THE AVERAGE OF THE DIVISOR FUNCTION OVER VALUES OF A QUADRATIC POLYNOMIAL

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1 THE AVERAGE OF THE DIVISOR FUNCTION OVER VALUES OF A QUADRATIC POLYNOMIAL SHENG-CHI LIU AND RIAD MASRI Abstract. We establish a uniform asymptotic formula with a power saving error term for the average of the divisor function τn := k n over values of the quadratic polynomial x + where D < 0 is a fundamental discriminant.. Introduction and statement of results It is an important problem in number theory to study the asymptotic distribution of averages like fp n n M as M where f is some arithmetical function and P Z[x] is a quadratic polynomial. A prototype for this problem is the classical divisor function τn := k n. For D a fixed integer such that D is not a perfect square, Hooley [H] established the asymptotic formula τn + D = a,d logmm + a,d M + O D logm 3 M 8 9, n M where a,d and a,d are explicit non-zero constants depending on D, and the implied constant in the error term depends on D here one sets τm = 0 if m 0. The error term in Hooley s result was improved by Deshouillers-Iwaniec [DI] and Bykovskii [By] to O ε,d M 3 +ε for any ε > 0 using methods from the spectral theory of automorphic forms. Assuming that D < 0 is a fundamental discriminant, we will establish a uniform asymptotic formula for the average of τn over the quadratic polynomial x + in which we save a power of both M and in the error term. This problem gives rise to considerable new challenges, particularly in the critical range M. Our main result is the following Theorem.. Let M and D < 0 be a fundamental discriminant. Then τn + = c,d log + c,d M n M + O ε M ε max 4 4, M where c,d := 8Lχ D, ζ Y X u / du

2 SHENG-CHI LIU AND RIAD MASRI and c,d := 8Lχ D, ζ with X = symbol. Y X + M and Y = logu + γ ζ u / du ζ + L χ D, Lχ D, log χ D 3 χ D + 4M. Here γ is Euler s constant and χ D is the Kronecker Remark.. The asymptotic formula in Theorem. is meaningful for with a power saving error term. M > 67 +ε There has recently been a great amount of interest in asymptotic formulas or upper bounds for averages like fn + D, n M with a particular emphasis on obtaining a wide range of uniformity in both M and D see e.g. [DFI], [FI, section 4.8], [FI], [M], [Te], [TT]. Such uniformity is often crucial for numbertheoretic applications. For example, Duke, Friedlander, and Iwaniec [DFI] established a uniform bound for averages of Weyl sums for quadratic roots and used it to study the distribution of prime points on the sphere and cycle integrals of the j function, among other things. Friedlander and Iwaniec [FI] later used this bound to study representations of integers by indefinite ternary quadratic forms. Namely, if rn denotes the number of representations of n as the sum of two squares and D > 0, they established a uniform asymptotic formula for the average rn + DF n M n M where F x is suitable smooth function supported on [M, M]. Their asymptotic formula is meaningful for M > D 33 with a power saving error term. To conclude the introduction we briefly explain how Theorem 8. which is the smooth version of Theorem. can be used to give an asymptotic formula with a power saving error term for the second moment Lχ, χ where χ varies over the ideal class group characters of an imaginary quadratic field Q D and Lχ, s is the L function of χ. A beautiful asymptotic formula for the second moment was first given by Duke, Friedlander, and Iwaniec [DFI] using a different method see also [Bl], [Te]. Let E z, / be the central derivative of the weight zero Eisenstein series for P SL Z and let Θ χ be the weight one theta series of level associated to χ. One has the Fourier expansion E z, = y log e γ y 4π + 4 y n τn cosπnxk 0 πny. The Rankin-Selberg L function of E z, / and Θ χ is given by LE χ, s = n a n χ n s, Res >

