THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION

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1 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG Abstract. In this paper, we use methods from the spectral theory of automorphic forms to give an asymptotic formula with a power saving error term for Andrews smallest parts function sptn. We use this formula to deduce an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n. Our method requires that we count the number of Heegner points of discriminant D < 0 and level N inside an expanding rectangle contained in a fundamental domain for Γ 0 N.. Introduction and statement of results.. Overview. In [A2], Andrews defined the smallest parts function sptn, which counts the number of smallest parts associated to partitions of a positive integer n. For example, spt4 = 0, which can be seen by considering the partitions of n = 4: 4, +, 2 + 2, 2 + +, The function sptn has many remarkable properties. Perhaps most notably, sptn satisfies congruences similar to Ramanujan s congruences for the partition function pn. For some examples of results in this direction, see the works [A2, ABL, FO, G, O]. Bringmann [B] gave an asymptotic formula for sptn in her work on partition statistics. In this paper, we will use methods from the spectral theory of automorphic forms to give an asymptotic formula for sptn with a power saving error term see Theorem.. An important input is recent work of Ahlgren and Andersen [AA] expressing sptn as the trace of a weak Maass form of weight zero for Γ 0 6 along a Galois orbit of Heegner points of discriminant 24n + and level 6 see.. Their expression for sptn is analogous to a formula of Bruinier and Ono [BO] for pn. Using work of Andrews [A, A2] which relates sptn to rank moments and marked Durfee symbols, we will also give an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n see Theorem.6. A large portion of this paper is devoted to counting the number of Heegner points inside an expanding rectangle contained in a fundamental domain for Γ 0 N see Proposition.2. This result is needed for the proofs of our main theorems, though it is of independent interest..2. Preliminaries. To state our main results, we fix the following notation and assumptions. Let N be a squarefree integer and D < 4 be an odd fundamental discriminant coprime to N such that every prime divisor of N splits in K := Q D. Fix a solution h mod of h 2 D mod 4N. Given a primitive integral ideal A of K, one can write A = Za + Z b + D 2, a = NA, b = b h A Z

2 2 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG with b h mod and b 2 D mod 4Na. Here NA denotes the norm of A. Then τ h A := b + D a defines a Heegner point on Y 0 N := Γ 0 N\H. It is known that τ h A Y 0N depends only on the ideal class of A and on h mod, so we denote it by τ h [A]. For details concerning these facts, see [GZ, Part II, Section ]. Let CLK be the ideal class group of K and h D be the class number. By Minkowski s theorem, we may choose a primitive integral ideal A in each ideal class [A] CLK such that NA 2 D. Having fixed such a choice A for each ideal class, we define the Galois orbit O D,N,h := τ h [A] : [A] CLK }. The set of all Heegner points of discriminant D and level N is then given by Λ D,N := O D,N,h. h mod h 2 D mod 4N Let F : H C be a Γ 0 N-invariant function. We define the trace of F by Tr D,N,h F := F τ. τ O D,N,h Next, we define the regularized integral of a weak Maass form of weight zero for Γ 0 N. Let F be the standard fundamental domain for SL 2 Z. For Y > 0, define the set FY := z F : Imz Y }. Then if a is a cusp of Γ 0 N and σ a SL 2 R is a scaling matrix such that σ a = a see e.g. [I2, 2.5 and ], we define the set F N Y := a σ a FY. Let M k! N resp. W kn denote the space of weakly holomorphic resp. weak Maass forms of weight k for Γ 0 N. Given a weak Maass form F W 0 N, we define the regularized integral F, reg := lim F x + iy dxdy Y F N Y y 2 voly 0 N, provided the limit exists. Note that this limit exists for weakly holomorphic forms F M 0N! W 0 N see [M2, Proposition 6.].

