Bivariate asymptotics for eta-theta quotients with simple poles
aa r X i v : . [ m a t h . N T ] F e b BIVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLEPOLES
GIULIA CESANA AND JOSHUA MALES
Abstract.
We employ a variant of Wright’s circle method to determine the bivariate asymptoticbehaviour of Fourier coefficients for a wide class of eta-theta quotients with simple poles in H . Introduction
Jacobi forms are ubiquitous throughout number theory and beyond. For example, they appear instring theory [12, 16], the theory of black holes [7], and the combinatorics of partition statistics [4].The Fourier coefficients of Jacobi forms often encode valuable arithmetic information. To describea motivating example, let a partition λ of a positive integer n be a list of non-increasing positiveintegers λ j with 1 ≤ j ≤ s that sum to n . The rank [9] of λ is given by the largest part minus thenumber of parts, and the crank [3] of λ is given by ( ℓ ( λ ) if λ contains no ones, µ ( λ ) − ω ( λ ) if λ contains ones.Here, ℓ ( λ ) denotes the size of λ , µ denotes the number of parts greater than one, and ω denotesthe number of ones in λ . We denote by M ( m, n ) (resp. N ( m, n )) the number of partitions of n with crank m (resp. rank m ). It is well-known that the generating function of M is given by (see[4, equation (2.1)]) X n ≥ m ∈ Z M ( m, n ) ζ m q n = i (cid:16) ζ − ζ − (cid:17) q η ( τ ) ϑ ( z ; τ ) , which is a Jacobi form and where ζ := e πiz for z ∈ C , and q := e πiτ with τ ∈ H , the upperhalf-plane throughout. Here, the Dedekind η -function is given by η ( τ ) := q Y n ≥ (1 − q n ) , and the Jacobi theta function is defined by ϑ ( z ; τ ) := iζ q Y n ≥ (1 − q n ) (1 − ζq n ) (cid:16) − ζ − q n − (cid:17) . Note that a similar formula can be found for the generating function of N as a mock Jacobi forminvolving an eta-theta quotient. In general Jacobi forms have a Fourier expansion of the form X n ≥ m ∈ Z a ( m, n ) ζ m q n . (1.1) Classical results in the theory of modular forms (see [5, Theorem 15.10] for example) give theasymptotic behaviour of the coefficients a ( m, n ) of Jacobi forms similar to (1.1) for fixed m .Many interesting examples of Jacobi forms arise as quotients of η and ϑ functions. As anilluminating example, for a k , b j ∈ N and n ∈ Z , consider the study of generalized theta quotients[10, equation (13)], ϑ ( a z ; τ ) ϑ ( a z ; τ ) · · · ϑ ( a k z : τ ) ϑ ( b z ; τ ) ϑ ( b z ; τ ) · · · ϑ ( b j z : τ ) η ( τ ) n which provide new constructions of Jacobi and Siegel modular forms. As highlighted by Gritsenko,Skoruppa, and Zagier, theta blocks also have deep applications to areas such as Fourier analysisover infinite-dimensional Lie algebras and the moduli spaces in algebraic geometry. In the presentpaper, we obtain the bivariate asymptotic behavior of the coefficients of a prototypical family ofsuch generalized theta blocks, while the steps presented here also offer a pathway to obtain similarresults for more general families.In [4] Bringmann and Dousse pioneered the use of new techniques in the study of the bivariateasymptotic behaviour of the Fourier coefficients and applied them to the partition crank function.In [8] Dousse and Mertens used these techniques to study the rank function. In particular, each ofthese papers used an extension of Wright’s circle method [18, 19] to obtain bivariate asymptoticsof N ( m, n ) and M ( m, n ), with m in a certain range depending on n .Recently, the second author of the present paper extended these techniques to an example ap-pearing in the partition function for entanglement entropy in string theory. In particular, [12]considered the eta-theta quotient ϑ ( z ; τ ) η ( τ ) ϑ (2 z ; τ ) =: X m ∈ Z n ≥ b ( m, n ) ζ m q m (1.2)with a simple pole at z = , since η ( τ ) has no zeros in H . Then the coefficients b ( m, n ) have thebivariate asymptotic behaviour given by the following theorem. Theorem 1.1. [12, Theorem 1.1]
For β := π q n and | m | ≤ β log( n ) we have that b ( m, n ) = ( − m + δ + β π (2 n ) e π √ n + O (cid:16) n − e π √ n (cid:17) as n → ∞ . Here, δ := 1 if m < and δ = 0 otherwise. The current paper serves to extend these results to a large family of eta-theta quotients withmultiple simple poles . Meromorphic eta-theta quotients appear in numerous places. For example,investigations into Vafa-Witten invariants [1, equation (2.5)] involve the functions iη ( τ ) N − ϑ (2 z ; τ ) , (1.3)which also appear in investigations into the counting of BPS-states via wall-crossing [17, equation(5.114)]. The asymptotics of this family of functions is studied in [6]. Other examples of similar A similar framework exists for those without poles by simply extending the results of [4, 8].
IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 3 shapes also arise as natural pieces of functions in investigations into BPS states, see e.g. [17, Section5.6.2].Throughout we consider an eta-theta quotient of the form f ( z ; τ ) := N Y j =1 η ( a j τ ) α j ϑ ( z ; τ ) ϑ ( bz ; cτ ) , (1.4)where a j , b, c ∈ N and α j ∈ Q and assume that b = c . In the language of [10], this is a family ofgeneralized theta quotients. We omit the dependency on these parameters for notational ease. Remarks. (1) The exposition presented here may be easily generalised to include products oftheta functions in both the numerator and denominator of f (thereby covering the case of[12]), although this becomes lengthy to write out for the general case.(2) We include a theta function in the numerator to allow us to assume that there are no polesof f at the lattice points 0 ,
1. However, using the techniques presented here and shiftingintegrals to not have end-points at 0 , b = c . If thisis not the case, a blend of techniques from the present paper and those in [12] would yieldsimilar results.We define the coefficients c ( m, n ) by f ( z ; τ ) =: X n ≥ m ∈ Z c ( m, n ) ζ m q n and investigate their bivariate asymptotic behaviour. To this end, we employ and extend thetechniques of [4], which also appear in [8, 12], using Wright’s circle method to arrive at the followingtheorem. Theorem 1.2.
Let h ℓ denote the ℓ -th pole of f in z ∈ (0 , . Define β = β ( n ) := π q n , Λ ℓ := 14 c − − N X j =1 α j a j + h ℓ − h ℓ , and D ℓ ( m, n ) := ( − bh ℓ + N P j =1 α j c b (cid:18) β π (cid:19) − N P j =1 α j N Y j =1 a − αj j . Assume that Λ ℓ ∈ (0 , for all ℓ , and | m | ≤ β log( n ) . Then c ( m, n ) = s X ℓ =1 D ℓ ( m, n )Λ − N P j =1 α j ℓ e − πimh ℓ e π √ ℓ n π (2Λ ℓ n ) + O n − + N P j =1 α j e π √ ℓ n as n → ∞ . GIULIA CESANA AND JOSHUA MALES
Remark.
Note that the restriction on Λ ℓ still leaves infinitely many choices. For example, (1.2) and(1.3) for 3 ≤ N ≤
14 each satisfy this condition.The paper is structured as follows. We begin in Section 2 by recalling relevant results that arepertinent to the rest of the paper. Section 3 deals with defining the Fourier coefficients of ζ m of f . Here we isolate the contribution from each pole by deformation of integrals defining the Fouriercoefficients. In Section 4 we investigate the behaviour of f toward the dominant pole q = 1. Wefollow this in Section 5 by bounding the contribution away from the pole at q = 1. In Section 6 weobtain the asymptotic behaviour of c ( m, n ) and hence prove Theorem 1.2. acknowledgements The authors would like to thank Kathrin Bringmann, Andreas Mono, and Larry Rolen for manyhelpful comments on an earlier version of the paper.2.
Preliminaries
Here we recall relevant definitions and results which will be used throughout the rest of thepaper.2.1.
Properties of ϑ and η . When determining the asymptotic behaviour of f we will requirethe modularity behaviour of both ϑ and η . It is well-known that ϑ has the following properties (seee.g. [14]), which encapsulate its behavior as a Jacobi form. Lemma 2.1.
The function ϑ satisfies ϑ ( z ; τ ) = − ϑ ( − z ; τ ) , ϑ ( z ; τ ) = − ϑ ( z + 1; τ ) , ϑ ( z ; τ ) = i √− iτ e − πiz τ ϑ (cid:18) zτ ; − τ (cid:19) . We also have the well-known triple product formula, yielding the summation representation of ϑ as (see e.g. [20, Proposition 1.3]) ϑ ( z ; τ ) = iq ζ X n ∈ Z ( − n q n n ζ n . (2.1)Furthermore, we have the following modular transformation formula of η (see e.g. [11]). Lemma 2.2.
We have that η ( τ ) = r iτ η (cid:18) − τ (cid:19) . A particular bound.