3 where a n χ := m k=n χ D m Na=k χaτk, the inner sum being over integral ideals a of norm Na = k in Q D. Using the identity LE χ, s = Lχ, s, the problem becomes equivalent to the asymptotic evaluation of LE χ,. χ After an application of the approximate functional equation [IK, Theorem 5.3] and orthogonality relations for the characters χ, we split the sum into diagonal and off-diagonal terms. The analysis of the diagonal term reduces to a standard contour shift and application of Burgess s [Bu] subconvexity bound for Lχ D, s, while the analysis of the off-diagonal term reduces to an asymptotic formula for a smooth average with the Fourier coefficients τn of the type considered in Theorem 8.. Acknowledgments. We would like to thank Matt Young for very helpful discussions, Professors Friedlander and Iwaniec for their interest in this work, and the referee for suggestions which improved the exposition. The second author was partially supported by the NSF grant DMS-6535 during the preparation of this work.. Outline of the proof In this section we outline the proof of Theorem 8., from which we will deduce Theorem. by un-smoothing. Let s 0 C and φ :, C be a C function with compact support. Define the absolutely convergent series Φ s0 φ; z := Imσz s 0 σz φ Imσz σ Γ where Γ = P SL Z. Bykovskii [By, Lemma 6] established the identity σ s0 n n + φ + = s0/ Tr DΦ s0 φ;. n Z where D < 0 is an integer, and σ s0 n := d n d s 0 Tr DΦ s0 φ; := Φ φ; z s0 Q #Γ Q z Q Λ D is the trace of Φ s0 φ, z over the CM points z Q Λ D of determinant D in the standard fundamental domain F for Γ see section 3. In Theorem 4. we establish an alternative version of Bykovskii s result [By, Theorem b] which gives the spectral expansion of Φ s0 φ; z in L F. We then substitute this expansion into., take the trace and let s 0 0 to obtain an identity of the form see Theorem 4.3 τn + φ n Z n + = M D φ + E D φ. 3

4 4 SHENG-CHI LIU AND RIAD MASRI To evaluate the main term M D φ, we establish a Chowla-Selberg type formula for any negative discriminant see Lemma 6.. To estimate the contribution of the discrete spectrum to error term E D φ, we use a period formula of Katok and Sarnak [KS] for Fourier coefficients of weight zero Maass forms, an explicit version of Waldspurger s formula due to Baruch and Mao [BM] for Fourier coefficients of half-integral weight Maass forms, and subconvexity bounds of Conrey-Iwaniec [CI] and Blomer-Harcos [BH] for twisted L functions on GL. To estimate the contribution of the continuous spectrum to error term E D φ, we use a period formula for Eisenstein series on P SL Z and a subconvexity bound of Burgess [Bu] for Dirichlet L functions. Considerable care must be taken in these arguments because of the presence of non-fundamental discriminants. 3. CM points Let D < 0 be an integer and let Q D be the set of positive definite, integral, binary quadratic forms QX, Y = ax + bxy + cy with even middle coefficient and determinant b ac = D recall that the determinant of Q is 4 where 0 mod 4 is the discriminant of Q. The two roots of the dehomogenized form QX, are complex conjugates. Let z Q := b + D a be the root in the complex upper half-plane H. The group Γ = P SL Z acts on Q D with finite quotient Q D /Γ of class number hd := #Q D /Γ. Let F be the standard fundamental domain for Γ in H. Each class in Q D /Γ contains exactly one form Q with z Q F. We denote the set of all such CM points by Λ D := {z Q F : Q Q D /Γ}. Let Γ Q < Γ be the stabilizer of Q. We define the trace of a Γ-invariant function h : H C by Tr Dh := hz Q. #Γ Q z Q Λ D For the analysis in this paper, we will also need traces defined using CM points associated to quadratic forms of any negative discriminant < 0, and in particular, the relationship between these traces and Tr D h when 0 mod 4 see e.g. the proof of Theorem 8.. Let < 0 be an integer with 0, mod 4 i.e., is negative discriminant and let Q be the set of positive definite, integral, binary quadratic forms Q X, Y = a X + b XY + c Y of discriminant b 4a c =. The two roots of the dehomogenized form Q X, are complex conjugates, and we let z Q := b + a be the root in H. The group Γ acts on Q with finite quotient Q /Γ of class number h := #Q /Γ, and each class in Q /Γ contains exactly one form Q with z Q F. We let Λ := { z Q F : Q Q /Γ } and define the trace Tr h := z Q Λ hz Q. #Γ Q Finally, suppose that 0 mod 4 and write = 4D for some integer D < 0. Then Q = Q D, Λ = Λ D and Tr h = Tr Dh. 3.

