Maass forms and the mock theta function f(q)
MMAASS FORMS AND THE MOCK THETA FUNCTION f p q q SCOTT AHLGREN AND ALEXANDER DUNN
Dedicated to George Andrews on the occasion of his 80th birthday.
Abstract.
Let f p q q : “ ` ř n “ α p n q q n be the well-known third order mock theta ofRamanujan. In 1964, George Andrews proved an asymptotic formula of the form α p n q “ ÿ c ď? n ψ p n q ` O (cid:15) p n (cid:15) q , where ψ p n q is an expression involving generalized Kloosterman sums and the I -Bessel func-tion. Andrews conjectured that the series converges to α p n q when extended to infinity, andthat it does not converge absolutely. Bringmann and Ono proved the first of these conjec-tures. Here we obtain a power savings bound for the error in Andrews’ formula, and we alsoprove the second of these conjectures.Our methods depend on the spectral theory of Maass forms of half-integral weight, andin particular on an average estimate for the Fourier coefficients of such forms which gives apower savings in the spectral parameter.As a further application of this result, we derive a formula which expresses α p n q withsmall error as a sum of exponential terms over imaginary quadratic points (this is similarin spirit to a recent result of Masri). We also obtain a bound for the size of the error termincurred by truncating Rademacher’s analytic formula for the ordinary partition functionwhich improves a result of the first author and Andersen when 24 n ´
23 is squarefree. Introduction
Let f p q q : “ ` ÿ n “ α p n q q n “ ` ÿ n “ q n p ` q q p ` q q ¨ ¨ ¨ p ` q n q be the famous third-order mock theta function of Ramanujan. One may consult for example[BO06], [BFOR17], [Duk14], and [Zag09] and the references therein for an account of thesubstantial body of research related to this and to other mock theta functions. Part of theimportance of the function f p q q arises from the fact that the coefficients α p n q are related toa fundamental combinatorial statistic. In particular, we have α p n q “ N e p n q ´ N o p n q , where these denote the number of partitions of even and odd rank respectively.Ramanujan recorded an asymptotic formula for α p n q in his last letter to Hardy in 1920; thiswas proved in 1951 by Dragonette [Dra52]. Andrews [And66] made a major breakthrough Date : March 11, 2019.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . N T ] M a r SCOTT AHLGREN AND ALEXANDER DUNN in his 1964 Ph.D. thesis by proving the remarkable formula (valid for any (cid:15) ą α p n q “ π p n ´ q t ? n u ÿ c “ p´ q t c ` u A c ´ n ´ c p `p´ q c q ¯ c I ˆ π ? n ´ c ˙ ` O (cid:15) p n (cid:15) q . (1.1)Here I is the I –Bessel function of order 1 {
2, and A c p n q is the generalized Kloosterman sum A c p n q : “ ÿ d p mod c qp d,c q“ e πis p d,c q e ˆ ´ dnc ˙ , c, n P N , (1.2)where s p d, c q is the Dedekind sum defined in (3.2) below and e p x q : “ exp p πix q .Andrews [And66, p. 456], [And03, §
5] conjectured that α p n q “ π p n ´ q ÿ c “ p´ q t c ` u A c ´ n ´ c p `p´ q c q ¯ c I ˆ π ? n ´ c ˙ , (1.3)and that the series does not converge absolutely. We note that by a result of Lehmer [Leh38,Theorem 8] we have the Weil-type bound ˇˇˇ A c ´ n ´ c p ` p´ q c q ¯ˇˇˇ ď ω o p c q ? c, (1.4)where ω o p c q is the number of distinct odd prime divisors of c (we give a refinement of thisresult in Section 9 below). This bound does not suffice to prove convergence.The formula (1.3) was proved by Bringmann and Ono [BO06] in 2006 using the theoryof harmonic Maass forms, and in particular the work of Zwegers [Zwe02] which packagedWatson’s transformation properties [Wat36] for f p q q in a three dimensional vector of real-analytic modular forms.Here we return to the question of obtaining an effective error estimate for the approxi-mation to α p n q by the truncation of the series (1.1). To this end, we define the error term R p n, N q by α p n q “ π p n ´ q ÿ c ď N p´ q t c ` u A c ´ n ´ c p `p´ q c q ¯ c I ˆ π ? n ´ c ˙ ` R p n, N q . (1.5)Then the result of Andrews gives R p n, ? n q ! (cid:15) n (cid:15) . Our first main result gives a power-saving improvement.
Theorem 1.1.
Suppose that n ´ is positive and squarefree. Then for all (cid:15) ą and γ ą we have R p n, γ ? n q ! γ,(cid:15) n ´ ` (cid:15) . As a corollary, we see that when n is sufficiently large, α p n q is the closest integer to thetruncated sum appearing in (1.1). It would be interesting to quantify what “sufficientlylarge” means here.It is also interesting to note that if one assumes the Ramanujan–Lindel¨of conjecture forthe coefficients of Maass cusp forms of weight 1 { R p n, γ ? n q ! γ,(cid:15) n ´ ` (cid:15) , AASS FORMS AND THE MOCK THETA FUNCTION f p q q while the analogue of the Linnik–Selberg conjecture for the sums of generalized Kloostermansums which arise in the proof would give R p n, γ ? n q ! γ,(cid:15) n ´ ` (cid:15) . We prove the second conjecture of Andrews mentioned above using a character sum iden-tity proved in Section 9 together with an equidistribution result of Duke, Friedlander andIwaniec [DFI95] for solutions of quadratic congruences to prime moduli.
Theorem 1.2.
The series (1.3) does not converge absolutely for any value of n . For squarefree values of 24 n ´
1, Masri [Mas16, Theorem 1.3] obtained an asymptoticformula of the form α p n q “ M p n q ` O (cid:15) ´ n ´ ` (cid:15) ¯ , where M p n q is a twisted finite sum of terms exp p π Im τ q as τ ranges over distinguishedGalois orbits of Heegner points on X p q . This relies on a general power saving bound fortraces of modular functions over such orbits, as well as a result of Alfes [Alf14] which relatesthe values α p n q to traces of certain real-analytic modular functions.As a consequence of Theorem 1.1 and the results in Section 9, we obtain an asymptoticformula for α p n q as a sum over a set of quadratic points in the upper half-plane H . Supposethat D ą
0, and define Q ´ D, : “ (cid:32) ax ` bxy ` cy : b ´ ac “ ´ D, | a, a ą ( . Then Γ p q acts on this set from the left and preserves b p mod 12 q . If Q “ r a, b, c s P Q ´ D, , define χ ´ p Q q “ ` ´ b ˘ , and let τ Q denote the root of Q p τ, q in H , so that gτ Q “ τ gQ for g P Γ p q . From this discussion the summands in the next theorem are well-defined. Theorem 1.3.
Suppose that n ´ is positive and squarefree. Then for all (cid:15) ą and γ ą we have α p n q “ i ? n ´ ÿ Q P Γ z Q ´ n, Im τ Q ą γ χ ´ p Q qp e p τ Q q ´ e p τ Q qq ` O γ,(cid:15) ´ n ´ ` (cid:15) ¯ . Our methods depend on bounds for sums of Kloosterman sums attached to a half-integralweight multiplier on Γ p q which are uniform with respect to all parameters. These dependin turn on the spectral theory of Maass forms, and in particular on average bounds forthe coefficients ρ p n q of such forms which are uniform both in the argument n and withrespect to the Laplace eigenvalue λ . We rely on three such bounds to treat various rangesof the parameters involved. The first two are recent results of Andersen-Duke and of thefirst author with Andersen. The third is a new bound which is important in obtaining theexponent ´ which appears in the theorems above. This result (which holds in any level)gives a significant improvement in λ aspect at the cost of a small loss in n aspect; it isrecorded as Theorem 4.3 below.The series (1.3) is reminiscent of Rademacher’s well-known series [Rad36, Rad43] for theordinary partition function p p n q : p p n q “ π p n ´ q ÿ c “ A c p n q c I ˆ π ? n ´ c ˙ SCOTT AHLGREN AND ALEXANDER DUNN (this does converge absolutely, in contrast with (1.3)). A classical problem is to estimatethe error associated with truncating this series, and the methods of this paper give animprovement for this estimate. In analogy with (1.5), define S p n, N q by p p n q : “ π p n ´ q ÿ c ď N A c p n q c I ˆ π ? n ´ c ˙ ` S p n, N q . Rademacher [Rad36, Rad43] proved that S p n, γ ? n q ! γ n ´ , and Lehmer [Leh38] improvedthis to S p n, γ ? n q ! γ n ´ log n . When 24 n ´
23 is squarefree, Folsom and Masri [FM10]proved that S ´ n, b n ¯ ! n ´ ´ δ for some δ ą . Recently the first author and Andersen [AA18] obtained the bound S ` n, γ ? n ˘ ! γ,(cid:15) n ´ ´ ` (cid:15) . As another application of Theorem 4.3, we obtain the following.
Theorem 1.4.