3 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION.. Asymptotics for sptn. Consider the weakly holomorphic form fz := E 2 z 2E 2 2z E 2 z + 6E 2 6z 2 ηzη2zηzη6z 2 M! 26, where E 2 z is the usual weight 2 quasi-modular Eisenstein series and ηz is the Dedekind eta function. Define the Maass weight raising operator := 2i z 4 2Imz : M! W 0 N. The image of fz under the operator is a weak Maass form in W 0 6 which we denote by P z. Bruinier and Ono [BO] proved the following formula expressing the partition function pn as a trace of the weak Maass form P z, pn = 24n Tr 24n,6,P.. Let ez := e 2iz. The third author [M, Theorem.6] combined. with methods from the spectral theory of automorphic forms to prove the following asymptotic formula for pn. Theorem. [M], Theorem.6. Let n Z + be such that 24n is squarefree. Then pn = e τ + 24n τ R 24n,6, 2Imτ h 24n + P, reg + O ε n ε, 24n where R 24n,6, := τ O 24n,6, : Imτ > n 76 }. Remark.2. Starting with a different formula for pn due to Bringmann and Ono [BrO], Folsom and the third author [FM] used spectral methods to give an asymptotic formula for pn. Theorem. can in some respects be viewed as a refinement of that result. Ahlgren and Andersen recently proved a formula analogous to. for Andrews smallest parts function sptn. Consider the weakly holomorphic form gz := E 4 z 4E 4 2z 9E 4 z + 6E 4 6z M 24 ηzη2zηzη6z 06,!.2 2 where E 4 z is the usual weight 4 Eisenstein series. Ahlgren and Andersen [AA, Theorem 2] proved that sptn = 2 Tr 24n,6,g P.. By combining. with spectral methods, we will prove the following asymptotic formula with a power saving error term for sptn.

4 4 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG Theorem.. Let n Z + be such that 24n is square-free. Then where sptn = e τ 2 2 τ R 24n,6, + h 24n + 2 R 24n,6, := τ R 24n,6, g, reg P, reg h 24n # τ O 24n,6, : + O ε n ε, + 24n 240 2Imτ 24n τ O 24n,6, : Imτ > 24 e τ 2 24n < Imτ + 24n 240 } 24n Remark.4. In Section 5 we plot the Heegner points in Λ 24n,6 for some small values of n. Note that R 24n,6, R 24n,6,, and moreover, R 24n,6, if and only if n 44, and R 24n,6, if and only if n 444. We now discuss the relation of Theorem. to some existing work. Define the symmetrized second rank moment function η 2 n := m Nn, m, 2 m Z }. 2 where Nn, m denotes the number of partitions of n with rank m. In [A], Andrews proved among many other things that η 2 n equals the number of 2-marked Durfee symbols associated to partitions of n see Section.4. In a subsequent paper, Andrews [A2, Theorem ] proved that Consider the generating function R 2 q := sptn = npn η 2 n..4 η 2 nq n, q := e 2iz. n=0 Bringmann [B, Theorem.] proved that, up to addition by a certain quasi-modular form, R 2 q is the holomorphic part of a harmonic weak Maass form of weight /2 for Γ By combining this result with the circle method, Bringmann [B, Theorem.2] gave an asymptotic formula for η 2 n, or equivalently, the number of 2-marked Durfee symbols associated to partitions of n, with an error term which is On +ε. Given.4 and the known asymptotics for pn, one immediately deduces an asymptotic formula for sptn with an error term which is On +ε. The error term in Theorem. saves a power of n. On the other hand, Bringmann s theorem relating R 2 q to harmonic weak Maass forms is a crucial input to the proof of the Ahlgren-Andersen formula., which was the starting point of our analysis.

5 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION 5.4. Asymptotics for rank moments and Durfee symbols. In their study of partition congruences, Atkin and Garvan [AtG] defined the k-th rank moment function N k n := m Z m k Nm, n. Observe that since Nm, n 0 for only finitely many m Z, the series N k n is finite. The symmetrized k-th rank moment function is defined by η k n := m + k 2 Nm, n. k m Z Given a partition λ = λ,..., λ t of an integer n, we say that λ has a Durfee square of side length s if s t is the largest integer such that the s-th part λ s s. In other words, the largest square contained in the Ferrers graph of λ is of size s s. The Durfee symbol associated to λ is an array consisting of two rows and a subscript. The first row consists of the partition obtained by counting the number of nodes in each column to the right of the Durfee square in the Ferrers graph of λ. The second row consists of the partition below the Durfee square. The subscript is the length of the side of the Durfee square. For example, the following figure illustrates the Durfee square and the Durfee symbol associated to the partition 2 = Figure. The Durfee square and the Durfee symbol associated to the partition 2 = Remark.5. Let R and R 2 denote the first and second rows of the Durfee symbol of λ, respectively. Then the largest part λ = s+#r and the total number of parts t = s+#r 2. Hence the rank of λ is rankλ = λ t = #R #R 2, the number of elements in the first row of the Durfee symbol minus the number of elements in the second row. Andrews [A] introduced a refinement of the Durfee symbol by marking the parts of the partitions appearing in the rows of the Durfee symbol using different copies of the integers. For a positive integer k, we designate k different copies of the positive integers by Z + =, 2,,... }, Z + 2 = 2, 2 2, 2,... },..., Z + k = k, 2 k, k,... }. The k-marked Durfee symbols associated to a partition λ are then defined as follows. Starting from the Durfee symbol associated to λ, we replace the parts in each row by the corresponding parts of the different copies of the integers Z + i according to the following rules. The sequence of entries and the sequence of subscripts in each row must be nonincreasing. 2 For each j k, at least one element of Z + j must appear as a part in the first row.