We require a bound on the size of P ( q ) := q η ( τ ) , away from the pole at q = 1. For this we use [4, Lemma 3.5]. Lemma 2.3.
Let τ = u + iv ∈ H with M v ≤ u ≤ for u > and v → . Then | P ( q ) | ≪ √ v exp (cid:20) v (cid:18) π − π (cid:18) − √ M (cid:19)(cid:19)(cid:21) . IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 5
In particular, with v = β π , u = βm − x π and M = m − this gives for 1 ≤ x ≤ πm β the bound | P ( q ) | ≪ n − exp πβ π − π − q m − . (2.2)2.3. I -Bessel functions. Here we recall relevant results on the I -Bessel function which may bewritten as I ℓ ( x ) = 12 πi Z Γ t − ℓ − e x ( t + t ) dt, where Γ is a contour which starts in the lower half plane at −∞ , surrounds the origin counterclock-wise and returns to −∞ in the upper half-plane. We are particularly interested in the asymptoticbehaviour of I ℓ , given in the following lemma (see e.g. [2, (4.12.7)]). Lemma 2.4.
For fixed ℓ we have I ℓ ( x ) = e x √ πx + O (cid:18) e x x (cid:19) as x → ∞ . The saddle point method.
We additionally require a bound for an integral of the form I ( g, h, t ) := Z Γ g ( z ) e th ( z ) dz, where Γ is some contour in the complex plane, g and h , which are independent of t , are analyticfunctions of z in some domain of the complex plane which contains Γ, and t is a real positivenumber. The asymptotic behavior of I is given in the following lemma (see e.g. [15, (3.29)]). Lemma 2.5.
For z a saddle point of h (i.e. h ′ ( z ) = 0 and h ′′ ( z ) = 0 ) we have that I ( g, h, t ) = g ( z ) (cid:18) − πth ′′ ( z ) (cid:19) e th ( z ) (cid:18) O (cid:18) t (cid:19)(cid:19) as t → ∞ and where we have to choose the square root in the right way (see [15, Chapter 6] ). Fourier Coefficients of f The goal of this section is to find the asymptotic behaviour of f . First note that f ( − z ; τ ) = f ( z ; τ )by Lemma 2.1, and so c ( − m, n ) = c ( m, n ). We therefore restrict our attention to the case m ≥ ζ m of f . Since we focus only on the case z ∈ [0 , h , ..., h s ∈ Q denote the poles of f in this range. Note that the distribution of the poles issymmetric on the interval [0 ,
1] (see Figure 1).Define the path of integration Γ ℓ,r byΓ ℓ,r := h − r if ℓ = 0 ,h ℓ + r to h ℓ +1 − r if 1 ≤ ℓ ≤ s − ,h s + r to 1 if ℓ = s, GIULIA CESANA AND JOSHUA MALES for some r > f ± m ( τ ) := s X ℓ =0 Z Γ ℓ,r f ( z ; τ ) e − πimz dz + s X ℓ =1 G ± ℓ,r , where G ± ℓ,r := Z γ ± ℓ,r f ( z ; τ ) e − πimz dz for a fixed pole h ℓ (1 ≤ ℓ ≤ s ). Here, γ + ℓ,r is the semi-circular path of radius r passing above thepole h ℓ and γ − ℓ,r is the semi-circular path passing below the pole h ℓ , see Figures 1 and 2.0 h h h h s − h s − h s Figure 1.
The path of integration taking γ + ℓ,r at each pole. γ + ℓ,r h ℓ r Figure 2.
The contour γ + ℓ,r for a fixed ℓ .In this case the Fourier coefficient of ζ m of f , for fixed m , is given by f m ( τ ) := lim r → + f + m ( τ ) + f − m ( τ )2 = lim r → + s X ℓ =0 Z Γ ℓ,r f ( z ; τ ) e − πimz dz ! + s X ℓ =1 G + ℓ,r + G − ℓ,r . (3.1)3.1. The integral G + ℓ . Next we consider the integral G + ℓ := lim r → + G + ℓ,r associated to the pole h ℓ . Shifting z z − h ℓ and parametrising the semi-circular path with z = re iθ yields G + ℓ = lim r → + ie − πimh ℓ Z π re iθ f (cid:16) re iθ + h ℓ ; τ (cid:17) e − πimre iθ dθ. Now we insert the Taylor expansion of e − πimre iθ and note that the terms of order f (cid:16) re iθ + h ℓ ; τ (cid:17) O (cid:16) r (cid:17) vanish for r → + since we have a simple pole at z = h ℓ . We therefore IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 7 consider lim r → + rf (cid:16) re iθ + h ℓ ; τ (cid:17) . Plugging in the definition of f we get N Y j =1 η ( a j τ ) α j lim r → + rϑ (cid:16) re iθ + h ℓ ; τ (cid:17) ϑ ( b ( re iθ + h ℓ ); cτ ) = N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) be iθ ϑ ′ ( bh ℓ ; cτ ) , (3.2)by L’Hˆopital’s rule, and where ϑ ′ = ∂∂r ϑ . Using Lemma 2.1 and the fact that [20, Proposition 1.3] ϑ ′ (0; τ ) = − πη ( τ ) we see that (3.2) equals ( − bh ℓ − πbe iθ N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) η ( cτ ) . Substituting back into the definition of G + ℓ gives G + ℓ = ie − πimh ℓ ( − bh ℓ − b N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) η ( cτ ) . By a similar argument, we have that G − ℓ = G + ℓ . Therefore we have that s X ℓ =1 G + ℓ + G − ℓ i s X ℓ =1 e − πimh ℓ ( − bh ℓ − b N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) η ( cτ ) . Remark.
In certain cases such as [12], this residue term may further reduce to an eta-quotient usingknown results for specific values of ϑ at specific arguments.In the following two sections we determine the asymptotic behavior of f towards and away fromthe dominant pole at q = 1, respectively. From now on we will let τ = iε π , ε := β (cid:16) ixm − (cid:17) , β := π q n and | m | ≤ β log( n ).4. Bounds toward the dominant pole
In this section we consider the behaviour of f m toward the dominant pole at q = 1. We firstconsider the residue terms R ℓ ( τ ) := ( − bh ℓ − b N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) η ( cτ ) . Fixing a choice of ℓ , we obtain the following result. Proposition 4.1.
Let τ = iε π and ≤ ℓ ≤ s . Define A ℓ ( m, n ) := ( − bh ℓ + N P j =1 α j c b (cid:16) ε π (cid:17) − N P j =1 α j N Y j =1 a − αj j . GIULIA CESANA AND JOSHUA MALES
Then for some N ℓ > we have R ℓ (cid:18) iε π (cid:19) = A ℓ ( m, n ) e π ε Λ ℓ (cid:18) O (cid:18) e − π ε N ℓ (cid:19)(cid:19) (4.1) as n → ∞ .Proof. Using Lemmas 2.1 and 2.2 we obtain R ℓ ( τ ) = ( − bh ℓ − √− icτ b √− iτ ϑ (cid:16) h ℓ τ ; − τ (cid:17) N Q j =1 η (cid:16) − a j τ (cid:17) α j η (cid:0) − cτ (cid:1) e − πih ℓτ N Y j =1 p − ia j τ ! α j . Plugging in the definitions of η and ϑ we obtain (after a little rewriting) R ℓ (cid:18) iε π (cid:19) = A ℓ ( m, n ) e π ε Λ ℓ (cid:18) − e − π hℓε (cid:19) × Y κ ≥ (cid:18) − e − π κε (cid:19) (cid:18) − e − π ε ( κ − h ℓ ) (cid:19) (cid:18) − e − π ε ( κ + h ℓ ) (cid:19) N Q j =1 (cid:18) − e − π κajε (cid:19) α j (cid:16) − e − π κcε (cid:17) . We next bound the asymptotic behaviour of the product over κ . Since c ∈ N we see that (cid:12)(cid:12)(cid:12) e − π κcε (cid:12)(cid:12)(cid:12) < κ ≥
1. Using the geometric series expansion and splitting α j into positive andnegative powers, labeled by γ j , δ j ∈ Q > , and a j into x j and y j respectively, we can rewrite theproduct as Y κ ≥ (cid:18) − e − π κε (cid:19) (cid:18) − e − π ε ( κ − h ℓ ) (cid:19) (cid:18) − e − π ε ( κ + h ℓ ) (cid:19) × X λ ≥ e − π λκcε N Y j =1 (cid:18) − e − π κxjε (cid:19) γ j N Y k =1 X ν ≥ e − π νκykε δ k . Since h ℓ < κ = 1, λ, ν = 0. Also note that e − π ε (1+ h ℓ ) < e − π ε < e − π ε (1 − h ℓ ) . Therefore either e − π xjε or e − π ε (1 − h ℓ ) is the second biggest order term. Including the extra factor1 − e − π hℓε and defining N ℓ := min (cid:16) x j , − h ℓ , h ℓ (cid:17) > (cid:3) Summing the residue terms immediately yields the following result.
Corollary 4.2.
We have that s X ℓ =1 G + ℓ + G − ℓ i s X ℓ =1 e − πimh ℓ A ℓ ( m, n ) e π ε Λ ℓ (cid:18) O (cid:18) e − π ε N ℓ (cid:19)(cid:19) . Next we consider the behaviour of f away from its poles. IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 9
Lemma 4.3.
Let τ = iε π , with < Re( ε ) ≪ , let z be away from the poles, let M > be anexplicitly given constant and let C ( z, τ ) := ( − N P j =1 α j (cid:16) ε π (cid:17) − N P j =1 α j c N Y j =1 a − αj j sinh (cid:16) π zε (cid:17) sinh (cid:16) π bzcε (cid:17) e π ε b z c − z − N P j =1 αj aj ! . Then we have that f (cid:18) z ; iε π (cid:19) = C (cid:18) z, iε π (cid:19) (cid:18) O (cid:18) e − π ε M (cid:19)(cid:19) as n → ∞ .Proof. Using Lemmas 2.1 and 2.2 and the definitions of η and ϑ a lengthy but straightforwardcalculation shows that f (cid:18) z ; iε π (cid:19) = C (cid:18) z, iε π (cid:19) Y κ ≥ (cid:18) − e − π κε (cid:19) (cid:18) − e π ε ( z − κ ) (cid:19) (cid:18) − e π ε ( − z − κ ) (cid:19) N Q j =1 (cid:18) − e − π κajε (cid:19) α j (cid:16) − e − π κcε (cid:17) (cid:16) − e π cε ( bz − κ ) (cid:17) (cid:16) − e π cε ( − bz − κ ) (cid:17) . In order to find a bound we inspect the asymptotic behaviour of the product over κ . We firstrewrite this as Y κ ≥ (cid:16) − e − π κε (cid:17) (cid:16) − e − π ε ( κ − z ) (cid:17) (cid:16) − e − π ε ( κ + z ) (cid:17) N Q j =1 (cid:18) − e − π κxjε (cid:19) γ j N Q k =1 P µ ≥ e − π µκykε ! δ k (cid:16) − e − π κcε (cid:17) (cid:16) − e − π cε ( κ − bz ) (cid:17) (cid:16) − e − π cε ( κ + bz ) (cid:17) , since | e − π κykε | < κ ≥
1. We also know that | e − π κcε | < | e − π cε ( κ + bz ) | < κ ≥ b, c ∈ N . Therefore we have that1 (cid:16) − e − π κcε (cid:17) (cid:16) − e − π cε ( κ + bz ) (cid:17) = X λ ≥ e − π λκcε X ξ ≥ e − π ξcε ( κ + bz ) . Up to this point our calculations are independent of the size of z . The remaining term is11 − e − π cε ( κ − bz ) . Let κ be the smallest κ ≥ κ − bz ) ≥
0. We may rewrite Y κ ≥ (cid:16) − e − π cε ( κ − bz ) (cid:17) = κ − Y κ =1 (cid:16) − e − π cε ( κ − bz ) (cid:17) Y κ ≥ κ X µ ≥ e − π µcε ( κ − bz ) . The first term is 1 (cid:16) − e − π cε ( κ − bz ) (cid:17) = − e π cε ( κ − bz ) X ν ≥ e π νcε ( κ − bz ) . Overall we see that, for 0 < Re( ε ) ≪ M := min (cid:16) x j , − z, bz +1 − κ c (cid:17) >
0, the product isof order 1 + O (cid:18) e − π ε M (cid:19) , which finishes the proof. (cid:3) Remark.
By separating into cases, one is able to obtain more precise asymptotics. However, this isnot required for the sequel and we leave the details for the interested reader.To find the contribution of the integrals in the right-hand side of (3.1), we use Lemma 4.3 anddefine the following integrals g m, := s X ℓ =0 Z Γ ℓ,r C ( z ; τ ) e − πimz dz, g m, := s X ℓ =0 Z Γ ℓ,r (cid:18) f (cid:18) z ; iε π (cid:19) − C ( z ; τ ) (cid:19) e − πimz dz. We first study the contribution of g m, and show the following proposition. Proposition 4.4.