Suppose that n ´ is positive and squarefree. Then for all (cid:15) ą and γ ą we have S p n, γ ? n q ! γ,(cid:15) n ´ ´ ` (cid:15) . We close with a brief outline of the contents of the paper. In the next section, we developsome background material on the spectral theory of automorphic forms and Kloostermansums. In Section 3 we develop the properties of a particular multiplier which is related tothe coefficients of f p q q .To prove Theorem 1.1 requires bounds for sums of Kloosterman sums which are uniform inall parameters. The analysis in Section 7 is similar to that of [AA18] (and is similar in spiritto the work of Sarnak and Tsimerman [ST09] in weight 0 on SL p Z q , although significantcomplications arise from the multiplier of weight ). The Kuznetsov trace formula is thebasic tool to relate sums of Kloosterman sums to the coefficients of Maass forms. Section 6contains a version of the Kuznetsov formula in the mixed sign case for half integral weightmultipliers, Section 7 contains the analysis which proves Theorem 1.1, and Section 8 containsa sketch of the proof of Theorem 1.4.In Section 4 we state three estimates for coefficients of Maass forms which are crucial forour work. The first is a mean value estimate which was recently proved by Andersen andDuke [AD18]. The second, which was proved in [AA18], is an average version of a well-knownresult of Duke [Duk88].The third is the average estimate Theorem 4.3 mentioned above; the proof of this resultis quite involved, and occupies Section 5. We make crucial use of a version of the Kuznetsovtrace formula which appears in a recent paper of Duke, Friedlander and Iwaniec [DFI12]. Wefollow the basic method of Duke [Duk88] but with a modified test function which leads to asavings with respect to the spectral parameter. Duke’s method relies in turn on estimates ofIwaniec [Iwa87] for averages of Kloosterman sums in level aspect. One of the terms arisingfrom the Kusnetsov formula is an infinite sum over weights (cid:96) of spaces of holomorphic cuspforms, and much of the technical difficulty arises from the need to bound the summandsuniformly in terms of (cid:96) . AASS FORMS AND THE MOCK THETA FUNCTION f p q q In Section 9 we prove the key identity Proposition 9.1 which expresses the Kloostermansums arising in (1.1) as Weyl-type sums, and we use this identity to prove Theorem 1.3.Finally, in the last section we prove Theorem 1.2.In the body of the paper we make the following convention: in equations which involvean arbitrary small positive quantity (cid:15) , the constants which are implied by the notation ! or O are allowed to depend on (cid:15) . Any other dependencies in the implied constants will beexplicitly noted. Acknowledgments
We thank Nick Andersen and Wadim Zudilin for their helpful comments. We also thankthe referee for comments which improved our exposition.2.
Background
We begin with some brief background material on Maass forms with general weight andmultiplier. For more details one may consult [AA18, Section 2], [DFI12, Section 2] (whereit is assumed that the cusp is singular), or [Sar84].Let k be a real number and let H denote the upper half-plane. For γ “ ˆ a bc d ˙ P SL p R q and τ “ x ` iy P H , we define j p γ, τ q : “ cτ ` d | cτ ` d | “ e i arg p cτ ` d q and the weight k slash operator by f ˇˇ k γ : “ j p γ, τ q ´ k f p γτ q , where we choose the argument in p´ π, π s .For each k , the Laplacian ∆ k : “ y ˆ B B x ` B B y ˙ ´ iky BB x commutes with the weight k slash operator.For simplicity we will work only with the groups Γ p N q for N P N and with weights k P Z ,although much of what is said here holds in more generality. Let Γ denote such a group. Wesay that ν : Γ Ñ C ˆ is a multiplier system of weight k if ‚ | ν | “ ‚ ν p´ I q “ e ´ πik , and ‚ ν p γ γ q j p γ γ , τ q k “ ν p γ q ν p γ q j p γ , τ q k j p γ , γ τ q k for all γ , γ P Γ.Given a cusp a , let Γ a : “ t γ P Γ : γ a “ a u denote the stabilizer in Γ and let σ a denote theunique (up to translation on the right) matrix in SL p R q satisfying σ a a and σ ´ a Γ a σ a “ Γ . Define α ν, a P r , q by the condition ν ˆ σ a ˆ ˙ σ ´ a ˙ “ e p´ α ν, a q . The cusp a is singular with respect to ν if α ν, a “
0. When a “ 8 we suppress the subscript. SCOTT AHLGREN AND ALEXANDER DUNN If ν is multiplier of weight k , then it is a multiplier in any weight k ” k mod 2, and ν is a multiplier of weight ´ k . If α ν “ α ν “
0, while if α ν ą α ν “ ´ α ν . For n P Z we define n ν : “ n ´ α ν ;then we have n ν “ ´p ´ n q ν if α ν ‰ ,n if α ν “ . (2.1)With this notation we define the generalized Kloosterman sum (at the cusp ) by S p m, n, c, ν q : “ ÿ ď a,d ă cγ “ p a bc d q P Γ ν p γ q e ˆ m ν a ` n ν dc ˙ . We have the relationships S p m, n, c, ν q “ S p ´ m, ´ n, c, ν q if α ν ą S p´ m, ´ n, c, ν q if α ν “
0. (2.2)Two important multipliers of weight are the eta-multiplier ν η on SL p Z q , given by η p γτ q “ ν η p γ q? cτ ` d η p τ q , γ “ ˆ a bc d ˙ P SL p Z q , and the theta-multiplier ν θ on Γ p q , given by θ p γτ q “ ν θ p γ q? cτ ` d θ p τ q , γ “ ˆ a bc d ˙ P Γ p q . Here η p τ q and θ p τ q are the two fundamental theta functions η p τ q : “ q ź n “ p ´ q n q ,θ p τ q : “ ÿ n “´8 q n , where we use the standard notation q : “ e p τ q “ e πiτ . For ν θ we have the formula ν θ ˆˆ a bc d ˙˙ “ ´ cd ¯ (cid:15) ´ d , where ` ‚‚ ˘ is the extended Kronecker symbol and (cid:15) d “ d ” p mod 4 q ,i if d ” p mod 4 q . From this we obtain ν θ p γ q “ ´ ´ d ¯ ν θ p γ q , γ “ ˆ a bc d ˙ P Γ p q . AASS FORMS AND THE MOCK THETA FUNCTION f p q q With dµ : “ dx dyy , define (cid:107) f (cid:107) “ ż Γ p N qz H | f | dµ. Denote by L k p N, ν q the space of L functions which satisfy f p γτ q “ j p γ, τ q k ν p γ q f p τ q for all γ P Γ p N q . (2.3)Let B k p N, ν q denote the subspace of L k p N, ν q consisting of smooth functions f such that f and ∆ k f are bounded on H . Then ∆ k has a unique self-adjoint extension to L k p N, ν q ,which we also denote by ∆ k . For each singular cusp a (and only at such cusps) there is anEisenstein series E a p z, s q . These provide the continuous spectrum, which covers r { , .The reminder of the spectrum is discrete and of finite multiplicity. We denote the discretespectrum by λ ď λ ď . . . , where we have λ ě | k | ˆ ´ | k | ˙ . One component of the discrete spectrum is provided by residues of the Eisenstein series E a p z, s q at possible simple poles s with ă s ď
1; the corresponding eigenvalues have λ ă .The remainder of the discrete spectrum arises from Maass cusp forms.Denote by r L k p N, ν q the subspace of L k p N, ν q spanned by eigenfunctions of ∆ k . If f P r L k p N, ν q has Laplace eigenvalue λ , then we write λ “ ` r , r P i p , i { s Y r , , and refer to r as the spectral parameter of f . Denote by r L k p N, ν, r q the subspace of suchfunctions. Let W κ,µ denote the usual W -Whittaker function. Then each f P r L k p N, ν, r q hasa Fourier expansion of the form f p τ q “ c p y q ` ÿ n ν ‰ ρ p n q W k sgn p nν q ,ir p π | n ν | y q e p n ν x q , (2.4)where c p y q “ $’&’% α ν ‰ , α ν “ r ě ρ p q y ` ir if α ν “ r P i p , { s ,with coefficients ρ p n q (see [Pro03, p. 3878] or [DFI12, p. 2509]) . Note that in the last case,we have ρ p q ‰ f arises as a residue.Complex conjugation gives an isometry (of normed spaces) r L k p N, ν, r q ÐÑ r L ´ k p N, ν, r q . If f P r L k p N, ν, r q , then using (2.4) with (2.1) and the fact that W κ,µ P R when κ P R and µ P R Y i R [DLMF, (13.4.4), (13.14.3), (13.14.31)], we find that the coefficients ρ c p n q of f c : “ f satisfy ρ c p n q “ ρ p ´ n q if α ν ą n ‰ ρ p´ n q if α ν “
