSF grant DMS-6535 during the preparation of this work.. Outline of the proof of Theorem. In this setion we give a brief outline of the proof of Theorem.. We want to establish an asymptoti formula as D for the trae fτ #Γ τ O τ D,N,h appearing in the Bruinier-Ono formula.. The image of a weakly holomorphi modular form f M N! under the operator = R /4π is a weight 0 weak Maass form on Γ 0 N. As a onsequene, the funtion f an be expressed as a finite linear ombination of ertain Maass-Poinaré series {F N,n } N n=, so it suffies to study the trae of F N,n. We will ompute the Fourier expansion of F N,n in the usps of Γ 0 N find that it has a part with linear exponential growth a part with polynomial growth. Now, the Galois orbit of Heegner points O D,N,h beomes quantitatively equidistributed with respet to the

7 normalized hyperboli measure on Γ 0 N\H as D. However, we annot diretly use this fat to obtain an asymptoti formula for the trae of F N,n beause the test funtion F N,n grows very rapidly in the usps hene is not admissible. We will overome this diffiulty using two different regularizations. We first prove an equidistribution theorem for Galois orbits of Heegner points in whih the test funtions are allowed to grow polynomially in the usps see Theorem 6.3. We then onstrut for eah η > 0 a ertain smooth Poinaré series P n,η whih regularizes the linear exponential growth of F N,n in the usps of Γ 0 N. This is inspired by a onstrution of Duke [D] to regularize the pole at of the j-funtion. Upon substituting the regularized funtion F N,n P n,η into the equidistribution theorem, we will obtain a smooth asymptoti formula for the trae of F N,η. Finally, using a Borherdstype integration along with a deliate analysis to un-smooth the main term bound as a funtion of η the Sobolev norm of F N,n P n,η appearing in the error term, we will obtain the desired asymptoti formula for the trae of F N,n see Theorem 9.. Note that our proof has some elements in ommon with [FM], though onsiderable new diffiulties arise beause of the presene of level in all of our arguments whih is ruial for our appliation to the partition funtion. 3. harmoni weak Maass forms In this setion we review some fats onerning harmoni weak Maass forms. For more details, see [BF]. Let k Z z = x + iy H. The weight k hyperboli Laplaian is defined by k := y x + y + iky x + i y. Let N be a positive integer. A weak Maass form of weight k on Γ 0 N is a smooth funtion f : H C satisfying: f k M = f for all M Γ 0 N. There is a omplex number λ suh that k f = λf. 3 There is a onstant C > 0 suh that fz = Oe Cy as y. An analogous ondition is required at all usps. Note. The slash operator k is defined as in Shimura s theory of half-integral weight forms. A weak Maass form is harmoni if k f = 0. Every harmoni weak Maass form has a Fourier expansion with non-holomorphi part holomorphi part f z := n<0 f + z := fz = f z + f + z N n= f nγ k, 4π n yqn + f nq n + + f nqn, N 0 is an integer Γa, t is the inomplete Gamma funtion. A harmoni weak Maass form with trivial non-holomorphi part is alled a weakly holomorphi modular form. Let M! kn H k N n=0 7

8 8 RIAD MASRI denote the spaes of weakly holomorphi modular forms harmoni weak Maass forms, respetively. We also require the notion of a vetor-valued weak Maass form. Let w = u + iv H, let Mp R be the metapleti two-fold over of SL R realized as the group of pairs M, φw M SL R φ : H C is a holomorphi funtion with φw = w+d. The multipliation is defined by M, φwm, φ w = MM, φm wφ w. Let Γ = Mp Z be the inverse image of SL Z under the overing map. The group Γ is generated by T = 0, S = 0 0, w. Given h Z/NZ let e h be the orresponding stard basis vetor for the group algebra C[Z/N Z]. The Weil representation ρ is the unitary representation of Γ on C[Z/NZ] defined in terms of the generators T S of Γ by h ρt e h = e e h, 4N ρse h = in h mod N e hh e h. N A vetor-valued weak Maass form of weight k with respet to Γ ρ is a smooth funtion g : H C[Z/NZ] satisfying: gmw = φw k ρm, φgw for all M, φ Γ. There is a omplex number λ suh that k g = λg. 3 There is a onstant C > 0 suh that gw = Oe Cv as v. An analogous ondition is required at all usps. A vetor-valued weak Maass form is harmoni if k g = 0. Every vetor-valued harmoni weak Maass form has a Fourier expansion gw = g w + g + w with non-holomorphi part g w := holomorphi part h g D, hγ k, π D v/nq D/4N e h mod N D<0 g + w := h N + g D, hq D/4N e h + mod N D= h + g D, hq D/4N e h. mod N D=0 Note that ± g D, h = 0 unless D h mod 4N. Let M! k,ρ H k,ρ denote the spaes of vetor-valued weakly holomorphi modular forms vetor-valued harmoni weak Maass forms, respetively.