5 5 4. A formula of Bykovskii Define the hyperbolic Laplacian := y x + y, z = x + iy H. Let {u j z} j= be an orthonormal basis of weight zero Hecke-Maass cusp forms for Γ with - eigenvalues λ j and let Ez, s := Imγz s, Res > γ Γ \Γ be the weight zero Eisenstein series for Γ. expansion The Hecke-Maass cusp form u j z has the Fourier where t j := λ j 4 u j x + iy = a j nw +it nx + iy, j n Z n 0 is the spectral parameter, W +it j x + iy = y K itj π y e πix is the Whittaker function and K s u is the modified Bessel function of the second kind. Let λ j n be the n-th Hecke eigenvalue of u j z. Then λ j = and a j ±n = a j ±λ j n n /. Iwaniec [I] and Hoffstein-Lockhart [HL] proved that Define the Dirichlet series t j ε ε a j coshπt j ε t j ε. 4. L j s := n Z n 0 a j n n s = a j n Z n 0 λ j n n s, Res >. Let s 0 C and φ :, C be a C function with compact support. Define the absolutely convergent series Φ s0 φ; z := Imσz s 0 σz φ. Imσz σ Γ Bykovskii calculated the spectral expansion of Φ s0 φ; z in [By, Theorem b]. The integral transform appearing in Bykovskii s formula is a somewhat complicated expression involving hypergeometric functions. We will establish the following alternative version of Bykovskii s formula in which the integral transform is expressed in a way which makes it easier to estimate. The proof is a modification and elaboration of Bykovskii s argument.

6 6 SHENG-CHI LIU AND RIAD MASRI Theorem 4.. For 0 < Res 0 < / we have where Φ s0 φ; z = 4Ez, + s 0 + 4Ez, s 0 + φu u / du G s0 φ; t j L j + s 0u j z j= + π ε R φuu s 0 u / du G s0 φ; tπ it ζ + it + s 0ζ it + s 0 Γ itζ it Ez, + itdt G s0 φ; t := π s 0 4πi Γ 4 + it + s 0 Γ 4 it + s 0 Γ s + it + s 0 Γ s + s 0 φ Γ s it Γ s it + s s it ds, ε > 0, t R and φ is the Mellin transform Proof. Define the function Then a short calculation yields Φ s0 φ P ; ; z = For c >, we have the formula πi Apply this formula with to obtain where the series c φs := φ P ; u := σ Γ 0< σz Imσz P 0 φuu s du. { P u, 0 < u P 0, u > P. Imσz s 0 σz Imσz P σz Imσz { X, X ss + Xs+ ds = 0, 0 X <. X = P σz Imσz. Φ s0 φ P ; ; z = πi c ss + P s+ Πz; s, s 0 ds, 4. Πz; s, s 0 := σ Γ Imσz s 0 σz s Imσz is absolutely convergent and holomorphic for Res 0 > 0 and Res > see [By, p. ].