0. (2.5)
SCOTT AHLGREN AND ALEXANDER DUNN
The Maass lowering operator L k : “ k ` iy BB τ “ k ` iy ˆ BB x ` i BB y ˙ gives a map r L k p N, ν, r q ÝÑ r L k ´ p N, ν, r q and satisfies (cid:107) L k f (cid:107) “ ˆ r ` p k ´ q ˙ (cid:107) f (cid:107) “ ´ λ ´ k ´ ´ k ¯¯ (cid:107) f (cid:107) . (2.6)From the last equation, we see that if f P r L k p N, ν q has the minimal eigenvalue | k | ´ ´ | k | ¯ ,then f p τ q is in the kernel of L k if k ě
0, and f p τ q is in the kernel of L ´ k if k ă
0. Let M k p N, ν q denote the space of holomorphic modular forms of weight k and multiplier ν onΓ p N q . Using (2.3), it follows that the function F p τ q : “ y ´ k f p τ q if k ě ,y k f p τ q if k ă M k p N, ν q if k ě M ´ k p N, ν q if k ă
0. Also, F p τ q is a cusp form if and only if f p τ q is a Maass cusp form.Suppose that f P r L k p N, ν, r q . Then using (2.4) and [AA18, (2.16)], we have the expansion L k f p τ q “ c p L q p y q ` ÿ n ν ‰ ρ p L q p n q W p k ´ q sgn p nν q ,ir p π | n ν | y q e p n ν x q , where ρ p L q p n q “ $&% ´ ˆ r ` p k ´ q ˙ ρ p n q if n ν ą ρ p n q if n ν ă
0, (2.7)and c p L q p y q “ ´ k ´ ´ ir ¯ c p y q . A multiplier on Γ p q and a formula for the coefficients of f p q q Here we relate the coefficients α p n q to Kloosterman sums attached to a multiplier ψ onΓ p q . For the eta-multiplier defined in the last section, we have a formula of Rademacher[Rad73, (74.11), (74.12)] which is valid for c ą ν η p γ q “ ?´ i e ´ πis p d,c q e ˆ a ` d c ˙ , (3.1)where s p d, c q is the Dedekind sum s p d, c q “ c ´ ÿ r “ rc ´ drc ´ Y drc ] ´ ¯ . (3.2)For c ą γ “ p a bc d q we have another convenient formula [Kno70, § ν η p γ q “ $’&’%´ dc ¯ e ´ “ p a ` d q c ´ bd p c ´ q ´ c ‰¯ if c is odd , ´ cd ¯ e ´ “ p a ` d q c ´ bd p c ´ q ` d ´ ´ cd ‰¯ if c is even. (3.3) AASS FORMS AND THE MOCK THETA FUNCTION f p q q We have ν η p˘p b qq “ e ` b ˘ . Finally, if c ą ν η p´ γ q “ iν η p γ q (this follows since γ and ´ γ act the same way on H ).For γ “ ˆ a bc d ˙ P Γ p q define ψ p γ q “ i c { ` ´ d ˘ ν η p γ q if c ” p mod 4 q ,i c { ν η p γ q if c ” p mod 4 q . (3.4)One can compute using (3.1) to see that the real-analytic form Ă M appearing on page 251 of[BO06] satisfies Ă M p γτ q “ ψ p γ qp cτ ` d q Ă M p τ q for all γ “ ˆ a bc d ˙ P Γ p q .(this can also be derived from [And66, Theorem 2.2]). By [Hej83, Proposition 2.1] it followsthat ψ is a multiplier of weight on Γ p q .For the cusp , we have ψ pp qq “ e ` ´ ˘ , so that α ψ “ . For the cusp 0 we may take σ “ ´ ´ {? ? ¯ . Then the formulas above give ψ ˆ σ ˆ ˙ σ ´ ˙ “ ψ ˆˆ ´ ˙˙ “ e ´ ´ ¯ . Lemma 3.1.
Let A c p n q and ψ be defined as in (1.2) , (3.4) . Then for c ą we have p´ q t c ` u A c ˆ n ´ c p ` p´ q c q ˙ “ e ´ ¯ S p , n, c, ψ q . Proof.
This follows from a case-by-case computation using (3.1) together with the fact that s p´ d, c q “ ´ s p d, c q . (cid:3) Since α p n q P Z , the formula (1.3) becomes α p n q “ π p n ´ q e ´ ´ ¯ ÿ c ą c ” p mod 2 q S p , n, c, ψ q c I ˆ π ? n ´ c ˙ . (3.5)We will work in the space r L p , ψ q . By the discussion above, neither cusp is singular, sothere are no Eisenstein series for this multiplier. Lemma 3.2.
For each r , the map τ ÞÑ τ gives an injection r L p , ψ, r q ÝÑ r L ´ , ´ ‚ ¯ ν θ , r ¯ . Proof.
It is enough to check the transformation law. Given f P r L p , ψ, r q , define g p τ q : “ f p τ q “ f ˇˇ ´ ?
24 00 1 {? ¯ . If γ “ p a bc d q P Γ p q with c ą
0, then we have g ˇˇ γ “ f ˇˇ γ ´ ?
24 00 1 {? ¯ , where γ “ ` a bc { d ˘ . Then a case by case computation using (3.4) and (3.3) shows that ψ p γ q “ ´ d ¯ ν θ p γ q . The identities e ` ´ d ˘ “ ` d ˘ (cid:15) d and ` ´ d ˘ (cid:15) d “ (cid:15) ´ d for odd d are useful for this computation. (cid:3) Three estimates for the coefficients of Maass forms
The proofs of our main results will depend on three different average estimates for theFourier coefficients of Maass forms.The first is a restatement of a recent result of Andersen and Duke [AD18, Theorem 4.1](we state this only in the case of the cusp ). Suppose that ν is a multiplier of weight forΓ p N q and that for n ν ‰ ÿ c ą c ” p mod N q | S p n, n, c, ν q| c ` β ! ν | n ν | (cid:15) (4.1)holds for some β P p , q .The result of [AD18] is stated for positive n in weights ˘ . To derive the statement belowwe use (2.1), (2.2), and (2.5). Note that the assumption (4.1) differs slightly from that of[AD18] in that n ν appears in place of n on the right side; an examination of the proof showsthat this is sufficient. This allows us to access the case when n “ n ν ă
0, which isimportant in our applications.Whenever we speak of an orthonormal basis t v j p τ qu for r L p N, ν q , we assume that each v j is an eigenform of ∆ with eigenvalue λ j and spectral parameter r j “ a λ j ´ { Proposition 4.1 (Andersen-Duke) . Suppose that ν is a multiplier on Γ p N q of weight which satisfies (4.1) . Let b j p n q denote the coefficients of an orthonormal basis t v j p τ qu for r L p N, ν q . Then we have | n ν | ÿ x ď r j ď x | b j p n q| e ´ πr j ! (cid:15),N ` x ` | n | β ` (cid:15) x ´ β log β x ˘ ¨ x ´ if n ν ą , x if n ν ă . The second estimate is a restatement of [AA18, Proposition 8.2].
Proposition 4.2.
Let D be an even fundamental discriminant and let N be a positive in-teger with D | N . Let b j p n q denote the coefficients of an orthonormal basis t v j p τ qu for r L ´ N, ´ | D |‚ ¯ ν θ ¯ . Then for square-free n ‰ and x ě we have | n | ÿ ď r j ď x | b j p n q| ch πr j ! (cid:15),N | n | ` (cid:15) x ´ sgn n . This follows directly from [AA18, Proposition 8.2] for n ą
0. If n ă
0, then using (2.6),we see that the map v j ÞÑ ´ r j ` ¯ ´ L v j AASS FORMS AND THE MOCK THETA FUNCTION f p q q gives an isometry between the subspaces of r L ´ N, ´ | D |‚ ¯ ν θ ¯ and r L ´ N, ´ | D |‚ ¯ ν θ ¯ spanned bythose forms with spectral parameter not equal to i {
4. Moreover, if we denote the coefficientsof ` r j ` ˘ ´ L v j by a j p n q , then (2.5) and (2.7) give a j p´ n q “ ˆ r j ` ˙ ´ b j p n q for n ă and usingpartial summation.Finally, we will prove an estimate which is slightly weaker in n aspect than Proposition 4.2but is significantly better in spectral parameter aspect. Theorem 4.3.
Let D be an even fundamental discriminant and let N be a positive in-teger with D | N . Let b j p n q denote the coefficients of an orthonormal basis t v j p τ qu for r L ´ N, ´ | D |‚ ¯ ν θ ¯ . Then for square-free n ‰ and x ě we have | n | ÿ | r j |ď x | b j p n q| ch πr j ! (cid:15),N | n | ` (cid:15) x ´ sgn n . The proof of this result is somewhat involved, and occupies the next section. We followthe basic strategy of Duke [Duk88], which in turn relies on bounds of Iwaniec [Iwa87] forsums of Kloosterman sums averaged over the level. We use a recent version of the Kusnetsovtrace formula due to Duke, Friedlander and Iwaniec [DFI12] which allows us to use testfunctions which lead to a significant savings with respect to the spectral parameter. Themost delicate part of the subsequent analysis involves a sum over holomorphic cusp forms ofarbitrarily large weight (cid:96) weighted by J -Bessel transforms, and much of the difficulty arisesin obtaining bounds which are uniform with respect to (cid:96) .4.1. Remarks.