6 6 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG For each j k, let M j denote the largest element of Z + j appearing as a part in the first row and let M 0 := and M k := s, where s is the length of the side of the Durfee square associated to λ. Then for i k, every element of Z + i appearing as a part in the second row must lie in the interval [M i, M i ]. A symbol formed from the Durfee symbol associated to λ by the above rules is called a k-marked Durfee symbol associated to λ. For example, there are six 2-marked Durfee symbols associated to the partition 7, 5,,,, 2 of 2, whose Durfee symbol and Durfee square are displayed in Figure. These 2-marked Durfee symbols are given in the following list: Let D k n denote the number of k-marked Durfee symbols associated to partitions of n. Andrews [A, Corollary ] proved that η 2k n = D k+ n,.5 thus providing a combinatorial interpretation of the 2k-th symmetrized rank moment function. Now, the identities.4 and.5 imply that D 2 n = npn sptn..6 Hence by combining.6, Theorem. and Theorem., we get the following asymptotic formula with a power saving error term for D 2 n. Theorem.6. Let n Z + be such that 24n is square-free. Then where D 2 n = e τ + cn 2 2 τ R 24n,6, + h 24n + 2 τ R 24n,6, cn P, reg g, reg h 24n n n # τ O 24n,6, : 2 + O ε n ε, cn := 6n 24n. 2Imτ 2 e τ 24n < Imτ } 24n 2

7 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION 7.5. Acknowledgments. We would like to thank Sheng-Chi Liu for helpful discussions regarding this work. The authors were supported in part by the NSF grants DMS-6255 R.M. and DMS Texas A&M U. mathematics REU. In addition, the second author was supported in part by the University of Costa Rica. 2. Traces of weakly holomorphic forms Let f M 0N! be a weakly holomorphic form of weight zero for Γ 0 N. Such a form fz has a Fourier expansion in the cusp at infinity given by M fz = a ne nz + anenz n=0 for some integer M 0. The following asymptotic formula for the trace of fz was proved by the third author in [M2, Theorem.]. Theorem 2. [M2], Theorem.. We have M Tr D,N,h f = a n e nτ + h D f, reg + O N,ε D n=0 τ R D,N,h n= ε. The first few terms in the Fourier expansion at infinity of the weakly holomorphic form g M! 06 defined by.2 are gz = e z ez Given 2., Theorem 2. immediately implies the following result. Proposition 2.2. Let n Z + be such that 24n is square-free. Then Tr 24n,6, g = e τ + h 24n + g, reg + 2 #R 24n,6, + O ε n τ R 24n,6, Remark 2.. The classical modular j-function jz = e z ez ε. is a weakly holomorphic modular form of weight zero for SL 2 Z. The values of jz at Heegner points are algebraic integers called singular moduli. Zagier [Za] proved that traces of singular moduli are Fourier coefficients of a weight /2 modular form for Γ 0 4 in Kohnen s plus space. Bruinier, Jenkins and Ono [BJO] studied the asymptotic distribution of these traces and conjectured the precise form of the limiting distribution. This conjecture was proved by Duke in [D].. Counting Heegner points in an expanding rectangle The appearance of #R 24n,6, in Proposition 2.2 leads us naturally to the problem of counting Heegner points in an expanding rectangle. We will need the following lemma. Lemma.. Let γ Γ \Γ 0 N with γ I. Then given a Heegner point τ O D,N,h, we have Imγτ 4N.