7 Now let φ :, C be a C function with compact support. calculation yields the identity Φ s0 φ; z = 7 Then a straightforward φ P Φ s0 φ P ; ; zdp. 4.3 Substitute 4. into 4.3, reverse the order of integration and integrate by parts twice with respect to P to get Φ s0 φ; z = Πz; s, s 0 πi φsds, 4.4 c where φ is the Mellin transform φs := 0 φp P s dp. From [By, Lemma ], if 0 < Res 0 < / and Res > then Πz; s, s 0 = s Γ π Γ s Ez, + s 0 + s Γ + s 0 π Γ s + s 0 Ez, s π s 0 + Γ s Γ s + s Γs, s 0 ; t j L j 0 + s 0u j z where j= + π s 0 π Γ s Γ s + s 0 R Γs, s 0 ; tπ it ζ + it + s 0ζ it + s 0 Γ itζ it Ez, + itdt, Γs, s 0, t := Γ 4 + it + s 0 Γ 4 it + s 0 Γ s 4 + it + s 0 Γ s 4 it + s 0. Next substitute 4.5 into 4.4 to get where and Φ s0 φ; z = πez, + s 0 I 0 φ + πez, s 0 I s0 φ + π s 0 L j 4πi + s 0u j zγ 4 + it j + s 0 Γ 4 it j + s 0 Iφ; t j j= + π s 0 π 4πi Iφ; t := R π it Γ 4 + it + s 0 Γ 4 it + s 0 Iφ; t ζ + it + s 0ζ it + s 0 Γ itζ it Ez, + itdt, I α φ := Γ s + α πi c Γ s + α φsds, α C c Γ s 4 + it + s 0 Γ s 4 it + s 0 Γ s Γ s + s 0 φsds. To complete the proof, we use Lemma 4. and the identity Γ s Iφ; t = + it + s 0 Γ s + s 0 φ Γ s it Γ s it + s s it ds, ε > 0 ε which follows by changing variables s s + + it and shifting the contour c to ε.

8 8 SHENG-CHI LIU AND RIAD MASRI Lemma 4.. We have I α φ = φuu α u / du. π Proof. Recall that Γ s + α has simple poles at s = n + α for n =, 0,... with residue n+ /n +!. Thus I α φ := Γ s + α πi c Γ s + α φsds = φ α n+ φ n + α Γ +. n +! Γ Using the identities Γ/ = π and along with the Taylor expansion we get Γ n+ + x / = + n=0 = π n+ n+ n! n +! n=0 Γ s + α πi c Γ s + α φsds = π Define the Eisenstein series Ẽz, s := m,n Z m,n 0,0 n+ n +! n+ n +!n! xn+, x < φuu α + n=0 = φuu α u / du. π y s n+ n +! n+ n +!n! u n du s, z = x + iy H, Res >. mz + n By Kronecker s first limit formula see e.g. [L, chapter 0], Ẽz, s = π + π γ log F z + Os 4.6 s where F z := log y ηz and ηz is the Dedekind eta function. Using the identity we obtain the expansion Ez, s = 3 π Ẽz, s = ζsez, s, s + 6 γ log ζ π ζ F z + Os. 4.7 Substitute 4.7 into Theorem 4., apply the identity see [By, Lemma 6] σ s0 n n + φ + = s0/ Tr DΦ s0 φ; n Z where D < 0 is an integer and σ s0 n := d s 0, d n and let s 0 0 to obtain the following analog of [By, Theorem 5 c].

9 Theorem 4.3. Let D < 0 be an integer and φ :, C be a C function with compact support. Then τn n + φ + = 4.8 n Z 4 Tr 48 D logu + γ log ζ φu u / du π π ζ 48 π Tr DF φu u / du + G 0 φ; t j L j Tr Du j j= + π R G 0 φ; tπ it ζ + it Γ itζ ittr DE, + itdt. Finally, we will repeatedly use the following estimate for G 0 φ; t. Lemma 4.4. For X < Y X and P, let φ :, C be a C function supported on [X, Y ] such that X A φ A, A = 0,,.... P Then G 0 φ; t X +ε P A + t A e π t. Proof. Integrating by parts A-times and using the bound for φ A yields φs X Res P A + s A The result now follows from 4.9 and Stirling s formula. 5. Periods of Eisenstein series Let < 0 be an integer with 0, mod 4 and write = f d for f Z + and d < 0 a fundamental discriminant. Following Zagier [Z, p. 09, eq. 6], we define the zeta function ζ s := Q, Res >. m, n s Q Q /Γ m,n Z /Γ Q Q m,n>0 Then a standard calculation yields the identity see e.g. [Z, pp. 80-8] s Tr E, s = c ζ s 5. ζs where c > 0 is a constant depending on #Γ Q which takes finitely many different values. On the other hand, Zagier [Z, Proposition 3 iii] showed that ζ s = L f,d sζslχ d, s where χ d := d is the Kronecker symbol, Lχ d, s = χ d nn s, Res > n=

10 0 SHENG-CHI LIU AND RIAD MASRI is the Dirichlet L function associated to χ d and L f,d s := µαχ d ασ s f/αα s α f is a finite Dirichlet series where µ is the Möbius function and The preceding facts imply σ ν m := β m β>0 Tr E, s = c ζs ζs β ν, m Z +. f s d L f,dslχ d, s. 5. Lemma 5.. Let < 0 be an integer with 0, mod 4 and write = f d for f Z + and d < 0 a fundamental discriminant. Then Tr E, + it ε + t 7 6 +ε f +ε d 7 6 +ε. Proof. This follows from 5., the bound see [T, Theorem 5.] ζ + it ε t 6 +ε, 5.3 a standard lower bound for ζs on Res = / see [T, eq ] and Burgess s [Bu] subconvexity bound Lχ d, + it ε + t d 3 6 +ε. 6. A formula for Tr F We will need the following Chowla-Selberg type formula. Lemma 6.. Let < 0 be an integer with 0, mod 4 and write = f d for f Z + and d < 0 a fundamental discriminant. Then Tr F = Tr γ log log L χ d, Lχ d, L f,d. 6. L f,d Proof. Recall that Then 5. yields the identity ζ s = c Ẽz, s = ζsez, s. s Tr Ẽ, s. Using the expansion s = + log s + Os and the Kronecker limit formula 4.6, we obtain the expansion ζ s = πc Tr s + γ log log Tr Tr F + Os.

11 Recall also that Then letting we obtain the expansion ζ s = L f,d sζslχ d, s. Z f,d s := L f,d slχ d, s, ζ s = Z f,d s + Z f,dγ + Z f,d + Os. By comparing coefficients in the two expansions of ζ s, we get Z f,d = πc Tr and Z f,d Together, these yield the identity or equivalently πc = Tr γ log log Tr Tr F. Z f,d Z f,d = γ log log Tr Tr F, Tr F = Tr γ log log L χ d, Lχ d, L f,d L f,d. Let 7. Periods of Maass forms and Waldspurger s formula k := iky x, k Z be the weight k hyperbolic Laplacian. Let u j z be an even Hecke-Maass cusp form of weight zero for Γ with -eigenvalue λ j = 4 + t j and u j, u j =. The theta lift g j τ := θu j, τ defined in [KS, p. 06] is a Maass form of weight for Γ 04 with / -eigenvalue 4 + t j 4. The Fourier expansion of g j τ is given by see [KS, p. 96, eq. 0.0] g j u + iv = ρ j nw signn, it j 4π n venu, τ = u + iv H. 4 n 0 In particular, one has ρ j n = 0 if n, 3 mod 4. If n < 0 with n 0, mod 4, one has the following period formula of Katok and Sarnak [KS], Write ρ j n = 4 π 3/4 n 3/4 Tr nu j. 7. Z f,d s = L f,d slχ d, s =: n= ε n n s,