For individual coefficients, this theorem implies that b j p n q ! N,(cid:15) λ ´ sgn n j ch ´ πr j ¯ | n | ´ ` (cid:15) . This is slightly weaker in n -aspect and significantly stronger in λ -aspect than the well-knownresult of Duke [Duk88, Theorem 5]. Baruch and Mao [BM10, Theorem 1.5] obtained abound which is stronger in n than Duke’s bound, but weaker in the spectral parameter. Theapproach of Baruch and Mao relies on subconvexity bounds for L -functions due to Blomer,Harcos, and Michel [BHM07] and a Kohnen-Zagier type formula which relates special valuesof these L -functions to the coefficients b j p n q . The subconvexity bounds were later improvedby Blomer and Harcos [BH08]. Following the approach of Baruch and Mao using theseimproved bounds should lead to a bound for the individual coefficients in which the exponent ´ “ ´ . . . . above is replaced by ´ “ ´ . λ j . Anaverage version of this bound in the style of Theorem 4.3 would improve the exponents inour applications.A very strong average version is available in special cases due to work of Young [You17].Young obtains estimates of the form ÿ T ď r j ď T ` L p u j ˆ χ q , q ! (cid:15) p q p T ` qq ` (cid:15) , where q is odd and square-free, χ q is a real character of conductor q , and t u j u is an or-thonormal basis of weight zero Maass cusp forms in level dividing q (note that there are — T terms in the sum). Unfortunately this does not apply to our situation. Andersen and Duke[AD18] use Young’s result as an input to obtain strong bounds for sums of Kloostermansums arising in the plus-space in level four (in which case the forms u j have level one), andobtain striking applications.Finally, we mention that the Lindel¨of hypothesis L p u j ˆ χ q , q ! (cid:15) pp ` | r j |q q q (cid:15) would leadto the pointwise bound b j p n q ! N,(cid:15) λ ´ sgn n j ch ´ πr j ¯ | n | ´ ` (cid:15) . This would of course lead to substantial improvements in the error bounds discussed in thispaper, as described in the Introduction.5.
Proof of Theorem 4.3
The Kusnetsov formula.
We follow the exposition of Duke, Friedlander and Iwaniec[DFI12, § ν is a multiplier on Γ p N q , and that α ν “ . Let Φ : r ,
8q Ñ R be a smooth function such thatΦ p q “ Φ p q “ J s denote the J -Bessel function. For s P C , define r Φ p s q “ ż J s p u q Φ p u q duu , and suppose that for some (cid:15) ą r Φ p it q ! t ´ ´ (cid:15) ch πt for t ě , (5.2) r Φ p (cid:96) q ! (cid:96) ´ ´ (cid:15) for (cid:96) ě . (5.3)Also define p Φ p t q “ i ´r Φ p it q cos π ´ k ` it ¯ ´ r Φ p´ it q cos π ´ k ´ it ¯¯ D k p t q sh πt , where D k p t q “ π Γ ´ ` k ` it ¯ Γ ´ ` k ´ it ¯ . For m, n ě K p N q Φ p m, n q : “ ÿ c ą c ” p mod N q c ´ S p m, n, c, ν q Φ ˆ π ? mnc ˙ . (5.4)The Kusnetsov formula expresses K p N q Φ p m, n q as the sum of three spectral terms. The firsttwo of these correspond to the discrete and the continuous spectrums. Let t u j p τ qu be anorthonormal basis for r L k p N, ν q , and denote the coefficients by ρ j p n q and the spectral param-eters by r j . For each singular cusp a , let ρ a p n, t q be the Fourier coefficients for the Eisenstein AASS FORMS AND THE MOCK THETA FUNCTION f p q q series attached to a . Define L p N q p Φ p m, n q : “ π ? mn ÿ j ě ρ j p m q ρ j p n q p Φ p r j q ch πr j , (5.5) M p N q p Φ p m, n q : “ π ? mn ÿ a π ż ρ a p m, t q ρ a p n, t q p Φ p t q ch πt dt. (5.6)The third term involves the holomorphic forms. For (cid:96) ” k p mod 2 q with (cid:96) ě
2, let S (cid:96) p N, ν q denote the space of holomorphic cusp forms of weight (cid:96) , level N and multiplier ν . Let B (cid:96) denote an orthonormal basis of this space with respect to the Petersson inner product x f, g y “ ż Γ p N qz H y (cid:96) f p τ q g p τ q dµ. Every f P B (cid:96) has a Fourier expansion f p τ q “ ÿ n “ a f p n qp πn q (cid:96) ´ e p nτ q . (5.7)The third spectral term is N p N q q Φ p m, n q “ ÿ (cid:96) ” k p mod 2 q (cid:96) ě q Φ p (cid:96) q Γ p (cid:96) q ÿ f P B (cid:96) a f p m q a f p n q , where q Φ p (cid:96) q : “ π ´ r Φ p (cid:96) ´ q e πi p (cid:96) ´ k q . (5.8)Define γ k : “ e ´ πik . Duke, Friedlander, and Iwaniec [DFI12, Theorem 2.5, (2.28), (2.29)] proved the following.
Proposition 5.1 (Duke–Friedlander–Iwaniec) . Suppose that ν is a multiplier of weight k on Γ p N q with α ν “ . Let Φ be a smooth function which satisfies (5.1) , (5.2) and (5.3) . Thenfor m, n ě we have γ k K p N q Φ p m, n q “ L p N q p Φ p m, n q ` M p N q p Φ p m, n q ` N p N q q Φ p m, n q . Start of the proof of Theorem 4.3.
We now suppose that D is an even fundamentaldiscriminant, that N ” p mod 8 q is a positive integer with D | N , and that p k, ν q “ ´ , ´ | D |‚ ¯ ν θ ¯ or p k, ν q “ ´ ´ , ´ | D |‚ ¯ ν θ ¯ “ ´ ´ , ´ ´| D |‚ ¯ ν θ ¯ . (5.9)For such a multiplier ν , we have the Weil bound [Wai17, Lemma 4] | S p n, n, c, ν q| ď τ p c qp n, c q c . (5.10)Suppose that µ P t´ , , u . For n P N and x ě K p N q µ p n ; x q : “ ÿ c ď xc ” p mod N q c ´ S p n, n, c, ν q e ˆ µnc ˙ . (5.11)Let P be a positive parameter (which will eventually be set to n ), and define Q “ Q p n, N, P q : “ t pN : p prime, P ă p ď P, and p (cid:45) nN u . (5.12) For the theta-multiplier, Iwaniec [Iwa87, Theorem 3] obtained a bound for the sums K p N q µ p n ; x q averaged over the level. This was extended to the case of twists by a quadratic characterby Waibel [Wai17, §
3] (this is also implicit in the work of Duke [Duk88]). Combining theseresults, we have
Proposition 5.2.
Suppose that N ” p mod 8 q , that µ P t´ , , u , that n ą is square-free, and that Q is as in (5.12) . Suppose that ν is one of the characters appearing in (5.9) .Then we have ÿ Q P Q ˇˇ K p Q q µ p n ; x q ˇˇ ! N,(cid:15) ! xP ´ ` xn ´ ` p x ` n q ´ x P ` n x P ¯) p nx q (cid:15) . We choose the smooth functionΦ p u q : “ b π u ´ J p u q . At u “ p u q “ u ` ¨ ¨ ¨ , so (5.1) is satisfied. Using the Weber–Schafheitlin integral [DLMF, (10.22.57)] we compute r Φ p it q “ ´ i t p ` t qp ` t qp ` t qp ` t q ch πt for t P R , and r Φ p (cid:96) q “ ´ (cid:96) p (cid:96) ´ q cos ` π(cid:96) ˘ p (cid:96) ´ qp (cid:96) ´ qp (cid:96) ´ q for (cid:96) ě . (5.13)Thus both (5.2) and (5.3) are satisfied (note that the cosine factor cancels the zeros in thedenominator of (5.13)).For k “ ˘ we have p Φ p t q “ ? t p ` t qp ` t qp ` t qp ` t q p ch πt q D k p t q sh p πt q . We have p Φ p t q ą t P R Y i p , { s . When k “ ´ , the factor D k p t q produces a poleof p Φ at t “ i {
4. However, this value does not occur in the sum (5.5), since this value ofthe spectral parameter r j corresponds to the minimal eigenvalue λ j “ , which does notarise by the discussion at the end of Section 2 since there are no holomorphic modular formsin negative weight. In this case every eigenvalue has λ j ě ´ ` ˘ ; this follows from thediscussion in [Sar84, §
3] and the lower bound ´ ` ˘ which is available in weight zero bythe work of Kim and Sarnak [Kim03, Appendix 2].Let n P N and let Q be as in (5.12). For each Q P Q , Proposition 5.1 gives L p Q q p Φ p n, n q ` M p Q q p Φ p n, n q “ γ k K p Q q Φ p n, n q ´ N p Q q q Φ p n, n q . From (5.5) and (5.6) we see that L p Q q p Φ p n, n q and M p Q q p Φ p n, n q are positive, from which L p Q q p Φ p n, n q ď γ k K p Q q Φ p n, n q ´ N p Q q q Φ p n, n q . (5.14) AASS FORMS AND THE MOCK THETA FUNCTION f p q q (a) p Φ p it q for t P r´ { , { s when k “ { (b) p Φ p it q for t P p´ { , { q when k “ ´ { Figure 1.