9 4. The Bruinier-Ono theta lift Let k Z z H. The Maass level raising lowering differential operators are defined by R k,z := i z + kimz, L k,z := iimz z. Bruinier Ono see Setions.3 3., Corollary 3.4 of [BO] defined a theta lift by Λf, w := L 3/,w Λ : H N H,ρ Γ 0 N\H R,z fzθw, z, φ KM, Θw, z, φ KM is a C[Z/NZ]-valued theta funtion onstruted from the Kudla-Millson Shwartz funtion φ KM see [KM] [BO, Setion.3]. As a funtion of z, the theta kernel Θw, z, φ KM is a Γ 0 N-invariant harmoni, -form on H. When restriting Λ to weakly holomorphi modular forms, one has see [BO, Theorem 3.5] Λ : M! N M!,ρ. 9 Define the operator 5. Differential operators weak Maass forms := πi z + πy. Then maps weak Maass forms of weight - on Γ 0 N to weak Maass forms of weight 0 on Γ 0 N. It turns out that the image f of a harmoni weak Maass form f H N under the operator an be expressed as a finite linear ombination of ertain Maass- Poinaré series. Let m Z +, define the Maass-Poinaré series see e.g. [B, Setion.3] F N,m z := πm Immγz / I 3/ πmimγze mreγz, z = x + iy H γ Γ \Γ 0 N I 3/ is the I-Bessel funtion of order 3/. It is known that F N,m is a weak Maass form of weight 0 on Γ 0 N. Now, suppose that f M! N has the Fourier expansion Then by [MP, Theorem.], fz = N n= f nq n + f = N n= f nenz. n=0 f nf N,n.

10 0 RIAD MASRI In partiular, the identity implies that 4π R f = 4π R = N n= f nf N,n. 5. As explained in Remark.3, a similar but more ompliated expression exists for any harmoni weak Maass form f H N, but we will not need this here. 6. quantitative equidistribution Let f, f : H C be Γ 0 N-invariant funtions FN be a fundamental domain for Γ 0 N. Define the Petersson inner produt f, f := f zf zdµz dµz := FN dxdy volfn y is the normalized hyperboli measure on FN. Let := y x + y be the weight 0 hyperboli Laplaian A be the omposition of with itself A-times A Z 0. For a usp a of Γ 0 N, let σ a SL R be a saling matrix suh that σ a = a see [I, p. 47]. Proposition 6.. Let g : H C be a C, Γ 0 N-invariant funtion, suppose that for eah usp a we have A gσ a z = Oe Cy, A = 0,,, for some onstant C > 0 depending on a A. Then gτ = h D gzdµz + O ɛ,n g D 6 +ɛ. τ O FN D,N,h Proof. Let {u m } be an orthonormal basis of Heke-Maass usp forms of weight 0 for Γ 0 N with -eigenvalues λ m = + 4 t m for m Z +. For eah usp a define the real-analyti Eisenstein series E a z, s := γ Γ a\γ 0 N Imσ a γz s, z H, Res >. Beause g satisfies the bound 6. we have the spetral expansion gz = g, + u m u m z + m= g, g, E a, 4π + it E a z, + itdt, a R