12 SHENG-CHI LIU AND RIAD MASRI and define the L series L, u j, s := n= ε nλ j n n s = α f α s One has the following explicit Waldspurger type formula. Theorem 7.. Let < 0 be an integer with 0, mod 4. Then n, f α = χ d nλ j nf n s, Res >. ρ j g j, g j = π Γ 3 4 it j Γ it j a j L, u j,. Proof. If = d is a fundamental discriminant this follows from Baruch and Mao [BM, Theorem.4]. If = f d for f > this follows by modifying [BM, Theorem.4] along the lines of the proof of Kohnen and Zagier [KZ, Corollary 4]. Lemma 7.. Let < 0 be an integer with 0, mod 4 and write = f d for f Z + and d < 0 a fundamental discriminant. Then ρ j ε f tj 5 + tj d ε d 5 6. Proof. By Theorem 7., the bound 4. and Stirling s formula, we have Define the L series ρ j ε g j, g j e π t j L, u j,. Lχ d u j, s := n= χ d nλ j n n s, Res >. Then one has the factorization see the proof of [KZ, Corollary 4] L, u j, = Lχ d u j, µαχ d nα λ j f/α α f. Using the Kim-Sarnak estimate [K, appendix ] λ j n n ε and the hybrid subconvexity bound of Blomer and Harcos [BH, eq..3] we obtain By [DFI, Proposition 3], Lχ d u j, ε + t j 3 + t j d ε d 3 8, L, u j, ε f + t j 4 + t j d ε d 3 8. g j, g j ε cosh π t j Combining the preceding estimates yields + t j +ε. ρ j ε f tj 5 + tj d ε d 5 6. Lemma 7.3. Let < 0 be an integer with 0, mod 4 and write = f d for f Z + and d < 0 a fundamental discriminant. Then Tr u j ε f tj 5 + tj d ε d 7 6 +ε.

13 3 Proof. By 7., Tr u j = 4 π 3/4 3/4 ρ j. The result now follows from Lemma The smooth average of τn over x + In this section we establish an asymptotic formula for the smooth average of τn over x +. Theorem 8.. Let D < 0 be a fundamental discriminant. For X < Y X and P, let φ :, C be a C function supported on [X, Y ] such that X A φ A, A = 0,...,. P Then τn n + φ + = c,d φ log + c,d φ + O ε X +ε P ε n Z where and c,d φ := 8Lχ D, ζ c,d φ := 8Lχ D, ζ φu u / du logu + γ ζ ζ + L χ D, Lχ D, log χ D φu u / du. 3 χ D Proof. Suppose that D < 0 is a fundamental discriminant, and let = 4D. Then 0 mod 4 is a negative discriminant, and if we write = f d for f Z + and d < 0 a fundamental discriminant, we have f = and d = D. Recall also that from 3., we have Tr Dh = Tr h for any Γ-invariant function h : H C. For notational convenience, we write Theorem 4.3 as τn n + φ + = M D φ + E D φ n Z where M D φ consists of the first and second terms on the right hand side of 4.8 and E D φ consists of the third and fourth terms on the right hand side of 4.8. To evaluate the main term M D φ, we use the identity see [Z, eq. 0] Tr D = π Lχ D, and the following identity which is implied by 6., Tr DF = π Lχ D, Then a straightforward calculation yields where γ log log4 L χ D, Lχ D, L,D L,D M D φ = c,d φ log + c,d φ c,d φ := 8Lχ D, ζ φu u / du.

14 4 SHENG-CHI LIU AND RIAD MASRI and c,d φ := 8Lχ D, ζ To evaluate observe that logu + γ ζ ζ + L χ D, Lχ D, + L,D φu u / du. L,D L,D L,D L,D s = µαχ D α α k α hence another straightforward calculation yields k s α s, L,D L,D = log χ D 3 χ D. We now estimate the error term E D φ. For u j z even, we have where Lu j, s := L j s = 4a j Lu j, s n= λ j n n s, Res >. Then Lemmas 4.4 and 7.3 yield Gφ, t j L j Tr Du j ε XP ε X 7 6 P 8 j= t j P XP ε Lu j, ε XP ε X P Similarly, using Lemmas 4.4 and 5., the bound 5.3, the lower bound for ζs on Res = / and Stirling s formula, we obtain Gφ, t π it ζ + it Γ itζ DE, ittr + itdt ε X +ε P ε. 8. R Finally, by combining these estimates we get E D φ = O ε XP ε X P Proof of Theorem. For X < Y X and P, let φ :, C be a C function such that 0 φ, φ is supported on [X, Y ], 3 φ on the interval [X + X P, Y X P ], and φ satisfies the bound φ A Using properties 3 and the trivial estimate X A, A = 0,...,. P τc ε c ε,