Plots of p Φ p t q for imaginary t .For each Q P Q , the functions ! r Γ p N q : Γ p Q qs ´ u j ) form an orthonormal subset of r L k p Q, ν q . Since r Γ p N q : Γ p Q qs ď p ` ! P , we find that L p Q q p Φ p n, n q ě L p N q p Φ p n, n qr Γ p N q : Γ p Q qs " L p N q p Φ p n, n q P .
Since | Q | — P { log P , summing (5.14) over Q gives1log P L p N q p Φ p n, n q ! ÿ Q P Q ˇˇˇ K p Q q Φ p n, n q ˇˇˇ ` ÿ Q P Q ˇˇˇ N p Q q q Φ p n, n q ˇˇˇ . (5.15)As in [Iwa87] and [Duk88], we choose P “ n . In the next two sections we bound the right-hand side of (5.15).5.3.
Treatment of N p Q q q Φ p n, n q . When k “ ˘ we have N p Q q q Φ p n, n q “ ÿ (cid:96) ” k p mod 2 q (cid:96) ą q Φ p (cid:96) q Γ p (cid:96) q ÿ f P B (cid:96) | a f p n q| . Recalling the normalization (5.7), we have Petersson’s formula (c.f. [DFI12, Proposition2.3]) ÿ f P B (cid:96) | a f p n q| “ p (cid:96) ´ q ¨˝ ` πi ´ (cid:96) ÿ c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯˛‚ . Recalling the definition (5.8), we obtain ÿ Q P Q ˇˇˇ N p Q q q Φ p n, n q ˇˇˇ ! P ÿ (cid:96) ” k p mod 2 q (cid:96) ą (cid:96) | r Φ p (cid:96) q|` ÿ (cid:96) ” k p mod 2 q (cid:96) ą (cid:96) | r Φ p (cid:96) q| ÿ Q P Q ˇˇˇˇˇ ÿ c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ . (5.16) The first sum on the right is ! P by (5.13). It is important to obtain bounds for the innerterm which are uniform in (cid:96) as well as n . For small (cid:96) we are able to control the dependenceon (cid:96) explicitly, and for large (cid:96) we exploit the decay of r Φ p (cid:96) q . Let β ą (cid:96) ď n β and (cid:96) ą n β separately.5.3.1. Small values of (cid:96) : (cid:96) ď n β . For these values of (cid:96) , we treat the three ranges1 ď c ď n { (cid:96) , n { (cid:96) ď c ď n and c ě n. In the first range we use an explicit representation for J (cid:96) ´ p z q when (cid:96) P N ´ N . For such (cid:96) and for k ě c (cid:96),k : “ Γ p (cid:96) ´ ` k q k !Γ p (cid:96) ´ ´ k q , and define the polynomial H (cid:96) p z q : “ e ´ πi p (cid:96) ´ q π (cid:96) ´ ÿ k “ i k c (cid:96),k z k . Using the first formula in [BE53, 7.11], a computation shows that for z P R and (cid:96) P N ´ N we have ? zJ (cid:96) ´ p πz q “ e p z q H (cid:96) ´ πz ¯ ` e p´ z q H (cid:96) ´ πz ¯ . (5.17)Recall the definition (5.11). Using (5.17) and partial summation, we obtain ÿ ď c ď n { (cid:96) c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ˆ πnc ˙ “ n ´ ? ˆ H (cid:96) ˆ π(cid:96) ˙ K p Q q ´ n ; n(cid:96) ¯ ` H (cid:96) ˆ π(cid:96) ˙ K p Q q´ ´ n ; n(cid:96) ¯˙ ´ n ´ π ? ż n { (cid:96) H (cid:96) ´ x πn ¯ K p Q q p n ; x q dx ´ n ´ π ? ż n { (cid:96) H (cid:96) ´ x πn ¯ K p Q q´ p n ; x q dx. (5.18)For 0 ď k ď (cid:96) ´ we have c (cid:96),k ď p (cid:96) q k k ! . Therefore H (cid:96) ` π(cid:96) ˘ ! (cid:96) Ñ 8 ), and we have H (cid:96) ´ x πn ¯ ! (cid:96) for 1 ď x ď n { (cid:96) . Using (5.18) and Proposition 5.2 with P “ n we obtain ÿ Q P Q ˇˇˇˇˇ ÿ ď c ď n { (cid:96) c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! ´ (cid:96) ´ n ` (cid:96) ´ n ¯ p (cid:96)n q (cid:15) . (5.19)Now consider the range n { (cid:96) ď c ď n . We have [DLMF, (10.6.1)]2 J (cid:96) ´ p z q “ J (cid:96) ´ p z q ´ J (cid:96) p z q . (5.20)We also have a uniform bound for the J -Bessel function due to L. Landau [Lan00]. AASS FORMS AND THE MOCK THETA FUNCTION f p q q Proposition 5.3 (Landau) . For v ą and x ą , we have | J v p x q| ď c x ´ , where c “ . . . . . Using Proposition 5.3 we find that ´ x ´ J (cid:96) ´ ´ πnx ¯¯ ! n ´ x ´ ` n x ´ . (5.21)Partial summation using this bound together with Proposition 5.2 and (5.21) and a carefulexamination of the error terms which arise yields ÿ Q P Q ˇˇˇˇˇ ÿ n { (cid:96) ď c ď nc ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! ´ (cid:96) n ` (cid:96) n ¯ p (cid:96)n q (cid:15) . (5.22)For the remaining range c ě n , the treatment of Iwaniec [Iwa87, pp. 400-401] appliesuniformly in (cid:96) . To see this, note that (5.20) and the inequality | J (cid:96) ´ p x q| ď x (cid:96) ´ Γ p (cid:96) ´ q for x ą ´ x ´ J (cid:96) ´ ´ πnx ¯¯ ! nx ´ for x ě n which is used in that argument. We conclude that for all (cid:96) we have ÿ Q P Q ˇˇˇˇˇ ÿ c ě nc ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! n ` (cid:15) . (5.23)From (5.13), (5.19), (5.22) and (5.23) we obtain ÿ (cid:96) ” k p mod 2 q ă (cid:96) ď n β (cid:96) | r Φ p (cid:96) q| ÿ Q P Q ˇˇˇˇˇ ÿ c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! ÿ ă (cid:96) ď n β (cid:96) ´ ´ (cid:96) n ` (cid:96) n ¯ p (cid:96)n q (cid:15) ! n ` β ` (cid:15) ` n ` β ` (cid:15) . (5.24)5.3.2. Large values of (cid:96) : (cid:96) ą n β . Let 0 ă γ ă ď c ď n γ , n γ ď c ď n, and c ě n. We begin with a simple lemma.
Lemma 5.4.
Let Q be as in (5.12) . For b ą ´ we have ÿ Q P Q ÿ c ď xc ” p mod Q q c b p n, c q ! (cid:15) x b ` n (cid:15) log P. Proof.
The inner sum is ď ÿ d | n d ÿ c ď xc ” p mod r d,Q sq c b ! x b ` ÿ d | n d r d, Q s ď x b ` p n, N q Q ÿ d | n d ! x b ` NQ n (cid:15) , where we have used the fact that such Q have Q “ pN with p (cid:45) n . Writing Q “ pN andsumming over p gives the lemma. (cid:3) In the first range, we estimate using the Weil bound (5.10), Proposition 5.3, and Lemma 5.4.We obtain ÿ Q P Q ˇˇˇˇˇ ÿ ď c ď n γ c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! ÿ Q P Q n ´ ÿ ď c ď n γ c ” p mod Q q c ´ p n, c q ! n γ ´ ` (cid:15) . (5.25)In the second range we use partial summation together with Proposition 5.2 and (5.21).Examining the error terms which arise from this computation, we obtain ÿ Q P Q ˇˇˇˇˇ ÿ n γ ď c ď nc ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! ´ n ` n ´ γ ` n ´ γ ` n ´ γ ` n ´ γ ` n ` γ ` n ´ ` γ ¯ n (cid:15) . (5.26)For the third range we recall that (5.23) holds for all (cid:96) .We choose γ “ to balance (5.25) and (5.26) (this is the value for which ´ γ “ γ ´ ). For this range of (cid:96) we obtain ÿ Q P Q ˇˇˇˇˇ ÿ n γ ď c ď nc ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! n ` (cid:15) , from which ÿ (cid:96) ” k p mod 2 q (cid:96) ą n β (cid:96) | r Φ p (cid:96) q| ÿ Q P Q ˇˇˇˇˇ ÿ c ” p mod Q q c ´ S p n, n, c, ν q J (cid:96) ´ ´ πnc ¯ˇˇˇˇˇ ! n ` (cid:15) ÿ (cid:96) ą n β (cid:96) ´ ! n ´ β ` (cid:15) . (5.27)Finally, we choose β “ to balance (5.24) and (5.27). From (5.16), we obtain ÿ Q P Q ˇˇˇ N p Q q q Φ p n, n q ˇˇˇ ! n ` (cid:15) . (5.28) AASS FORMS AND THE MOCK THETA FUNCTION f p q q Treatment of K p Q q Φ p n, n q . From the definition (5.4) we have ÿ Q P Q ˇˇˇ K p Q q Φ p n, n q ˇˇˇ ! n ´ ÿ Q P Q ˇˇˇˇˇ ÿ c ď nc ” p mod Q q c ´ S p n, n, c, ν q J ´ πnc ¯ˇˇˇˇˇ ` n ´ ÿ Q P Q ˇˇˇˇˇ ÿ c ą nc ” p mod Q q c ´ S p n, n, c, ν q J ´ πnc ¯ˇˇˇˇˇ . Here the situation is simpler since the J -Bessel function has fixed order. For the first term,we use the expression (5.17) and partial summation with Proposition 5.2, together with thefact that ´ x H ` x πn ˘¯ ! x ´ ` x n ´ for 1 ď x ď n . For the second term, (5.20) and[DLMF, (10.14.4)] give ´ J ` πnx ˘¯ ! n x ´ for x ě n . We conclude that ÿ Q P Q ˇˇˇ K p Q q Φ p n, n q ˇˇˇ ! n ` (cid:15) . (5.29)5.5. Proof of Theorem 4.3.
From (5.15), (5.28) and (5.29) we have L p N q p Φ p n, n q ! n ` (cid:15) . Until now we have assumed that N ” p mod 8 q , but we may drop this assumption usingpositivity after replacing N by 2 N if necessary. The last inequality gives n ÿ j ě | ρ j p n q| p Φ p r j q ch πr j ! n ` (cid:15) . (5.30)From the asymptotics of the Gamma function [DLMF, (5.11.9)], for k “ ˘ we have p Φ p t q " t k ´ for t ě . (5.31)By the discussion in Section 5.2, there are positive constants c and c such that for each r j P r , s Y i p , { s which appears in the sum (5.30) we have c ď p Φ p r j q ď c . (5.32)So for every r j which appears in the sum, we have p Φ p r j q ´ ! λ ´ k j . Suppose that x ě
1. We have x k ´ p Φ ´ p r j q ! ď r j ď x by (5.31). The same boundholds for | r j | ď n ÿ | r j |ď x | ρ j p n q| ch πr j ! x ´ k n ` (cid:15) . (5.33)Suppose now that k “ as in the statement of Theorem 4.3. When n ą
0, the theoremfollows directly from (5.33). When n ă Kuznetsov trace formula in the mixed-sign case
We give a version of the Kusnetsov trace formula in the mixed sign case which is suitablefor our applications. Suppose that φ : r ,
8q Ñ C is four times continuously differentiableand satisfies φ p q “ φ p q “ , φ p j q p x q ! (cid:15) x ´ ´ (cid:15) p j “ , . . . , q as x Ñ 8 , (6.1)for some (cid:15) ą
0. Define the transform q φ p r q : “ ch πr ż K ir p u q φ p u q duu (6.2)(we use the notation φ instead of Φ for the rest of the paper to avoid potential confusionwith the transform defined in the last section).Suppose that N is a positive integer and that ν is a multiplier of weight for Γ p N q with the property that no cusp is singular with respect to ν . In this case the Kusnetsovformula has a relatively simple expression since there are no Eisenstein series and there isno contribution from cusp forms due to the mixed sign of the arguments. A proof of thefollowing result is given in Section 4 of [AA18] in the case when ν is the multiplier on SL p Z q associated to the Dedekind eta function (in this case there is no residual spectrum). Thegeneral case follows by this argument with only cosmetic changes. Blomer [Blo08] has proveda version of this formula for the twisted theta-multiplier. Proposition 6.1.
Let ν be a multiplier of weight for Γ p N q such that no cusp is singularwith respect to ν . Let ρ j p n q denote the coefficients of an orthonormal basis t v j p τ qu for r L p N, ν q . Suppose that φ satisfies conditions (6.1) . If m ν ą and n ν ă then ÿ c ą c ” p mod N q S p m, n, c, ν q c φ ˜ π a m ν | n ν | c ¸ “ ? i a m ν | n ν | ÿ r j ρ j p m q ρ j p n q ch πr j q φ p r j q . We describe a family of test functions. Given a, x ą
0, let T ą T ď x { , T — x ´ δ with 0 ă δ ă { . (6.3)Let φ “ φ a,x,T : r ,
8q Ñ r , s be a smooth function (as in [ST09] and [AA18]) satisfying(i) The conditions in (6.1),(ii) φ p t q “ a x ď t ď ax ,(iii) φ p t q “ t ď a x ` T and t ě ax ´ T ,(iv) φ p t q ! ` ax ´ T ´ ax ˘ ´ ! x aT ,(v) φ and φ are piecewise monotone on a fixed number of intervals.We require bounds for q φ which are recorded in [AA18, Theorem 6.1]. Proposition 6.2.
Let a,x,T be as above and let φ “ φ a,x,T . Then q φ p r q ! $’&’% r ´ e ´ r for 1 ď r ď a x ,r ´ for max ` , a x ˘ ď r ď ax , min ´ r ´ , r ´ xT ¯ for r ě max ` ax , ˘ . AASS FORMS AND THE MOCK THETA FUNCTION f p q q Proof of Theorem 1.1
Let ψ be the multiplier defined in (3.4). Theorem 1.1 will follow from a uniform estimatefor sums of Kloosterman sums attached to ψ . We note that many of the terms x (cid:15) could bechanged to log x factors if necessary, but for simplicity we do not keep track of these here. Theorem 7.1.
Suppose that n ´ is positive and squarefree and that ă δ ă { . For X ě and (cid:15) ą we have ÿ c ď Xc ” p mod 2 q S p , n, c, ψ q c ! δ,(cid:15) ´ n ` (cid:15) X δ ` n ` (cid:15) ` X ´ δ ¯ X (cid:15) . (7.1)This follows in turn from an estimate over dyadic ranges. Proposition 7.2.
Suppose that n ´ is positive and squarefree and that ă δ ă { . For x ě and (cid:15) ą we have ÿ x ď c ď xc ” p mod 2 q S p , n, c, ψ q c ! δ,(cid:15) n ` (cid:15) x ´ ` n x δ ` (cid:15) ` x ´ δ ` (cid:15) . We show that Proposition 7.2 implies Theorem 7.1. Corollary 9.2 below gives a Weilbound for the individual Kloosterman sums, so the initial segment 1 ď c ď n contributes O ´ n ` (cid:15) ¯ to (7.1). One can then break the interval n ď c ď X into O p log X q dyadicintervals x ď c ď x with n ď x ď X { Proof of Proposition 7.2.
Let t u j p τ qu be an orthonormal basis for r L p , ψ q with Fourier co-efficients ρ j p n q and eigenvalues λ j “ ` r j . Recall that α ψ “ , so that n ψ “ n ´ , anddefine a : “ π ? ? n ψ . Let φ “ φ a,x,T : r ,
8q Ñ R be a smooth test function with the properties listed in Section 6.Using Corollary 9.2 and recalling the definition (6.3) of T we obtain ˇˇˇˇˇˇˇ ÿ c ą c ” p mod 2 q S p , n, c, ψ q c φ ´ ac ¯ ´ ÿ x ď c ď xc ” p mod 2 q S p , n, c, ψ q c ˇˇˇˇˇˇˇ ď ÿ x ´ T ď c ď x x ď c ď x ` T | S p , n, c, ψ q| c ! δ x ´ δ log x. (7.2)We now estimate the smoothed sums. Proposition 6.1 gives ÿ c ą c ” p mod 2 q S p , n, c, ψ q c φ ´ ac ¯ “ ? i ? ? n ψ ÿ r j ρ j p q ρ j p n q ch πr j q φ p r j q . (7.3)If f p τ q P r L p , ψ q has minimal eigenvalue λ “ , then by the discussion in Section 2 y f p τ q P M ` , ` ‚ ˘ ν θ ˘ . By the Serre-Stark basis theorem [SS77], this space is spannedby θ p τ q “ ` ¨ ¨ ¨ . In view of the Fourier expansion (2.4) this minimal eigenvalue does not occur, so each form u j appearing in (7.3) is cuspidal. Applying a lift of Sarnak [Sar84, § L p , q with eigenvalue 4 λ j ´ ą λ j ÞÑ λ j ´ corresponds to the map r j ÞÑ r j ).The image of u j under this lift must therefore be a cusp form (since the residual spectrumoccurs only when λ “ r j ą .
1. It follows that r j ą for all j in the sum (7.3).After this discussion we break the sum (7.3) into three ranges dictated by the behavior of q φ in Proposition 6.2. These are0 . ă r j ď a x , max ´ , a x ¯ ă r j ă ax , and r j ě max ´ , ax ¯ . Let v j p τ q : “ u j p τ q . After multiplication by a fixed constant, Lemma 3.2 shows that t v j p τ qu is an orthonormal subset of r L ` , ` ‚ ˘ ν θ ˘ . Denote the Fourier coefficients of v j p τ q by b j p n q , so that ρ j p n q “ b j p n ´ q . (7.4)By a straightforward argument using the Weil bound (5.10) for the sums S ` n, n, c, ` ‚ ˘ ν θ ˘ ,(4.1) is satisfied with the choice β “ ` (cid:15) . Using (7.4) and Proposition 4.1 we obtain ρ j p q a ch πr j “ b j p´ q a ch πr j ! r j , (7.5)while Proposition 4.2 and (7.4) give ρ j p n q a ch πr j “ b j p n ´ q a ch πr j ! n ´ ` (cid:15) r j . (7.6)Using Proposition 6.2, (7.5) and (7.6) we obtain ? n ψ ÿ . ă r j ď a x ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j q φ p r j q ˇˇˇˇˇ ! n ` (cid:15) ¨˝ ÿ . ă r j ď | q φ p r j q| ` ÿ ď r j ď a x r j e ´ rj ˛‚ ! n ` (cid:15) . (7.7)To obtain the last estimate we use the fact that the first sum on the right contains onlyfinitely many terms, so it is O p q by (6.2) and [DLMF, (10.25.3)]. The second sum is also O p q by Weyl’s law for the multiplier ψ [Hej76, Theorem 2.28, p. 414].Note that the results in Section 4 apply to the orthonormal collection t v j p τ qu by positivity.Using the Cauchy-Schwarz inequality, Proposition 4.1, Theorem 4.3, and Proposition 6.2, weobtain ? n ψ ÿ a x ď r j ă ax ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j q φ p r j q ˇˇˇˇˇ ! ¨˝ ÿ a x ď r j ă ax | b j p´ q| ch πr j ˛‚ ˆ ¨˝ n ψ ÿ a x ď r j ă ax | b j p n ´ q| ch πr j | q φ p r j q| ˛‚ ! n ` (cid:15) ´ ax ¯ ! n ` (cid:15) x ´ . (7.8) AASS FORMS AND THE MOCK THETA FUNCTION f p q q We record two estimates which are required for the range r j ě ax . Proposition 4.1 gives ? n ψ ÿ ď r j ď X ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j ˇˇˇˇˇ ! ¨˝ ÿ ď r j ď X | b j p´ q| ch πr j ˛‚ ¨˝ n ψ ÿ ď r j ď X | b j p n ´ q| ch πr j ˛‚ ! ´ X ` n ` (cid:15) X ¯ log X, (7.9)while the same argument using Proposition 4.1 and Proposition 4.2 together gives ? n ψ ÿ ď r j ď X ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j ˇˇˇˇˇ ! ´ X n ` (cid:15) ¯ log X. (7.10)Let A ě max ` ax , ˘ and consider the dyadic range A ď r j ď A . Using the Cauchy–Schwarz inequality, (7.9), (7.10) and Proposition 6.2 we obtain ? n ψ ÿ A ď r j ď A ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j q φ p r j q ˇˇˇˇˇ ! min ´ A ´ , A ´ xT ¯ ¨ min ´ A ` n ` (cid:15) A, n ` (cid:15) A ¯ log A ! min ´ A , A ´ xT ¯ ¨ min ´ ` n ` (cid:15) A ´ , n ` (cid:15) A ¯ log A ! n ` (cid:15) min ´ A , A ´ xT ¯ log A, where we have usedmin p B ` C, D q ď min p B, D q ` min p C, D q and min p B, C q ď ? BC for positive B, C and D . Summing over dyadic integrals yields ? n ψ ÿ r j ě max p a { x, q ˇˇˇˇˇ ρ j p q ρ j p n q ch p πr j q q φ p r j q ˇˇˇˇˇ ! n ` (cid:15) ´ xT ¯ ` (cid:15) ! δ n ` (cid:15) x δ ` (cid:15) . (7.11)Proposition 7.2 follows from (7.2), (7.7) (7.8) and (7.11). (cid:3) We need a simple lemma before proving Theorem 1.1. By [DLMF, (10.30.1)], for fixed ν, M ą ď z ď M we have I ν p z q ! ν,M z ν . (7.12) Lemma 7.3.
Suppose that b, β ą . Then for t { b ě β we have I ´ bt ¯ ! β ´ bt ¯ and ´ I ´ bt ¯¯ ! β b t ´ ` b t ´ . Proof.
The first inequality follows directly from (7.12). From the identity I p t q “ I p t q ` t I p t q we obtain ´ I ´ bt ¯¯ “ ´ bt ˆ I ´ bt ¯ ` t b I ´ bt ¯˙ , and the second bound follows. (cid:3) Proof of Theorem 1.1.
From Section 3 we have R p n, N q “ π p n ´ q e ´ ´ ¯ ÿ c ą Nc ” p mod 2 q S p , n, c, ψ q c I ˆ bc ˙ , (7.13)where b : “ π ? n ´ . Let S p n, X q : “ ÿ c ď Xc ” p mod 2 q S p , n, c, ψ q c . By partial summation we have ÿ c ą Nc ” p mod 2 q S p , n, c, ψ q c I ´ bc ¯ “ S p n, N q I ´ b N ¯ ´ ż N S p n, t q ´ I ´ bt ¯¯ dt. (7.14)Fix γ ą N : “ γ ? n . Theorem 7.1 and Lemma 7.3 imply that S p n, N q I ´ b N ¯ ! γ,δ ´ n ` δ ` n ` n ´ δ ¯ n (cid:15) . We choose δ “ to obtain S p n, N q I ´ b N ¯ ! γ n ` (cid:15) . (7.15)With this choice of δ , the integral in (7.14) also satisfies the bound (7.15). By (7.13) we have R p n, N q ! γ n ´ ` (cid:15) , and Theorem 1.1 is proved. (cid:3) Proof of Theorem 1.4
One can follow the argument in [AA18, § § t u j p τ qu be anorthonormal basis for r L p , ν η q with spectral parameters r j and Fourier coefficients ρ j p m q .Here each u j is cuspidal and r j ą j [AA18, Corollary 5.3]. Let v j p τ q : “ u j p τ q P L ` , p ‚ q ν θ ˘ ; after scaling by a fixed constant, t v j u is an orthonormal set. If the coeffi-cients are denoted by b j p n q , then we have ρ j p n q “ b j p n ´ q . AASS FORMS AND THE MOCK THETA FUNCTION f p q q Suppose that 24 n ´
23 is negative and squarefree. We have n η “ n ´ . Arguing as in(7.8) using Theorem 4.3 we obtain b | n η | ÿ a x ă r j ă ax ˇˇˇˇˇ ρ j p q ρ j p n q ch πr j q φ p r j q ˇˇˇˇˇ ! ¨˝ ÿ a x ă r j ă ax | b j p q| ch πr j ˛‚ ˆ ¨˝ | n η | ÿ a x ă r j ă ax | b j p n ´ q| ch πr j | q φ p r j q| ˛‚ ! n ` (cid:15) x ´ . Following the argument in [AA18] with this estimate and making the choice δ “ yieldsTheorem 1.4. 9. Character sums and the proof of Theorem 1.3
To translate the error estimates of Theorem 1.1 to the setting of Theorem 1.3 requires areinterpretation of the Kloosterman sums S p , n, c, ψ q in terms of Weyl-type sums. To setnotation, for c P N , define F c p n q : “ ÿ x p mod 24 c q x ” ´ n p mod 24 c q ˆ ´ x ˙ e ´ x c ¯ . (9.1)Then we have Proposition 9.1. If c is odd then F c p n q “ . Furthermore we have F c p n q “ c c e ´ ´ ¯ S p , n, c, ψ q . As a consequence we obtain an improved Weil-type bound for the Kloosterman sums S p , n, c, ψ q . Computations suggest that this bound is sharp. For example, when c “ n “ . . . . . Corollary 9.2.