Assume that | x | ≤ and let G ( m, n ) := ( s + 1)( − N P j =1 α j (cid:16) ε π (cid:17) − N P j =1 α j c sinh (cid:18) πim b c − (cid:19)q − b c sinh (cid:18) πimb b c − (cid:19) N Y j =1 a − αj j . Then for r → + , and with τ = iε π we have that g m, = G ( m, n ) e π ε c − − N P j =1 αj aj ! (1 + O ( ε )) as n → ∞ .Proof. For simplicity we define g m, ,ℓ := Z Γ ℓ,r C ( z ; τ ) e − πimz dz. Plugging in the definition of C ( z ; τ ) and some rearranging gives us g m, , = ( − N P j =1 α j (cid:16) ε π (cid:17) − N P j =1 α j c N Y j =1 a − αj j e − π ε N P j =1 αj aj × h − r Z sinh (cid:16) π zε (cid:17) sinh (cid:16) π bzcε (cid:17) e π ε (cid:16) b z c − z − imεzπ (cid:17) dz. IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 11
Using the saddle point method detailed in Section 2.4 with input data Γ = Γ ,r , g ( z ) = sinh (cid:16) π zε (cid:17) sinh (cid:16) π bzcε (cid:17) , t = π ε , and h ( z ) = b z +14 c − z +14 − imεzπ we obtain the asymptotic approximation r επ e π ε m ε π (cid:18) b c − (cid:19) (cid:16) − b c + (cid:16) b c − (cid:17)(cid:17) + c − sinh (cid:18) πim b c − (cid:19)q − b c sinh (cid:18) πimb b c − (cid:19) (cid:16) O (cid:16) ε π (cid:17)(cid:17) . (4.2)Noting that for each Γ ℓ,r the same calculation holds we immediately see that g m, = G ( m, n ) e π ε c − − N P j =1 αj aj ! (1 + O ( ε )) , where we use that e m ε = 1 for ε → m is fixed. (cid:3) We bound the contribution of g m, by noting the following trivial lemma. Lemma 4.5.
For | x | ≤ , we have that | g m, | ≪ g m, . Noting that h ℓ < h ℓ we immediately see that (4.1) is asymptotically larger in absolute valuethan g m, . Gathering the asymptotic behaviour of the integral parts of (3.1) we obtained from theProposition 4.4 and Lemma 4.5, and using Corollary 4.2 yields f m ( τ ) = i s X ℓ =1 e − πimh ℓ A ℓ ( m, n ) e π ε Λ ℓ + O (cid:18) e π ε (Λ ℓ − N ℓ ) (cid:19) + G ( m, n ) e π ε c − − N P j =1 αj aj ! . Therefore, near the dominant pole at q = 1 we have shown the following theorem. Theorem 4.6.
For | x | ≤ we have that f m (cid:18) iε π (cid:19) = i s X ℓ =1 e − πimh ℓ A ℓ ( m, n ) e π ε Λ ℓ + O e π β max c − − N P j =1 αj aj , Λ ℓ − N ℓ ! as n → ∞ . Bounds away from the dominant pole
In this section we investigate the contribution of f m away from the dominant pole, and showthat it forms part of the error term. We begin by inspecting the residue term R ℓ . Lemma 5.1.
For ≤ x ≤ πm β we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ℓ (cid:18) iε π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ B ( m, n ) exp π √ n Λ ℓ − π πcβ + N X k =1 πδ k y k β ! − q m − with B ( m, n ) := r πβ n − N P k =1 δk N Y j =1 πx j β ! γj . Proof.
Recall that R ℓ ( τ ) = ( − bh ℓ − b N Y j =1 η ( a j τ ) α j ϑ ( h ℓ ; τ ) η ( cτ ) . We begin with the term Q Nj =1 η ( a j τ ) α j . As in [12] we write N Y j =1 η ( a j τ ) α j = N Y j =1 η ( x j τ ) γ j N Y k =1 q − ykδk P ( q y k ) δ k . In the same way as [12], using Lemma 2.2 we see that η (cid:18) ix j ε π (cid:19) γ j ≪ πx j β ! γj e − π γj xjβ . Using (2.2) we also obtain that | P ( q y k ) | ≪ n − exp πy k β π − π − q m − . Therefore we find N Y j =1 η (cid:18) ia j ε π (cid:19) α j ≪ N Y j =1 πx j β γj e − N P j =1 π γj xjβ × N Y k =1 n − δk exp πδ k y k β π − π − q m − . (5.1)It remains to consider the behaviour of ϑ ( h ℓ ; τ ) η ( cτ ) . Using Lemma 2.1 we find that | ϑ ( h ℓ ; τ ) | = 1 √− iτ e − πiτ ( h ℓ − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X κ ∈ Z ( − κ e πiτ (2 h ℓ κ − κ − κ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We see that X κ ∈ Z ( − κ e − π ε κ ( κ +1 − h ℓ ) = 1 + ∞ X κ =1 ( − κ e − π ε κ ( κ +1 − h ℓ ) + ∞ X κ =1 ( − κ e − π ε κ ( κ − h ℓ ) , IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 13 where the exponents in both sums are strictly negative since h ℓ ∈ (0 , κ = 1 terms, namely e − π ε (2 − h ℓ ) (resp. e − π hℓε ). Therefore we obtain the bound (cid:12)(cid:12)(cid:12)(cid:12) ϑ (cid:18) h ℓ ; iε π (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≪ r πβ e − π β ( h ℓ − ) (cid:18) e − π β min((1 − h ℓ ) ,h ℓ ) (cid:19) . (5.2)With (2.2) we also obtain η (cid:18) icε π (cid:19) − = q − c P ( q c ) ≪ n − exp πcβ π − π − q m − . (5.3)Combining (5.1), (5.2) and (5.3) finishes the proof. (cid:3) Summing the residue terms over ℓ immediately yields the following corollary. Corollary 5.2.
We have that s X ℓ =1 G + ℓ + G − ℓ ≪ s X ℓ =1 B ( m, n ) exp π √ n Λ ℓ − π πcβ + N X k =1 πδ k y k β ! − q m − . Next we bound the integral part of (3.1). We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X ℓ =0 Z Γ ℓ,r f ( z ; τ ) e − πimz dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ s X ℓ =0 Z Γ ℓ,r (cid:12)(cid:12)(cid:12) f ( z ; τ ) e − πimz (cid:12)(cid:12)(cid:12) dz. Consider (cid:12)(cid:12)(cid:12) f ( z ; τ ) e − πimz (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 η ( a j τ ) α j ϑ ( z ; τ ) ϑ ( bz ; cτ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) away from the dominant pole. We see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 η ( a j τ ) α j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ϑ ( z ; τ ) ϑ ( bz ; cτ ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 πx j β γj e − N P j =1 π γj xjβ N Y k =1 n − δk × exp πδ k y k β π − π − q m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) ϑ ( z ; τ ) ϑ ( bz ; cτ ) (cid:12)(cid:12)(cid:12)(cid:12) , using (5.1). Plugging in (2.1), using Lemma 2.1, and some rearranging leads to (cid:12)(cid:12)(cid:12)(cid:12) ϑ ( z ; τ ) ϑ ( bz ; cτ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) q − c (cid:12)(cid:12)(cid:12) | ϑ ( z ; τ ) | (cid:12)(cid:12)(cid:12)(cid:12) P κ ∈ Z ( − κ q c κ κ ζ bκ (cid:12)(cid:12)(cid:12)(cid:12) ≪ r πβ e − π β min z ∈ Γ ℓ,r ( z − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X κ ∈ Z ( − κ e − π ε ( κ +(1 − z ) κ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ β − e − π β min z ∈ Γ ℓ,r ( z − ) ≪ n e − π β min z ∈ Γ ℓ,r ( z − ) . With this we have, for r → + , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s X ℓ =0 Z Γ ℓ,r f ( z ; τ ) e − πimz dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ s X ℓ =0 N Y j =1 πx j β γj n − N P k =1 δk e − π β min z ∈ Γ ℓ,r ( z − ) × exp N X k =1 πδ k y k β π − π − q m − − N X j =1 π γ j x j β . Note that this is smaller than the contribution of the residue terms determined in Corollary 5.2.Hence, away from the dominant pole in q we have shown the following proposition. Proposition 5.3.
For ≤ x ≤ πm β we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f m (cid:18) iε π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ s X ℓ =1 B ( m, n ) exp π √ n Λ ℓ − π πcβ + N X k =1 πδ k y k β ! − q m − for n → ∞ . The Circle Method
In this section we use Wright’s variant of the Circle Method to complete the proof of Theorem1.2. We start by noting that Cauchy’s Theorem implies that c ( m, n ) = 12 πi Z C f m ( τ ) q n +1 dq, where C := { q ∈ C | | q | = e − β } is a circle centred at the origin of radius less than 1, with the pathtaken in the counter-clockwise direction. Making a change of variables, changing the direction ofthe path of the integral, and recalling that ε = β (1 + ixm − ) we have c ( m, n ) = β πm Z | x |≤ πm β f m (cid:18) iε π (cid:19) e εn dx. Splitting this integral into two pieces, we have c ( m, n ) = M + E where M := β πm Z | x |≤ f m (cid:18) iε π (cid:19) e εn dx, and E := β πm Z ≤| x |≤ πm β f m (cid:18) iε π (cid:19) e εn dx. Next we determine the contributions of each of the integrals M and E , and see that M contributesto the main asymptotic term, while E is part of the error term. IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 15
The major arc.
Considering the contribution of M , we obtain the following proposition. Proposition 6.1.
We have that M = s X ℓ =1 D ℓ ( m, n )Λ − N P j =1 α j ℓ e − πimh ℓ e π √ ℓ n π (2Λ ℓ n ) + O n − + N P j =1 α j e π √ ℓ n as n → ∞ .Proof. Making the change of variables v = 1 + ixm − and then v p Λ ℓ v we obtain that M = s X ℓ =1 D ℓ ( m, n )Λ − N P j =1 α j ℓ e − πimh ℓ P − N P j =1 α j , ℓ − iβ π p Λ ℓ O e π β max c − − N P j =1 αj aj , Λ ℓ − N ℓ ! im − √ Λ ℓ Z − im − √ Λ ℓ e β √ Λ ℓ vn dv = s X ℓ =1 D ℓ ( m, n )Λ − N P j =1 α j ℓ e − πimh ℓ P − N P j =1 α j , ℓ + O βe π β max c − − N P j =1 αj aj , Λ ℓ − N ℓ ! + π √ ℓ n , where P s,k := im − √ Λ ℓ Z − im − √ Λ ℓ v s e π q kn ( v + v ) dv. One may relate P s,k to I -Bessel functions in exactly the same way as [4, Lemma 4.2], making theadjustment for p Λ ℓ where necessary, to obtain that P s,k = I − s − π r kn ! + O exp π r kn (cid:18) m − (cid:19)!! . Using the asymptotic behaviour of the I -Bessel function given in Lemma 2.4 we obtain P − N P j =1 α j , ℓ = e π √ ℓ n π (2Λ ℓ n ) + O e π √ ℓ n (8 π Λ ℓ n ) ! + O e π √ ℓ n m − ! , and therefore M = s X ℓ =1 D ℓ ( m, n )Λ − N P j =1 α j ℓ e − πimh ℓ e π √ ℓ n π (2Λ ℓ n ) + O β − N P j =1 α j e π √ ℓ n (8 π Λ ℓ n ) + O β − N P j =1 α j e π √ ℓ n m − ! + O βe π √ n max c − − N P j =1 αj aj , Λ ℓ − N ℓ ! + π √ ℓ n . It is clear that the first O -term is the dominant one sincemax c − − N X j =1 α j a j , Λ ℓ − N ℓ < Λ ℓ ≤ p Λ ℓ . This finishes the proof. (cid:3)
The error arc.
Finally, we bound E as follows. Proposition 6.2.
We have E ≪ M as n → ∞ .Proof. By Proposition 5.3 we have E ≪ β πm s X ℓ =1 B ( m, n ) exp π √ n Λ ℓ − π πcβ + N X k =1 πδ k y k β ! − q m − × e βn Z ≤| x |≤ πm β e βnixm − dx ≪ π s X ℓ =1 B ( m, n ) exp π √ n (Λ ℓ + 1) − π πcβ + N X k =1 πδ k y k β ! − q m − , where we trivially estimate the final integral. Using (Λ ℓ + 1) ≤ p Λ ℓ the result follows immediatelyby comparing to M and therefore also finishes the proof of Theorem 1.2. (cid:3) Further questions
We end by briefly commenting on some related questions that could be the subject of furtherresearch.(1) Here we only discussed the case of eta-theta quotients with simple poles. A natural questionto ask is: does a similar story hold for functions with higher order poles? The situation isof course expected to be more complicated, in particular finding Fourier coefficients withthe method presented here seems to be much more difficult. One could attempt to build aframework by following the definitions of Fourier coefficients given in [7, Section 8].For example, in [13] Manschot and Rolon study a Jacobi form with a double pole relatedto χ y -genera of Hilbert schemes on K3. They obtain bivariate asymptotic behaviour in asimilar flavour to those here. Can one extend this family?(2) Although the functions considered in the present paper provide a wide family of results, itshould be possible to extend the method to other related families of functions. In particular,it would be instructive to consider similar approaches for prototypical examples of mockJacobi forms. IVARIATE ASYMPTOTICS FOR ETA-THETA QUOTIENTS WITH SIMPLE POLES 17
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University of Cologne, Department of Mathematics and Computer Science, Division of Mathemat-ics, Weyertal 86-90, 50931 Cologne, Germany
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