15 we find that M n M τn + = n Z τn + φ n + + O ε X +ε P +ε, with X = + M and Y = + 4M. By Theorem 8. we have τn n + φ + = c,d φ log + c,d φ + O ε X +ε P ε. n Z Next we un-smooth the coefficients c,d φ and c,d φ. Using properties 3, a straightforward estimate yields φu logu u / du = Y X logu u / du + OX +ε P. The same estimate holds with logu replaced by. Then using the upper bound we obtain L j χ D, ε D ε, j = 0, c,d φ = c,d + O ε X +ε P ε, c,d φ = c,d + O ε X +ε P ε, where c,d and c,d are defined as in the statement of Theorem.. Combining the preceding estimates yields τn + = c,d + c,d log M n M Finally, to optimize the error term we choose + O ε X +ε P +ε + O ε X +ε P ε + O ε X +ε P +ε. P = 68 max, References [BM] E. Baruch and Z. Mao, A generalized Kohnen-Zagier formula for Maass forms. J. Lond. Math. Soc. 8 00, 6. [Bl] V. Blomer, Non-vanishing of class group L-functions at the central point. Ann. Inst. Fourier Grenoble , [BH] V. Blomer and G. Harcos, Hybrid bounds for twisted L functions. J. Reine Angew. Math , [Bu] D. A. Burgess, On character sums and L-series. Proc. London Math. Soc. 96, 93 06; II, ibid , [By] V. A. Bykovskii, Spectral expansions of certain automorphic functions and their number-theoretic applications. Russian Automorphic functions and number theory, II. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. LOMI , English translation in: Journal of Soviet Mathematics , 8. [CI] B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic L-functions. Ann. of Math , [DI] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math /83, [DFI] W. Duke, J. Friedlander, and H. Iwaniec, Class group L functions. Duke Math. J , 56. M. 5

16 6 SHENG-CHI LIU AND RIAD MASRI [DFI] W. Duke, J. Friedlander, and H. Iwaniec, Weyl sums for quadratic roots. Int. Math. Res. Not. IMRN 0 doi: 0.093/imrn/rnr. [FI] J. Friedlander and H. Iwaniec, Opera de cribro. American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 00. xx+57 pp. [FI] J. Friedlander and H. Iwaniec, Small representations by indefinite ternary quadratic forms, preprint. [HL] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero with an appendix by D. Goldfeld, J. Hoffstein and D. Lieman, Ann. of Math , 6 8. [H] C. Hooley, On the number of divisors of a quadratic polynomial. Acta Math , [I] H. Iwaniec, Small eigenvalues of Laplacian for Γ 0N. Acta Arith , [IK] H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 004. xii+65 pp. [KS] S. Katok and P. Sarnak, Heegner points, cycles and Maass forms. Israel J. Math , [K] H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL. With appendix by Dinakar Ramakrishnan and appendix by Kim and Peter Sarnak. J. Amer. Math. Soc , [KZ] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of critical strip. Invent. Math , [L] S. Lang, Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics,. Springer-Verlag, New York, 987. xii+36 pp. [M] R. Masri, The asymptotic distribution of traces of cycle integrals of the j function, Duke Math. J., 6 0, [Te] N. Templier, Heegner points and Eisenstein series. Forum Math. 3 0, [Te] N. Templier, A nonsplit sum of coefficients of modular forms. Duke Math. J. 57 0, [TT] N. Templier and J. Tsimerman, Non-split sums of coefficients of GL-automorphic forms, Israel J. Math , [T] E. C. Titchmarsh, The theory of the Riemann zeta-function. Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 986. x+4 pp. [Z] D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. Modular functions of one variable, VI Proc. Second Internat. Conf., Univ. Bonn, Bonn, 976, pp Lecture Notes in Math., Vol. 67, Springer, Berlin, 977. [Z] D. Zagier, Eisenstein series and the Riemann zeta function. Automorphic forms, representation theory and arithmetic Bombay, 979, pp. 7530, Tata Inst. Fund. Res. Studies in Math., 0, Tata Inst. Fundamental Res., Bombay, 98. Department of Mathematics, Washington State University, Pullman, WA address: scliu@math.wsu.edu Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX address: masri@math.tamu.edu