11 whih onverges pointwise absolutely uniformly on ompat subsets of FN. Summing over the spetral expansion yields gτ = h D g, + u m W m D + τ O D,N,h m= g, g, E a, 4π + it W a D, tdt, a R 6. the hyperboli Weyl sums are defined by W m D := u m τ W a D, t := τ O D,N,h τ O D,N,h E a τ, + it. To estimate the ontribution of the disrete spetrum, it suffies to onsider only L - normalized Heke-Maass newforms for Γ 0 N see e.g. the proof of [HM, Theorem 6]. By a formula of Waldspurger/Zhang see e.g. [W, W] [Z], for suh a newform u m one has u m τ τ O D,N,h = D D am Λu m, Λu m χ D,, 6.3 D is a positive onstant whih takes only finitely many different values, a m is the first Fourier oeffiient of u m, ΛΠ, s := L Π, slπ, s is the ompleted L funtion. Note that the term a m appears on the right h side beause in the Waldspurger/Zhang formula the Heke-Maass newform is arithmetially normalized. By a subonvexity bound of Jutila Motohashi [JM] we have Lu m, ɛ,n λ 6 +ɛ m, by a hybrid subonvexity bound of Blomer Haros [BH, Theorem ] we have Lu m χ D, ɛ,n λ 7 4 +ɛ m D 8 +ɛ. Then using the following estimate of Hoffstein Lokhart [HL] a m ɛ,n λ ɛ me π t, the fat that the ontribution from the infinite parts of the L funtions in 6.3 is e π tm by Stirling s formula, we obtain the estimate W m D ɛ,n λ 3 4 +ɛ m D 6 +ɛ. 6.4 Following the argument in [HM, setion 6.4], one an redue the estimate of W a D, t to an analogous estimate for τ O D,, Eτ, + it Ez, s is the real-analyti Eisenstein series for SL Z. One has the identity see e.g. [GZ, p. 48] τ O D,, Eτ, s = s D s/ ζs ζslχ D, s.

12 RIAD MASRI Then using a stard lower bound for ζs, the subonvexity bound ζ + it 4 + t +ɛ, the following hybrid subonvexity bound of Heath-Brown [HB] we obtain Lχ D, + it ɛ 4 + t 3 3 +ɛ D 4 6 +ɛ, W a D, t ɛ 4 + t ɛ D 6 +ɛ. 6.5 Using the bound 6., a repeated appliation of Stokes theorem see e.g..8] yields the identities g, u m = λ A m A g, u m g, E a, + it = 4 + t A A g, E a, + it. By Parseval s formula see [IK, 5.7], A g, u m + 4π a m= R A g, E a, + it dt = A g. [I, Lemma Then by the Cauhy-Shwarz inequality we have g, u m λ 3 4 +ɛ m g 6.6 R m= g, E a, + it 4 + t ɛ dt g. 6.7 Finally, the proposition follows by ombining 6. with the estimates Definition 6.. Let g : H C be a C, Γ 0 N-invariant funtion α be a real number. We say that g has uspidal growth of power α if for eah usp a of Γ 0 N there exists a onstant a C suh that A gσ a z a y α = Oe Cy, A = 0,,,... for some onstant C > 0 depending on a A. We now ombine Proposition 6. with a suitable regularization to establish the following Theorem 6.3. Suppose that F has uspidal growth of power α < 5/8. Then F τ = h D F zdµz τ O FN D,N,h + O ɛ,n F T0 D 6 +ɛ + O ɛ,n D α+ɛ + O N D α +ɛ,

13 F T0 is a regularized version of F for a ertain onstant T 0 > independent of D see 6.9 { α :=, α 6 α, < α < Proof. For T >, define ψ T t := t α χt/t χ : R + [0, ] is a C funtion suh that { 0, t < χt =, t >. Define the inomplete Eisenstein series let E b ψ T z := γ Γ b \Γ 0 N η T z := b ψ T Imσ b γz b E b ψ T z. Then by [I, 3.7] 8. we have 0, < y < T η T σ a z = a y α χy/t, T y T a y α, y > T. It follows that for y > T the regularized funtion satisfies the bound 6.. Now, let F T z := F z η T z T = T D := + max { D N, 4N π Then by Lemma 6.4 we have η T τ = 0 for all τ O D,N,h, thus F τ = F T τ. 6.8 τ O D,N,h τ O D,N,h Let T 0 be a onstant independent of D with T > T 0 > deompose the regularized funtion as }. F T z = F T0 z + η T z 6.9 η T z := η T0 z η T z. 3

14 4 RIAD MASRI Sine F T0 satisfies the bound 6., it follows from Proposition 6. that F T0 τ = h D F T0 zdµz + O ɛ,n F T0 D 6 +ɛ. τ O FN D,N,h Then by , to omplete the proof it suffies to show that η T τ = h D η T0 zdµz + O ɛ,n D α+ɛ + O N D α +ɛ. τ O FN D,N,h By [I, Theorem ] we have η T z = η T, + b ψt0 πi + it ψ T + it E b z, + itdt, It follows that η T τ = h D τ O D,N,h ED, T := πi b FN R ψt := b b 0 ψtt s+ dt. η T0 zdµz h D η T zdµz + ED, T, FN R ψt0 + it ψ T + it W b D, tdt. By [KMY, Lemma 5.6], for all B > 0 we have ψ T0 + it ψ T + it + t B dt Cα, T, 6.0 R Cα, T := { logt, α T α, α >. Sine T < + 4N π D, we ombine to obtain ED, T = {O ɛ,n D 6 +ɛ, α O ɛ,n D 6 α 4 +ɛ, 8 Finally, a straightforward estimate yields h D η T zdµz = Oh DT α = O ɛ,n D α +ɛ, we used FN h D ɛ D +ɛ.

15 5 Lemma 6.4. For γ Γ a \Γ 0 N τ O D,N,h we have { Imσa γτ max Imτ, 4N } { D max π N, 4N π }. Proof. Reall that we hose O D,N,h so that a Heegner point τ O D,N,h has the form with τ = τ h A = b + D Na a D. 6. π a Write σa b γ = d SL R. Then we have Imσa γτ = Imτ τ + d = D τ + d Na. If = 0, then d = see [I,.5-.7] so that D Imσa γτ N reall that a = N K/Q A. On the other h, if 0 then by 8. we have so that τ + d = b Na + D d + D Na 4N a. Then using 6. we obtain Imσa γτ 4N a D D Na = Na 4N D π. Write 7. Fourier expansion of F N,m z F N,m z = γ Γ \Γ 0 N p m γz p m z := ψ m Imze mz ψ m t := πm mti 3/ πmtemit, t R +. Then by [I, p. 60] the Fourier expansion of F N,m in the usp b is given by F N,m σ b z = δ,b ψ m ye mz + enx S,b m, n; A m n,, y, n Z R +

16 6 RIAD MASRI S,b m, n; is the Kloosterman sum S,b m, n; := a d with group of integral translations { b B := A m n,, y := R B\σ Γ 0 Nσ b /B y ψ m t + y } : b Z e md + na e m t + iy nt dt. In Lemma 7. we will evaluate the integral A m n,, y. Then inserting this evaluation into the Fourier expansion yields F N,m σ b z = δ,b πm 3/ yi 3/ πmye mx + C b my + n Z n 0 C b m, n yk 3/ π n yenx C b m := π3 m 3 S,b m, 0; 3 4 R + πm 3/ S,b m, n; 4π m n J 3, n < 0 C b m, n := R + πm 3/ S,b m, n; 4π mn I 3, n > 0. R + Finally, using the identities ti3/ t = π e t t + e t + t we obtain F N,m σ b z = C b my + δ,b m π tk3/ t = e t +, t e mz + E b m, x, y 7. πmy

17 7 m E b m, x, y := δ,b e πmy + e mx πmy + C b m, n n / e π n y + n Z n 0 enx. π n y Lemma 7.. We have πm 3/ 4π m n yk3/ π n yj 3, n < 0 A m n,, y = π 3 m 3 y, 3 4 n = 0 Proof. Using the identity πm 3/ yk3/ πnyi 3 4π mn M 0,3/ u = 7/ Γ5/ ui 3/ u, n > 0. M 0,3/ is the usual M-Whittaker funtion of order 0, 3/, we find that ψ m t = C mm 0,3/ 4πmtemit C := π π 7/ Γ5/ =. Therefore A m n,, y = m I mn,, y, I m n,, y := R 4πmy M 0,3/ t + y e mt t + y nt dt. By a simple hange of variables, one an show that I m n,, y equals the integral I in [B, p. 33] with the hoies k = 0 s =. Then using the evaluation of the integral I given there, we have m/ n 4π m n C W 0,3/ 4π n yj 3, n < 0 I m n,, y = m C 3 4 y, n = 0 m/n 4π mn C W 0,3/ 4πnyI 3, n > 0, C := πγ4 Γ = π, C 3 := 4π3 Γ4 3Γ = 8π 3,