We have | S p , n, c, ψ q| ď ω o p c q d c p , c q , where ω o p c q is the number of distinct odd prime divisors of c .Proof. The congruence x ” ´ n p mod p (cid:96) q has at most four solutions if p “ | c and Lehmer’sbound (1.4), which is stronger when 3 (cid:45) c . (cid:3) Before turning to the proof of Proposition 9.1 we recall the definition of the Gauss sum G p a, b, c q : “ ÿ x p mod c q e ˆ ax ` bxc ˙ c ą , a, b P Z and a ‰ . For any d | p a, c q , replacing x with x ` c { d gives G p a, b, c q “ e ˆ bd ˙ G p a, b, c q . So G p a, b, c q “ d | b , in which case we have G p a, b, c q “ d ¨ G ˆ ad , bd , cd ˙ . If 4 | c and p a, c q “
1, then replacing x by x ` c { G p a, b, c q “ b is odd. Thesefacts together with standard evaluations [BEW98, § G p a, b, c q “ $’’&’’% e ´ ´ ab c ¯ p ` i q (cid:15) ´ a ? c ` ca ˘ if b is even and 4 | c,e ´ ´ ab c ¯ (cid:15) c ? c ` ac ˘ if c is odd , b is odd and 4 | c. (9.2) Proof of Proposition 9.1.
For convenience, set D n : “ n ´ . Suppose that c is odd. If x is a solution to x ” ´ D n p mod 24 c q then x ` c is also asolution, and the corresponding terms have opposite sign. Therefore F c p n q “ F c p n q “ c c e ´ ´ ¯ p´ q t c ` u A c ˆ n ´ c p ` p´ q c q ˙ . (9.3)By work of Selberg and Whiteman [Whi56] we have A c p n q : “ c c ÿ x p mod 24 c q x ”´ D n p mod 24 c q ˆ x ˙ e ´ x c ¯ . For convenience we define M c p n q : “ A c ˆ n ´ c p ` p´ q c q ˙ , so that M c p n q “ c c ÿ x p mod 48 c q x ”´ D n ` c p `p´ q c q p mod 48 c q ˆ x ˙ e ´ x c ¯ . We will make use of the following identities for x P Z : ˆ ´ x ˙ “ i ? ´ e ´ ´ x ¯ ´ e ´ x ¯ ´ e ´ x ¯ ` e ´ ´ x ¯¯ , ˆ x ˙ “ ? ˆ e ´ x ¯ ´ e ˆ x ˙ ´ e ˆ ´ x ˙ ` e ´ ´ x ¯˙ . (9.4)Since F c and M c each have period 2 c in n , it suffices to establish the identity for the Fouriertransforms p F c and x M c . We have p F c p h q “ c ÿ n p mod 2 c q F c p n q e ˆ hn c ˙ “ c ÿ x p mod 48 c q ˆ ´ x ˙ e ´ x c ¯ e ˆ h p ´ x q c ˙ , AASS FORMS AND THE MOCK THETA FUNCTION f p q q and using (9.4) we obtain p F c p h q “ i ? c e ˆ h c ˙´ G p´ h, p ´ c q , c q ´ G p´ h, p ` c q , c q´ G p´ h, p ` c q , c q ` G p´ h, p ´ c q , c q ¯ . (9.5)Similarly we have x M c p h q “ ? c e ˆ h p ` p´ q c q ˙ ÿ x p mod 48 c q ˆ x ˙ e ´ x c ¯ e ˆ h p ´ x q c ˙ , from which x M c p h q “ ? c e ˆ h c ` h p ` p´ q c q ˙´ G p´ h, p ` c q , c q´ G p´ h, p ` c q , c q ´ G p´ h, p ´ c q , c q ` G p´ h, p ´ c q , c q ¯ . (9.6)Suppose that p h, c q “ h by hh ” p mod 48 c q . Using (9.2) we find that p F c p h q “ i p ` i q? c e ˆ h ` h c ` hc ˙ˆ c ´ h ˙ (cid:15) ´ ´ h ˆ e ˆ ´ h ˙ ´ e ˆ h ˙ ´ e ˆ h ˙ ` e ˆ ´ h ˙˙ and that x M c p h q “ ` i ? e ˆ h ` h c ` hc ˙ e ˆ h ˙ e ˆ h p ` p´ q c q ˙ˆ c ´ h ˙ (cid:15) ´ ´ h ¨ ˆ e ˆ ´ h ˙ ´ e ˆ h ˙ ´ e ˆ h ˙ ` e ˆ ´ h ˙˙ . The result follows when p h, c q “ c p mod 4 q .Note that each of the Gauss sums G p´ h, b, c q appearing in (9.5) and (9.6) has p b, c q “ p h, c q “ p F c p h q “ x M c p h q “ p h, c q “
3. Suppose that this is the case and that c ” p mod 3 q . Setting h “ h {
3, weobtain p F c p h q “ i ? c e ˆ h c ˙´ G p´ h , p ´ c q , c q ´ G p´ h , p ` c q , c q ¯ “ ? i ? c p ` i q (cid:15) ´ ´ h ˆ c ´ h ˙ e ˜ h c ` h c ¸ e ˜ h p c ´ q ¸ and x M c p h q “ ? c e ˆ h c ` h p ` p´ q c q ˙´ G p´ h , p ` c q , c q ´ G p´ h , p ´ c q , c q ¯ “ ? p ` i q (cid:15) ´ ´ h ˆ c ´ h ˙ e ˜ h c ` h c ¸ e ˆ h p ` p´ q c q ˙ e ˜ h p c ` q ¸ . Comparing these expressions gives (9.3). When c ” p mod 3 q the situation is similar andwe omit the details. (cid:3) We turn to the proof of Theorem 1.3. For D ą Q ´ D, : “ (cid:32) ax ` bxy ` cy : b ´ ac “ ´ D, | a, a ą ( . Each Q “ r a, b, c s P Q ´ D, has a unique root τ Q P H given by τ Q “ ´ b ` ?´ D a . Matrices g “ ` α βγ δ ˘ P Γ p q act on these forms by gQ p x, y q : “ Q p δx ´ βy, ´ γx ` αy q . This action preserves b p mod 12 q , and for g P Γ p q we have g τ Q “ τ gQ . For Q “ r a, b, c s P Q ´ D, define χ ´ p Q q “ ˆ ´ b ˙ . Let Γ Ď Γ p q be the subgroup of translations. Since p t qr a, b, ‚s “ r a, b ´ ta, ‚s ,there is a bijectionΓ z Q ´ D, ÐÑ (cid:32) p a, b q : a ą , ď b ă a, b ” ´ D p mod 48 a q ( . Then we have
Proposition 9.3.
Suppose that γ ą and that n is a positive integer. Then πD n e ´ ´ ¯ ÿ c ď ? Dnγ c ” p mod 2 q S p , n, c, ψ q c I ˆ π ? D n c ˙ “ i ? D n ÿ Q P Γ z Q ´ Dn, Im τ Q ě γ χ ´ p Q qp e p τ Q q ´ e p τ Q qq . Proof of Proposition 9.3.
Let A p n, γ q denote the sum on the left side of the proposition.Using Proposition 9.1 and the identity I p x q “ c π sh x ? x , we find that A p n, γ q “ i ? D n ÿ a ď ? Dnγ F a p n q ´ e π ? Dn a ´ e ´ π ? Dn a ¯ . Pairing the terms b and b ` a in (9.1) gives F a p n q “ ÿ b p mod 24 a q b ”´ D n p mod 48 a q ˆ ´ b ˙ e ˆ b a ˙ , Therefore A p n, γ q “ i ? D n ÿ Q P Γ z Q ´ Dn, Im τ Q ě γ χ ´ p Q qp e p τ Q q ´ e p τ Q qq . (cid:3) AASS FORMS AND THE MOCK THETA FUNCTION f p q q Proof of Theorem 1.3.
By (3.5) we have α p n q “ A p n, γ q ` R ˆ n, ? D n γ ˙ . Theorem 1.3 follows from Proposition 9.3 and Theorem 1.1. (cid:3)
Proof of Theorem 1.2
Recall the definition (9.1) and fix n ą
0. We will consider the quantities F p p n q where p ě ´ ´ np ¯ “
1. For such a p let m p satisfy48 m p ” ´ n p mod p q . (10.1)In the sum defining F p p n q we may take x “ pj ˘ m p , where p j ” ´ n p mod 48 q .For simplicity, define (cid:15) n,p P t , u by1 ` (cid:15) n,p ” p p ´ n q p mod 48 q . Then we have F p p n q “ ÿ j ” ` (cid:15) n,p p mod 48 q x “ pj ˘ m p ˆ ´ x ˙ e ´ x c ¯ “ ˆ ´ p ˙ cos ˆ πm p p ˙ ÿ j ” ` (cid:15) n,p p mod 48 q ˆ ´ j ˙ e ˆ j ˙ . Evaluating the last sum, we find that F p p n q “ ? i p´ q n ˆ ´ p ˙ e ˆ p ´ ˙ cos ˆ πm p p ˙ . Let S be the set of primes p ě m p satisfying (10.1) with0 ă m p p ď . For p P S , we have F p p n q " t p P S : p ď X u " π p X q . By Proposition 9.1 and the fact that I p x q " x as x Ñ
0, we find that ÿ c ď Xc ” p mod 2 q c ´ | S p , n, c, ψ q| I ˆ π ? n ´ c ˙ " ÿ p ď Xp P S p . Theorem 1.2 follows.
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