8 8 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG Proof. If γ Γ \Γ 0 N with γ I, then a b γ = SL c d 2 Z with c 0 hence c 2. Recall that τ O D,N,h has the form with Now, we have Since c 2, we get Then. implies that Imγτ = cτ + d 2 = τ = τ h A = b + D a a = NA 2 D.. Imτ cτ + d = D 2 cτ + d 2 a. 2 cb a + d + Imγτ 4N 2 a 2 D c D a 2 D 4N 2 a 2. D a = a D 4N. Let Lχ D, s be the Dirichlet L function associated to the Kronecker symbol χ D. Proposition.2. For b > 0 and 0 < δ < /2, define 4N R D,N,h,b,δ := τ O D,N,h : + D b Imτ D Assume that Lχ D, 2 + it ε + t B+ε D 4 δ +ε for some 0 B < and 0 < δ /4. Then for any δ 2 > b, we have #R D,N,h,b,δ = h D voly 0 N where 4N + D b D 2 δ + O N,ε D 2 b+ε + O N,ε D 2 δ b+ε 2 δ + O N,ε D 2 f δ,δ,b+ε + O N,B,ε D 2 f 2δ,δ,δ 2,B+ε, f δ, δ, b := δ 2 + δ b, 4 f 2 δ, δ, δ 2, B := B + δ + δ B + δ 2 B }..2

9 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION 9 Remark.. The asymptotic formula in Proposition.2 is meaningful for any b, δ satisfying the inequalities 0 < b < δ, 2 b + 4 δ < δ < 2, B + 4 δ + B + δ 2 + B < δ < There exists a δ satisfying the second and third inequalities if 2 b + 4 δ < 2 b < δ and B δ + B + δ 2 + B < 2 2 δ 2 < δ B +. The Lindelöf hypothesis implies that.2 holds with B = 0 and δ = /4. In this case, the asymptotic formula is meaningful for any b, δ such that 0 < b < /4 and 2δ 2 < δ < /2 where b < δ 2 < /4. Recently, M. Young [Y] proved that.2 holds with B = /6 and δ = /2, i.e., Weyl-subconvexity in both the t and D-aspects see [Y, Theorem., eqn..8] and the discussion following [Y, Theorem.]. In this case, the asymptotic formula is meaningful for any b, δ such that 0 < b < /2 and max2b +, δ 2} < δ < /2 where b < δ 2 < /4. Proof of Proposition.2: Let λ, λ 2 : R [0, ] be C functions such that 0, t 0 λ t =, t, λ, t 0 2t = 0, t. Define the functions t 4N φ t := λ D b t D/2 δ, φ 2 t := λ 2 D b Then φ : R [0, ] is a C function which is supported on 4N, D/2 δ [ 4N + D b, D/2 δ ], and satisfies the bound, φt = φ D,N,b,δ t := φ tφ 2 t. + D b, equals on φ A D Ab, A = 0,, 2,.... where the implied constant is independent of D, N and δ. To ease notation, we set R b,δ := R D,N,h,b,δ. Define the incomplete Eisenstein series g φ z := φimγz. Then by Lemma., we have τ O D,N,h g φ τ = τ O D,N,h = #R b,δ + γ Γ \Γ 0 N γ Γ \Γ 0 N γ Ī φimγτ + τ O D,N,h φimτ τ / R b,δ φimτ..4

10 0 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG The real-analytic Eisenstein series E z, s := γ Γ \Γ 0 N Imγz s, Res > has a meromorphic continuation to C with a simple pole at s = with residue /voly 0 N see [I, Theorem. and Proposition 6.]. Then by [I, 7.2], we have g φ z = voly 0 N φ + φ 2 2i + ite z, + itdt, 2 where φs := 0 R φuu s+ du is the Mellin transform of φ. Sum over the Heegner points τ O D,N,h to get g φ τ = h D voly 0 N φ + φ 2i + itw 2 D, tdt, τ O R D,N,h where W D, t := A straightforward calculation yields φ = 4N τ O D,N,h E τ, 2 + it. + D b D + /2 δ OD b..5 We will estimate φ 2i + itw 2 D, tdt R by dividing the integral over R into two regions. First, we integrate by parts A-times and use the bound. to get A φ + it 2 N D Ab D 4 + δ D 2 δ t Next, we combine.2 with an argument similar to that in [M2, Proposition 2.] to get W D, t ε + t B+ε D 2 δ +ε.7 for some 0 B < and 0 < δ /4. Now, fix δ 2 > b, and divide the integral over R into the regions t D 2 δ+δ 2 and t D 2 δ+δ 2. Since A can be chosen to be arbitrarily large, by applying.6 and.7 in the first region, we get I := 2i t D 2 δ+δ 2 for any C > 0 here we used δ 2 > b. φ 2 + itw D, tdt N,B,ε D C