18 8 RIAD MASRI K 3 J 3 are the usual K J-Bessel funtions of order 3, respetively, W 0,3/ is the usual W -Whittaker funtion of order 0, 3/. Using the identity W 0,3/ u = u/πk 3/ u K 3/ is the usual K-Bessel funtion of order 3/, we have The result now follows after simplifiation. W 0,3/ 4π n y = n yk 3/ π n y. 8. Poinaré series Let λ : R [0, ] be a C funtion suh that { 0, t 0 λt =, t. Let η > 0 define P m,η z := ψ m,η t := λ Then define the regularized funtion Proposition 8.. For y > 4N π γ Γ \Γ 0 N t 4N π η ψ m,η Imγze mγz, m. πmt F N,m,η z := F N,m z P m,η z. + η we have F N,m,η σ b z = C b my + E b m, x, y. In partiular, the regularized funtion F N,m,η has uspidal growth of power α =. Proof. We have the Fourier expansion P m,η σ b z = δ,b ψ m,η ye mz + n Z A m,η n,, y := R enx y ψ m,η t + y R + S,b m, n; A m,η n,, y, 8. e m t + iy nt The funtion ψ m,η : R [0, m] is C satisfies 0, t 4N, π ψ m,η t = m, t 4N π πmt + η. Moreover, sine { min R + : dt. σ a Γ 0 Nσ b } 8.

19 for all usps a, b of Γ 0 N see [I, eqs..8.3], we have for y 4N π y t + y 4N π. It follows from 8. that δ,b ψ m,η ye mz, y 4N π P m,η σ b z = m δ,b e mz, y 4N π πmy + η. 8.3 The proposition now follows from the Fourier expansion 7.. Define the trae Theorem 9.. We have Tr D F N,m = m τ O D,N,h Imτ> 4N π +D traes of weak maass forms Tr D F N,m := τ O D,N,h F N,m τ. as D through all D satisfying the hypothesis H. Here N,m := F N,m zdµz is a Borherds-type regularized integral see Setion 3. e mτ + h D N,m + O ɛ,n D 76 +ɛ πmimτ reg Proof. Assume that 0 < η < 998 we will eventually hoose η to be very small as a funtion 000 of D. By Proposition 8., the regularized funtion F N,m,η has uspidal growth of power α =. Moreover, this growth is uniform in η for y > 4N + 998, hene the same hoie of π 000 onstant T 0 := 4N orresponding funtion η π 000 T 0 given by 0, < y < 4N π 000 η T0 σ b z = C b my 4N χy/, y 4N π 000 π 000 C b my, y > 4N π 000 an be used to regularize eah funtion F N,m,η as in the proof of Theorem 6.3. Upon substituting F N,m,η into Theorem 6.3, we obtain the asymptoti formula Tr D F N,m = Tr D P m,η + h D F N,m,η zdµz 9. FN + O ɛ,n F N,m,η,T0 D 6 +ɛ + O ɛ,n D 6 +ɛ + O N D +ɛ, F N,m,η,T0 z := F N,m,η z η T0 z. 9

20 0 RIAD MASRI By Lemma 9. we have Tr D P m,η = Imτ> 4N π ψ m,η Imτe mτ. Split the sum on the right h side into the ranges Imτ 4N + η Imτ > 4N + η, π π define R N,m,η D := Tr D F N,m m e mτ. πmimτ Then 9. an be written as R N,m,η D = h D FN Imτ> 4N π +η F N,m,η zdµz + O ɛ,n F N,m,η,T0 D 6 +ɛ + O ɛ,n D 6 +ɛ + O N D +ɛ + ψ m,ηimτe mτ. 4N π <Imτ 4N π +η In Lemma 3. we will show that F N,m,η zdµz = FN reg F N,m zdµz =: N,m the right h side is a Borherds-type regularized integral note that the right h side is independent of η. A straightforward estimate yields 4N π <Imτ 4N π +η ψ m,ηimτe mτ m e Λ N,h,η D := {τ O D,N,h : By Lemma. we have the estimate πm 4N π #Λ N,h,η D, 4N π < Imτ 4N π + η}. #Λ N,h,η D = O N ηh D + O ɛ,n η D 6 +ɛ. Moreover, by Lemma. we have the estimate Combining the preeding estimates yields F N,m,η,T0 = O N,m η 0. R N,m,η D = h D N,m + O ɛ,n η 0 D 6 +ɛ + O N ηh D. If we let η = D b for some b > 0, then we used R N,m,η D = h D N,m + O ɛ,n D 6 0b+ɛ + O ɛ,n D b+ɛ h D ɛ D +ɛ.

21 The exponent is optimized when 0b = b, or b = /76, thus 6 R N,m,η D = h D N,m + O ɛ,n D 76 +ɛ. Lemma 9.. We have Tr D P m,η = ψ m,η Imτe mτ. Imτ> 4N π Proof. Write Tr D P m,η = P m,η τ + P m,η τ =: I + II. Imτ 4N π Imτ> 4N π Let γ Γ \Γ 0 N fix a Heegner point τ O D,N,h with Imτ 4N/π. Then by Lemma 6.4, Imγτ 4N/π. Sine ψ m,η t = 0 for t 4N, it follows that π P m,η τ = ψ m,η Imγτe mγτ = 0, γ Γ \Γ 0 N thus I = 0. On the other h, if we fix a Heegner point τ O D,N,h with Imτ > 4N/π, then by 8.3 we have thus P m,η τ = ψ m,η Imτe mτ, II = ψ m,η Imτe mτ. Imτ> 4N π 0. proof of Theorems..6 Proof of Theorem.. By ombining the Bruinier-Ono period formula., the identity 5., Theorem 9. we obtain + Λ f D, h = 4N D Tr D 4π R f = 8N D = D N n= f ntr D F N,n Imτ> 4N π +D 76 M f,n z := 4N N n= M f,n τ + C f,n h D D + O ɛ,nd f nn e nz πnimz ɛ, Reall that we assumed η , whih is equivalent to D for the hoie η = D /76. Sine D 4 by assumption, the latter inequality is satisfied.

22 RIAD MASRI C f,n := 8N N f n N,n = 8N N n= reg n= for the last equality we again used 5.. f nf N,n zdµz = N π R fzdµz, reg Proof of Theorem.6. Reall the Bruinier-Ono formula.3 for the partition funtion, pn = 4 + Λ fp 4n,, for the weakly holomorphi modular form f p M 6! defined by. we have N = fp =. Then by Theorem. we obtain + Λ fp 4n, = 4n τ O 4n,6, Imτ> 4 π +4n 76 M fp,6z = 4 e z πimz C fp,6 = 6 π reg h 4n + M fp,6τ + C fp,6 + O ɛ n ɛ, 4n R f p zdµz. The theorem now follows after multiplying by /4.. Proof of Lemma. In this setion we establish the following estimate by modifying the argument in [D, p ]. Lemma.. For eah number 0 < η we have #Λ N,h,η D = O N ηh D + O ɛ η D 6 +ɛ. Proof. Let 0 < η φ η : R [0, ] be a C funtion suh that φ η is supported on 4N π η, 4N π + η. φ η = on [ 4N π, 4N π + η]. 3 φ η satisfies the bound Define φ A η η A, A = 0,,.. g η z := γ Γ \Γ 0 N φ η Imγz.

23 3 Then we have τ O D,N,h g η τ = τ O D,N,h γ Γ \Γ 0 N γ Ī φ η Imτ τ O D,N,h = + τ O D,N,h τ Λ N,h,η D #Λ N,h,η D. φ η Imγτ + τ O D,N,h τ / Λ N,h,η D φ η Imτ τ O D,N,h φ η Imτ The real-analyti Eisenstein series E z, s has a meromorphi ontinuation to C with a simple pole at s = with residue /volfn see [I, Theorem.3 Proposition 6.3]. Then by [I, eq. 7.] we have Thus g η z = τ O D,N,h g η τ = volfn φ η + πi φ η s := Now, a straightforward estimate yields Moreover, by 6.5 we have It follows that 0 R φ η + ite z, + itdt, φ η uu s+ du. volfn φ η h D + φ η πi + itw D, tdt. R φ η η. W D, t ɛ 4 + t ɛ D 6 +ɛ. τ O D,N,h g η τ = O N ηh D + O ɛ C η D C η := R φ η + it 4 + t ɛ dt. We integrate by parts times use the bound. to obtain 6 +ɛ, whih yields C η η. φ η + it η j= 3 + it j,

24 4 RIAD MASRI Lemma.. For eah number 0 < η < Proof. Fix Y > 0 define. Proof of Lemma. we have A F N,m,η,T0 N,m η 4A+, A =,,.... P Y := {z = x + iy : 0 < x <, y Y } LY := {z = x + iy : 0 < x < }. One an hoose a fundamental domain D for Γ 0 N suh that D a Y := σ a P Y D = DY a D a Y DY := D\ a D a Y has ompat losure is adjaent to eah uspidal zone D a Y along the horoyle σ a LY see [I, Setion.]. Using the SL R-invariane of the measure dµz we find that A F N,m,η,T0 := A F N,m,η,T0 z dµz I + II, D I := A F N,m,η,T0 z dµz D 4N π II := b P 4N π A F N,m,η,T0 σ b z dµz. First we estimate II. By 7., we have reall that η < 998 m F N,m σ b z δ,b ψ 4N m,ηye mz, y < 4N + η π π F N,m,η,T0 σ b z = C b my 4N + E b m, x, y, + η y < 4N π π 000 C b my 4N χy/ + E b m, x, y, π y < 4N π 000 E b m, x, y, y 4N π By splitting the y-integral in II into the different ranges onsidered in., we obtain III := b II = b 4N π +η 4N π 4N π 0 0 A F N,m,η,T0 σ b z dµz = III + O, A F N,m σ b z δ,b m ψ m,ηye mz dµz.

25 5 By linearity of the triangle inequality, IV := b Using the estimate we obtain We onlude that 4N π N π V := b 0 4N π N π III IV + V + O, A F N,m σ b z δ,b m A ψ m,η ye mz dµz 0 δ,b m A ψ m,η ye mz dµz. max x,y [0,] [ 4N π, 4N π ] A ψ m,η ye mz η A, IV η A V η A. II η A. Next we estimate I. Observe that D4N/π an be ontained in a retangle R N := [ B N, B N ] [C N, 4N π ] for some B N 0 < C N 3/. Sine ψ m,η t = 0 for t 4N/π, by 8. we have F N,m,η,T0 z = F N,m z f m,η x, y η T0 z, f m,η x, y := n Z enx C N S m, n; A m,η n,, y. By linearity of three appliations of the triangle inequality, we have I A F N,m z f m,η x, y η T0 z dµz VI + VII + O, R N VI := A F N,m z + A η T0 z A f m,η x, y dµz R N VII := A f m,η x, y dµz. R N Using the estimate we have max A f m,η x, y η 4A+,. x,y R N VI η 4A+ VII η 4A+.

26 6 RIAD MASRI We onlude that I η 4A+. It remains to establish the estimate.. Define y Φ m,η,,y u := ψ m,η e u + y so that Sine A m,η n,, y = Φ m,η,,y u = 0 for u integrating by parts A + -times yields In partiular, we have A m,η n,, y = A f m,η x, y = n Z I m,η,n, x, y := R πin A+ C N For larity, we first assume that A =. Then I m,η,n, x, y = y enx CN πin C N Φ 4 Using the estimate [I,.37] the estimates we obtain m, u + iy Φ m,η,,y ue nudu. C N C N C N, Φ A+ m,η,,y ue nudu. S m, n; A I m,η,n, x, y, enx CN πin A+ Φ A+ m,η,,y ue nudu. CN m,η,,yue nudu y enx CN πin 4 S m, n;, C N max u [ CN, Φ 4 m,η,,yu m,,y η 4 CN ] max u [ CN, yφ 4 m,η,,yu m,,y η 6, CN ] max f m,η x, y η 6. x,y R N yφ 4 m,η,,yue nudu. The preeding argument generalizes in a straightforward way to A, the estimate. follows.

27 3. Regularized integrals First we reall the notion of a regularized integral in the sense of Borherds [Bo] Harvey-Moore [HMo]. Let F be the stard fundamental domain for SL Z. Then a fundamental domain for Γ 0 N is given by FN := σf. For a fixed Y >, define the trunated domains σ Γ 0 N\SL Z F Y := {z F : Imz Y } F Y N := σ Γ 0 N\SL Z We then define the regularized integral F N,m zdµz := lim Y reg For eah η > 0 define the funtion the assoiated Poinaré series P m,η,y z := ψ m,η,y t := γ Γ \Γ 0 N F Y N σf Y. F N,m zdµz. { ψ m,η t, t Y 0, t > Y, ψ m,η,y Imγze mγz. Lemma 3.. For z F Y N we have P m,η,y z = P m,η z. Proof. By definition of ψ m,η,y we have P m,η z = P m,η,y z + γ Γ \Γ 0 N Imγz>Y ψ m,η Imγze mγz. Let γ Γ \Γ 0 N z F Y N. Then γz = Az for some A SL Z z F Y, thus Imγz = ImAz < Imz < Y. It follows that hene P m,η,y z = P m,η z. Lemma 3.. We have FN #{γ Γ \Γ 0 N : Imγz > Y } = 0, F N,m,η zdµz = lim F N,m zdµz. Y F Y N Proof. Sine F N,m,η := F N,m P m,η L FN, F N,m,η zdµz = lim F N,m z P m,η z dµz. Y FN F Y N 7

28 8 RIAD MASRI Now, by Lemma 3. we have P m,η zdµz = F Y N F Y N P m,η,y zdµz. We laim that if z FN \ F Y N, then P m,η,y z = 0. Let γ Γ \Γ 0 N z a b FN \ F Y N. Then γz = Az for some A = SL d Z z = x + iy F \ F Y, i.e., z F with y > Y. We have Imγz = ImAz = y z + d. If = 0 then d =, so that Imγz = y > Y. Sine ψ m,η,y t = 0 for t > Y, it follows that P m,η,y z = 0. On the other h, if 0 then, so that Imγz = y x + d + y y < Y < reall Y >. Sine ψ m,η,y t = ψ m,η t = 0 for t <, it follows that P m,η,y z = 0, whih ompletes the proof of the laim. By the laim we have P m,η,y zdµz = P m,η,y zdµz. F Y N Moreover, unfolding yields Thus we onlude that FN FN FN P m,η,y zdµz = 0. F N,m,η zdµz = lim F N,m zdµz. Y F Y N Remark 3.3. If F N,m = R f for a harmoni weak Maass form f H N, then F N,m zdµz = α a + f,a 0, reg a α a is the width of the usp a + f,a 0 is the onstant term of the holomorphi part of the Fourier expansion of f in the usp a of Γ 0 N. In partiular, for the onstants C f,n Cf p,6 in Theorems..6, respetively, we have C f,n = N α a f,a 0 π C f p,6 = 4π the last sum is over the 4 usps of Γ 0 6. a α a fp,a0, We thank Jan Bruinier for a very helpful orrespondene regarding this fat. a

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30 30 RIAD MASRI [R] H. Rademaher, On the expansion of the partition funtion in a series. Ann. of Math , [R] H. Rademaher, Topis in analyti number theory. Edited by E. Grosswald, J. Lehner M. Newman. Die Grundlehren der mathematishen Wissenshaften, B 69. Springer-Verlag, New York-Heidelberg, 973. ix+30 pp. [W] J.-L. Waldspurger, Sur les oeffiients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl , [W] J.-L. Waldspurger, Sur les valeurs de ertaines fontions L automorphes en leur entre de symétrie. Compos. Math , [Za] D. Zagier, Ramanujan s mok theta funtions their appliations after Zwegers Bringmann- Ono. Séminaire Bourbaki. Vol. 007/008. Astérisque No , Exp. No. 986, viiviii, [Z] S. Zhang, Gross Zagier formula for GL. Asian J. Math. 5 00, Department of Mathematis, Texas A&M University, Mailstop 3368, College Station, TX address: masri@math.tamu.edu