11 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION On the other hand, by applying.6 with A = 0 and.7 in the second region, we get II := φ 2 2i + itw D, tdt N,B,ε D 2 f 2δ,δ,δ 2,B+ε, where t D 2 δ+δ 2 f 2 δ, δ, δ 2, B := B + δ + δ B + δ 2 B Combining the estimates for I and II yields φ 2 2i + itw D, tdt = O N,B,ε D 2 f 2δ,δ,δ 2,B+ε..8 Using.5,.8, and the bound we get R τ O D,N,h g φ τ = h D ε D 2 +ε,.9 h D voly 0 N 4N + D b D /2 δ + O N,ε D 2 b+ε + O N,B,ε D 2 f 2δ,δ,δ 2,B+ε. From.4, we see that it remains to estimate the contribution of τ / R b,δ φimτ. Observe that τ / R b,δ φimτ = 4N # φimτ + Imτ< 4N +D b τ O D,N,h : + # τ O D,N,h : D /2 δ <Imτ D/2 δ 4N Imτ < 4N + D b D /2 δ } D/2 δ < Imτ +D b φimτ + D b }.0 =: R b + R b,δ.. The numbers R b and R b,δ can be estimated using a modification of the proof of.0. We sketch the estimate of R b,δ, leaving the simpler estimate of R b to the reader. Let ψ : R [0, ] be a C function which is supported on D/2 δ equals on [ D/2 δ, D/2 δ + D b ], and satisfies the bound D b, D/2 δ + 2D b, ψ A D Ab, A = 0,, 2,....2 where the implied constant is independent of D, N and δ. Define the incomplete Eisenstein series g ψ z := ψimγz. γ Γ \Γ 0 N

12 2 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG A minor variant of.4 shows that Again, we have τ O D,N,h g ψ τ = where ψ is the Mellin transform of ψ. A straightforward estimate yields #R b,δ τ O D,N,h g ψ τ. h D voly 0 N ψ + ψ 2i + itw 2 D, tdt, R ψ D b.. Now, integrate by parts A-times, then use.2 and the expansion to get + X α = + αx + O α X 2, X <, α R ψ 2 + it A,N D A b + t D A 2 δ A 2..4 Then using.4 with A = 2 and.7, we get ψ 2 2i + itw D, tdt = O N,ε D 2 f δ,δ,b+ε,.5 where From.,.5 and.9, we get A similar argument shows that R f δ, δ, b := δ 2 + δ b. 4 #R b,δ = O N,ε D 2 b+ε + O N,ε D 2 f δ,δ,b+ε..6 #R b = O N,ε D 2 b+ε + O N,ε D 2 δ b+ε..7 Finally, by combining.4,.0,.,.6 and.7, for δ 2 > b we get #R b,δ = h D voly 0 N + D b D /2 δ 4N + O N,ε D 2 b+ε + O N,ε D 2 δ 2b+ε + O N,ε D 2 f δ,δ,b+ε + O N,B,ε D 2 f 2δ,δ,δ 2,B+ε.

13 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION Combine. and. to get 4. Proof of Theorem. sptn = 2 Tr 24n+,6,g 24n pn. 2 Now, by Proposition 2.2 and Theorem., we have sptn = e τ 2 2 τ R 24n,6, τ R 24n,6, h 24n + + g, reg P, reg + #R 24n,6, 2 + O ε n ε. e τ 4. 2Imτ On the other hand, by Proposition.2 with the choices B = /6, δ = /2, b = /240, δ 2 = /29 and δ = 8/478, we get #R 24n,6,, 240, = h 24n n n O ε n ε, 4.2 where we used voly 0 6 = 4 see Remark 4. for an explanation of these choices of parameters. Since #R 24n,6, = #R 24n,6,, 240, # τ O 24n,6, : 24n < Imτ } 24n, 2 the result follows by combining 4. and 4.2. Remark 4.. To get 4.2, we inserted the best unconditional constants B = /6 and δ = /2 due to M. Young [Y]. Then given b = /240, we chose δ 2 so that δ could be made small see Remark.. Of course, we could choose δ 2 and δ so that the exponent in the error term is smaller, at the expense of increasing the size of δ. 5. Heegner points of small discriminant We used SageMath to plot the Heegner points in Λ 24n,6 on a fundamental domain for Γ 0 6 for some small values of n. Figures 2 and display the Heegner points in Λ 59,6 and O 59,6,, respectively.

14 4 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG Figure 2. Heegner points in Λ 59,6. Figure. Heegner points in O 59,6,. Figure 4 displays the Heegner points τ Λ 9999,6 with Imτ > , where the points τ R9999,6, appear as circles these are the points in Theorems. and.6 when n = Imz = Figure 4. Heegner points τ Λ 9999,6 with Imτ >

15 THE ASYMPTOTIC DISTRIBUTION OF ANDREWS SMALLEST PARTS FUNCTION 5 References [AA] S. Ahlgren and N. Andersen, Algebraic and transcendental formulas for the smallest parts function. Preprint available at: arxiv: [ABL] S. Ahlgren, K. Bringmann and J. Lovejoy, l-adic properties of smallest parts functions. Adv. Math , [A] G. E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math , 7 7. [A2] G. E. Andrews, The number of smallest parts in the partitions of n. J. Reine Angew. Math , 42. [AtG] A. O. L. Atkin and F. Garvan, Relations between the ranks and cranks of partitions. Ramanujan J , [B] K. Bringmann, On the explicit construction of higher deformations of partition statistics. Duke Math. J , [BrO] K. Bringmann and K. Ono, An arithmetic formula for the partition function. Proc. Amer. Math. Soc , [BJO] J. H. Bruinier, P. Jenkins, and K. Ono, Hilbert class polynomials and traces of singular moduli. Math. Ann , 7 9. [BO] J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. Adv. Math , [D] W. Duke, Modular functions and the uniform distribution of CM points. Math. Ann , [FM] A. Folsom and R. Masri, Equidistribution of Heegner points and the partition function. Math. Ann , [FO] A. Folsom and K. Ono, The spt-function of Andrews. Proc. Natl. Acad. Sci. USA , [G] F. Garvan, Congruences for Andrews smallest parts partition function and new congruences for Dyson s rank. Int. J. Number Theory [GZ] B. Gross and D. Zagier, Heegner points and derivatives of L series. Invent. Math , [I] H. Iwaniec, Introduction to the spectral theory of automorphic forms. Biblioteca de la Revista Matemática Iberoamericana. Revista Matematica Iberoamericana, Madrid, 995. xiv+247 pp. [I2] H. Iwaniec, Topics in classical automorphic forms. Graduate Studies in Mathematics, 7. American Mathematical Society, Providence, RI, 997. xii+259 pp. [M] R. Masri, Fourier coefficients of harmonic weak Maass forms and the partition function. American Journal of Mathematics 7 205, [M2] R. Masri, Singular moduli and the distribution of partition ranks modulo 2. Mathematical Proceedings of the Cambridge Philosophical Society, to appear. [O] K. Ono, Congruences for the Andrews spt function. Proc. Natl. Acad. Sci. USA 08 20, [Y] M. P. Young, Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets. Journal of the European Mathematical Society, to appear. Preprint available at: arxiv: v2 [Za] D. Zagier, Traces of singular moduli. Motives, polylogarithms and Hodge theory, Part I Irvine, CA, 998, 2 244, Int. Press Lect. Ser.,, I, Int. Press, Somerville, MA, Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH address: jmbanks0@student.ysu.edu Department of Mathematics, Mailstop 68, Texas A&M University, College Station, TX address: adrianbs@math.tamu.edu address: masri@math.tamu.edu

16 6 JOSIAH BANKS, ADRIAN BARQUERO-SANCHEZ, RIAD MASRI, YAN SHENG Mathematics & Computer Science, Mail Stop: -002-AC, Emory University, Atlanta, GA 022 address: