AN EXTENSION OF THE HARDY-RAMANUJAN CIRCLE METHOD AND APPLICATIONS TO PARTITIONS WITHOUT SEQUENCES

Transcription

1 AN EXTENSION OF THE HARDY-RAMANUJAN CIRCLE METHOD AND ALICATIONS TO ARTITIONS WITHOUT SEQUENCES KATHRIN BRINGMANN AND KARL MAHLBURG Abstract. We develop a generalized version of the Hardy-Ramanujan circle method in order to derive asymptotic series expansions for the products of modular forms and moc theta functions. Classical asymptotic methods including the circle method do not wor in this situation because such products are not modular, and in fact, the error integrals that occur in the transformations of the moc theta functions can and often do mae a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified. To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order moc theta function χ and a modular infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the moc theta function component appear in the exponential main term.. Introduction.. artition functions and asymptotics for harmonic Maass forms. We begin with a general discussion of the history of the study of asymptotics for combinatorial generating functions and q-series before stating our main results in Section.. Recall that an integer partition is a decomposition of a positive integer into the sum of wealy decreasing nonnegative integers, and that an overpartition is a partition in which the first occurrence of a part may also be overlined. Denote the number of integer partitions of n by pn, the number of partitions into distinct parts by Qn, and the number of overpartitions by pn see [] and [4] for more 000 Mathematics Subject Classification. 8, 05A7, C0. The first author was partially supported by NSF grant DMS and by the Alfried Krupp prize. The second author was partially supported by NSA Grant

2 KATHRIN BRINGMANN AND KARL MAHLBURG combinatorial bacground. The corresponding generating functions are. pnq n = q =, n q; q n 0 n 0 + q n = q; q, Qnq n = n 0 n 0 n 0 pnq n = n 0 + q n q n = q; q q; q, where we use standard notation for the rising q-factorials a n = a; q n := n i=0 aqi. In particular, the overpartitions are a convolution product of ordinary partitions and distinct parts partitions, so. pn = n pqn. =0 An important question in the theory of partitions is to determine exact formulas or asymptotics for functions such as pn and its relatives. Indeed, since the generating functions in. are essentially meromorphic modular forms, these are special cases of the general question of determining the coefficients of modular forms. In fact, since many partition functions also have coefficients that grow monotonically, the Hardy-Ramanujan Tauberian Theorem [7] for eta-quotients shows that as n, the following asymptotics hold:. pn 4n eπ n, Qn 4 4 n eπ n, /4 pn 8n eπ A ey implication of Hardy and Ramanujan s result [7] is that the coefficients in a convolution product of modular forms such as the overpartition function will satisfy a logarithmic asymptotic of the form.4 log pn log pn + log Qn, so that both summands from. mae a predictable contribution to the overall asymptotic. Building on Hardy and Ramanujan s earlier developments, Rademacher and Zucerman later proved much more precise results about the coefficients of modular forms using the circle method, culminating in exact asymptotic series expansions for functions lie pn []. Such expansions loo much lie the one seen in Theorem., although the series for modular forms involve only Bessel functions rather than principal part integrals. n.

3 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES Another important example in the study of coefficients of hypergeometric series and automorphic forms is Ramanujan s third order moc theta function.5 fq = αnq n := +, q; q n 0 n n which is famously not a modular form []. Instead, recent wors of the first author, Ono, and Zwegers [0,, 5] show that fq is best understood as the holomorphic part of a harmonic Maass form of half-integral weight. In terms of the practical application of the circle method, this means that the modular transformations of fq yield another automorphic q=series object plus a Mordell-type integral see Section ; these integrals were absorbed into the error terms of the asymptotic series expansion for αn obtained by Dragonette [5] and Andrews []. Recent wor of the first author and Ono essentially allows one to calculate exact series expansions for all of the above examples, and indeed for any harmonic Maass form of weight at most / without using the circle method although there are technical convergence issues in the case of weight equal to / []. The series are derived from real analytic Maass-oincaré series that are uniquely determined by the automorphic transformations and principal parts of the Maass forms. For example, this allowed the first author and Ono to completely prove the Andrews-Dragonnette conjecture, giving an exact formula for the coefficients αn in [0]. However, these very precise results do not apply to products of harmonic Maass forms as the space of such automorphic forms is not closed under multiplication, and there has been recent interest in many functions of this type that arise in the study of probability, mathematical physics, and partition theory. Our main result uses calculations based on the circle method to find the asymptotic series expansion for such functions. Since our current state of nowledge does not include oincaré series for functions in the space of harmonic Maass forms tensored with modular forms and one should not necessarily even expect that such a basis exists, our method yields the best nown asymptotics in this situation. In particular, we will use the important example of partitions without sequences to illustrate the application of our general results throughout the rest of the paper, but we emphasize that our approach of identifying principal part integrals in Sections and 4 is widely applicable to other products of moc modular forms and modular forms. q n.. artitions without sequences and the statement of the main results. In [], Andrews considered partitions that do not contain any consecutive integers as parts, which had recently arisen in connection with certain probability models as well as in the study of threshold growth in cellular automata [8] also see [6]. Adopting his notation, let p n be the number of such partitions of size n. He derived the generating function.6 G q := n 0 p nq n = q ; q q ; q χq,

4 4 KATHRIN BRINGMANN AND KARL MAHLBURG where χq := q n q; q n q ; q n 0 n is another of Ramanujan s third-order moc theta functions []. Holroyd, Liggett and Romi [8] used clever combinatorial arguments to show that n.7 rn p n rj, where rn are the coefficients of the infinite product from.6, namely ξq := n 0 j= rnq n = q ; q q ; q. This means that in the convolution product G q = ξqχq, the exponential growth of the coefficients p n is entirely due to the ξq factor, despite the fact that χq lie any of the moc-theta functions also has coefficients that grow exponentially. In other words, there must be a great deal of cancellation when these two series are multiplied. Holroyd et al. also identified the main exponential growth factor of p n, using Tauberiantype estimates to show that log rn π n. In fact, a much stronger statement also follows from those authors observations. The circle method or a Maass-oincaré series decomposition as in [] applied to the weight / eta-quotient ξq gives the more precise estimate rn c e π n for some explicit constant c as in Section 4. Applying this to.7 and using Ingham s n Tauberian theorem from [9] shows that Holroyd et al. s wor actually implies asymptotic bounds of the form.8 n e π n p n e π n. n Andrews [] improved upon Holroyd et al s results and even the stronger implication.8 by determining the cusp expansion of G q as q. He rewrote the function by using a moc theta identity, replacing ξq by the sum of an eta-quotient and a different moc theta function. In particular, he obtained the decomposition.9 G q = where fq is as previously defined in.5. equation.9 by g q := q 6 ; q 6 q ; q fq +, 4q ; q q ; q 4q; q q ; q q 6 ; q 6 q 6 ; q 6 4q ; q q ; q fq, g q := We denote the two terms on the right-side of q ; q 4q; q q ; q q 6 ; q 6. If q = e s, Andrews proved that as s 0, G q has the asymptotic behavior s.0 G q 6π e π 9s + e π 9s,

5 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 5 where the two terms come from g q and g q, respectively. However, it requires more than the cuspidal estimate of.0 to determine p n precisely and it would not be enough to merely consider the other cusps. The chief technical issue is that although G q essentially has weight zero modular transformation properties, it is not an automorphic form or a holomorphic part thereof. Therefore it does not lie in the standard framewor of of the circle method and/or oincaré series, in which a modular or harmonic Maass form is determined by its principal part, or cusp expansions. In fact, the theory developed by the first author, Ono, and Zwegers explains that χq has an associated non-holomorphic part that is necessary to construct a completed harmonic Maass form. The principal part integrals that arise in our main theorem should be viewed as arising from the intermixing of the non-holomorphic part of χq with the principal part of the eta-quotient. We assume the definition of ω h, from Section. For positive integers h, h, n,, and ν, with hh mod, define the roots of unity ζh, n,, ν := ν e πi hn+h ν + ν and α r h, := ω h, ω h,r, ω h,r,r,,r ω 6h 6,r, 6,r. Furthermore, for any b > 0, define the integral I b,,ν n := x cosh πiν /6 π I bn x πx b dx, where I x is a modified Bessel function of the first ind, which can be defined by the integral representation.. We have the following asymptotic expansion.

6 6 KATHRIN BRINGMANN AND KARL MAHLBURG Theorem.. Let N := [n / ]. The asymptotic expansion for p n is given by p n = π 6 α h, ζh, n,, νi 6n,,νn 6 + 5π 6 6n + π 8 6n + π 6 n 0 h< N h,= 6,= 0 h< N h,= 6,= 0 h< N h,= 6,= 0 h< N h,= 6,= α h, α h, ω h, ω h, ω 6h, ω h, ν mod ν mod ν mod e πihn ζh, n,, νi 5 6,,νn ζh, n,, νi 8,,νn π I n + Olog n. We can also isolate part of the leading exponential term to obtain the leading terms of the asymptotic expansion for p n explicity. Theorem.. For any 0 < c < /8, as n we have p n = 4 n + /4 8 n e π n + O e π Remar. Recalling.7 and the subsequent discussion, this asymptotic is equivalent to p n C n 4 rn up to a constant scaling, which can be compared with.8. This is maredly different from the behavior of modular partition functions seen in. and.4; although the coefficients of χq grow asymptotically with some exponential factor e Cn the same is true for any moc theta function due to their nontrivial principal parts [, 5], the exponential growth of p n is the same as that of rn. Remar. The second asymptotic term for p n arises from one of the principal part integrals in Theorem., and the proof in Section 4 shows that there are also products of modular forms and moc theta functions in which the dominant exponential term in the coefficient asymptotics arises from such an integral and thus from the non-holomorphic part of the moc theta function. Although Theorem. is a consequence of Theorem. the two terms correspond to the terms = in the two sums with 6, =, it is not an immediate corollary, as it requires some analysis to identify the principal part of the integrals I b,,ν n. We now describe the structure of the paper. In Section, we record the modular transformation laws for G q. Section contains the proofs of several technical integral estimates. We n n +c.

7 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 7 then apply the circle method in Section 4 and prove the asymptotic expansion of Theorem.. In Section 5 we analyze the exponentially dominant terms in the expansion to prove Theorem.. Acnowledgments The authors than Todd Kemp for helpful discussions about principal part integrals. Moreover they than the referee for many helpful comments.. Modular transformation properties In this section we determine the modular transformations for G q. First, adopt the notation q := q; q. If h and are coprime positive integers, then define h so that hh mod if is even, we can assume this congruence holds modulo 4 and if is odd we may assume that 8 h. Furthermore, we introduce a complex variable z with Rez > 0 such that q = e πi h+iz, and define q := e πi h +iz. The classical modular transformation for q can then be written as. q = ω h, z / e πz z q, where ω h, := e πish,, and sh, is the standard Dedeind sum []. If r is a positive integer, the transformation for q r follows from.. To compactly write the formula, we first let g r := r, and define ρ r := r g r, r := g r. We also set q r := e πi r h r+iz /ρ r with h r defined so that h rρ r h mod r. The transformation law is then. q r = ω hρr,r ρ r z / e π z r ρr ρrz q r. We write q r only when it is clear that h and are fixed. This nonstandard notation is appealing because we can and will always select h such that ρ r h since ρ r, r =, and thus q r = q gr/ρr. The eta-quotient component of g q is ξq := q q. Since ξq is an eta-quotient that q 6 is essentially modular with respect to a congruence subgroup of level 6, we need only consider the different transformations for all possible values of 6,. We begin with the case 6 ; we have. ξq = ω h, Next, if 6, =, then ω h, ω h,/6.4 ξq = ω h, ω h, ω h, z / e π z z ξq. z / e πz + 5π 6z q q /. q /

8 8 KATHRIN BRINGMANN AND KARL MAHLBURG Similarly, if 6, =, then.5 ξq = ω h,ω h, ω h, Finally, if 6, =,.6 ξq = ω h,ω h, ω 6h, z / e πz + π 6z z / e πz + π 8z q. q / q / q / q /. Now we turn to fq, whose transformation law was studied by Andrews [], and is essentially of level. If is even we have fq = + e πi h h 4 ω h, z πz z.7 e fq + ω h, z e where.8 I,ν z := R πz ν mod cosh q /6 ν e πih ν +ν I,ν z, e πzx πiν 6 dx. πzx We note that there is a typo regarding the term e πz z in this transformation as it is stated in Andrews Theorem. although the correct formula is stated in the proof; we have also replaced his by an even. Throughout we will use the residues ν in all of our calculations. If is odd, then we have fq = e πih 4 ω h, z e π z πz ω q.9 + z πz e ωh, ν mod ν e πih ν where q nn+ ωq := q; q n=0 n+ is another one of Ramanujan s third-order moc theta functions. If 6, then the transformation law of g is given by.0 g q = q q q 6 4 q = ω h, ω h, ω h, 6 ω h, πih ν I,ν z, g q.

9 If 6, =, then. g q = 4 If 6, =, then. g q = Finally, if 6, =, then. g q = ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 9 ω h, ω h, ω h, e π ωh, 9z q q q /. q / ω h, ω h, ω h, q q e π ω z q h, q / ω h, ω h, ω q q 6h, q 6 e π ωh, 8z. q. Integral estimates In this section we prove some of the technical bounds that we will need in order to apply the circle method and prove Theorem.. Specifically, we show that integrating the transformation laws from Section naturally leads to Bessel functions and our I b,,ν. Throughout we let 0 h < N with h, =, and z = N iφ with ϑ h, Φ ϑ h,. Here ϑ h, := +, ϑ h, := +, where h < h < h are adjacent Farey fractions in the Farey sequence of order N. From the theory of Farey fractions it is nown that. j =,. + j N + Lemma.. If b R, ν Z, with 0 < ν, let J b,,ν z := ze πb z I,ν z, and define the principal part truncation of J b,,ν as J b,,νz := b cosh e πb z x πiν /6 πx b dx. As z 0, we have the following asymptotic behavior: If b 0, then J b,,ν z π πν /6. If b > 0, then J b,,ν z = J b,,ν z + E b,,ν, where the error term satisfies for 0 < ν E b,,ν π πν /6..

10 0 KATHRIN BRINGMANN AND KARL MAHLBURG Here all of the implied constants are allowed to depend on b. Remar. We use the terminology principal part truncation for the integral defining J b,,ν as it is indeed a principal part integral, whose distribution is concentrated around x = 0. The phrase also serves as a reminder that we will view these integrals as a continuous analogue of the principal part of a q-series in the circle method. roof. We begin by maing the substitution x x/az in I,ν z, where a is some undetermined real constant which we will select later. This means that. J b,,ν z = a e πb e πx a z e πx a z z S cosh πiν /6 dx = πx a e πb z a R cosh πiν /6 dx, πx a where S is the line through the origin defined by arg±x = argz. The last equality follows from the facts that e s is entire and coshs has poles only at imaginary values of s. We also need the simple observation that for a fixed z the magnitude of the integrand can be bounded by e Cx as x for some constant C > 0. Thus the integral along a circular path of radius R that joins S and R vanishes as R, and Cauchy s Theorem then allows us to shift S bac to the real line. We next require the bound. Rez = N N 4 + Φ N + N, which grows as. This means that the asymptotic behavior of e bz as and therefore J b,,ν z as well depends on the sign of b. If b 0, then we may tae a = for simplicity and proceed similarly as in [, 8]. If α 0 and β R, we use the simple bound π.4 coshα + iβ = coshα cosβ + i sinhα sinβ cosβ sin β. For 0 < β < π, this yields a simple uniform bound throughout the range ν <, namely that.5 πiν /6 cosh πx π sin πν /6 π πν /6. Combining a simple bound for the Gaussian error function with. and.5 then completes the proof of, giving J b,,ν z e πx Rez z dx cosh πiν /6 πx π πν /6.6 R π πν /6.

11 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES In the case that b > 0, we follow. and write.7 J b,,ν z = a cosh R b e π x z a πiν /6 dx. πx a The asymptotic behavior of z recall. implies that the integral in.7 naturally splits at b = x a. We therefore set a = where E b,,ν z := b for convenience, which gives J b,,ν z = J b,,νz + E b,,ν z, b x > One easily sees that.8 E b,,ν z b cosh cosh e πb z x πiν /6 πx b e πb z x πiν /6 πx b dx. dx. Maing the substitution x x + and mimicing the arguments that led to.6 gives.9 E b,,ν z b e πb z x x dx cosh πiν /6 b π πν /6, which completes the proof of. roposition.. If b > 0 and n N, then h, ϑ h, e πnz J b,,νz dφ = with E b,,ν N 0 πb 6n x cosh πiν /6 πx+ π I bn x πx b dx + E b,,ν = πb 6n I b,,νn + E b,,ν, π πν /6. Here all the implied constants may depend on b. roof. We begin by symmetrizing the outer integral, writing.0 h, ϑ h, = N N ϑ h, N N In the second and third integrals of.0 the boundary errors, the range is bounded away from zero as Φ. In this range,. implies that N Rez and thus e πb z = O b. h,.

12 KATHRIN BRINGMANN AND KARL MAHLBURG This means that even though b is positive, the bound from Lemma. part still applies. The integrals are over an interval of length at most, so the overall boundary contribution is N O b π πν /6. N Now we expand the integral for J, recalling that Φ = i z variable z: N and switching to the. N N e πnz J b,,νz dφ = = b i N + i N N i N N N e πnz cosh b cosh e πb x z + πn z πiν /6 πx b e πb z x πiν /6 πx b dx dz. dx dφ Next, we utilize a standard contour shift in the complex z-plane in order to better recognize the main term as a Bessel function. Let Γ be the counterclocwise circle that passes through the points ± i and is tangent to the imaginary axis at the origin. The radius of this circle N N is c = +, and if z = u + iv, then the circle s equation is N u + v = cu. This implies that Rez = u = < for all nonzero points of Γ. u +v c In., the integrand s only pole in the z-variable is at z = 0, and thus Cauchy s theorem allows us to shift the original straight-line path to Γ, which is the portion of Γ that is to the right of Rez =. Let Γ N denote the arc of Γ to the left of this line. Along Γ, both z and z have bounded real parts, and the numerator of the integrand is O b. Therefore.4 and.5 imply that b Γ Γ cosh e πb x z + πn z πiν /6 πx b π πν /6 dx dz dx dz N π πν /6.

13 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES Excluding the error terms, we have now replaced the integral of J by b i e πb x z + πn z dx dz cosh πiν /6 Γ = b i cosh πx b πiν /6 πx b Γ e πb x z + πn z dz dx. The change of variables Z = πb x z taes Γ to the vertical line Rez = γ = πb c x > 0 and gives πb b i cosh x πiν /6 πx b γ+i γ i Z e Z+ π nb x Z dz dx. Finally, we obtain the claimed formula by applying the integral representation for the modified Bessel functions I σ here σ = π nb x, namely. πi γ+i γ i t r e σt +t dt = σ r Ir σ. We next turn to the contribution of the non-holomorphic part. roposition.. Assuming the notation above, we have for r > 0 and as n : h, e π nz+ z r π r 4π dφ = n I nr + O. N ϑ h, roof setch. The proof follows as in the wor of Rademacher and Zucerman [, ]; their setup was also used in the preceding proof of roposition.. Note that the integral in this result comes from the holomorphic part of the harmonic Maass form and has the same shape as in the case of classical modular forms. We sip the details of the proof here as it is essentially a nown result, and is significantly easier than the above proof of roposition. due to the absence of the ν-parameter and the J function. 4. The circle method and the proof of Theorem. 4.. Set up. To prove Theorem., we use the Hardy-Ramanujan method. By Cauchy s Theorem we have for n > 0 p n = G q dq, πi q n+ C

14 4 KATHRIN BRINGMANN AND KARL MAHLBURG where C is an arbitrary path inside the unit circle that loops around 0 in the counterclocwise direction. We chose the circle with radius r = e π/n with N := n /, and use the parametrization q = e π/n +πit with 0 t. This gives p n = 0 G e π N +πit e πn N πint dt. We let h,, ϑ h,, ϑ h, be defined as in Section. We decompose the path of integration into paths along the Farey arcs ϑ h, Φ ϑ h,, where Φ = t h. Thus 4. p n = 0 h< N h,= = 0 h< N h,= e πihn e πihn h, ϑ h, h, ϑ h, G e πi h+iz e πnz dφ ] [g e πi h+iz + g e πi h+iz e πnz dφ, where z = N iφ as before. For notational convenience we group the terms based on the divisibility properties of, writing p n = , where d denotes the sum over all terms 0 h < N with h, = and 6, = d. 4.. Estimation of 6. Our strategy for estimating 6, as well as the other sums, is inspired by the classical circle method, where the asymptotic contributions are largely determined by the leading powers of q. Specifically, since q = e π Rez and we are considering z 0, positive exponents will be absorbed into the global error. In contrast, negative exponents will lead to principal part integrals or Bessel functions depending on whether or not the term involves a Mordell integral using roposition. and Lemma., respectively.

15 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 5 Combining the transformations.,.7, and.0, we have 4. 6 = S 6 + S 6 + S 6 := h, 6 + h, 6 + h, 6 ω h, ω h, ω h, + e πih πihn ω h, 6 ω h, ω h, ω h, 6 e πihn ω h, h, ϑ h, ω h, ω h, ω h, e πihn ω h, 6 e πnz ν mod h, g q dφ h, ϑ h, ν e πih ν +ν e πnz g q dφ e πnz z e π z ξq I,ν zdφ. ϑ h, Throughout the remainder of the paper, we write h, as a shorthand for the summation conditions in 4.. We now estimate each of the S 6i and show that they are part of the error term. Using the trivial bound for all of the roots of unity we obtain the estimate 4. S 6 h, 6 h, ϑ h, e πnz g q dφ = h, 6 h, ϑ h, e πn N g q dφ. Furthermore,. implies that g q is uniformly bounded over the outer sum. Therefore, by 4. we have 4.4 S 6 h, 6 e πn N N eπn/n = O. The same arguments also imply that S 6 = O.

16 6 KATHRIN BRINGMANN AND KARL MAHLBURG This leaves S 6, which is complicated by the presence of the error integral I,ν z within the integrand. Again we use that in the domain of integration ξ is uniformly bounded. We have S 6 h, 6 e πn/n h, 6 h, ν mod ϑ h, ν= ϑ h, e πnz h, ξq ze π z I,ν z dφ J,,νz dφ. Lemma. implies that 4.5 S 6 e πn/n h, 6 h, 6 h, 6 N N π ν= h, ν= ϑ h, π πν /6 log = O log N. πν /6 dφ dφ Overall, we have proven that 6 = Olog N.

17 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES Estimation of. By.4,.7, and., we have = S + S + S := h, 6,= h, 6,= + h, 6,= ω h, ω h, ω h, + ω h, ω h, ω h, ω h, e πihn ωh, ω h, ω h, ω h, ω h, e πihn e πih πihn h, ϑ h, ν mod h, ϑ h, e πnz + π 9z h, ϑ h, ν e πih ν +ν e πnz z e 5π 6z e πnz + π 9z q q q q / dφ q / q q / I,ν zdφ. q / q / q / fq dφ Although S and S both have e π/z to the positive exponent /9 and thus would individually mae a nontrivial asymptotic contribution according to Lemma., we now show that the leading term cancels for all h and in S + S, leaving only negative exponents which lie in the asymptotic error term. For odd h, we use the explicit Dedeind sum evaluation [] 4.6 ω h, = h exp πi 4 h h + h h + h h. Recalling that at the beginning of Section we chose h so that h, it is not hard to see that 4.7 Indeed 4.7 is equivalent to the identity ω h, ω h, ω h, + e πih = ω h, ω h, ω h, ω h, ω h,. ω 4 h, ω h, = / e πih. By inserting 4.6 we find that the left-hand side is ω 4 h, ω h, = exp 4πi h h + 4 exp πi 4 6h h / + h h h/ h + / 6h h / + h h Now 4.7 follows by using the fact that h h mod 4 and is even. = + e πih +.

18 8 KATHRIN BRINGMANN AND KARL MAHLBURG Using 4.7, S + S simplifies to 4.8 S + S = 4 h, 6,= ω h, ω h, ω h, e πihn ωh, h, ϑ h, e πnz e π 9z q dφ, where 4.9 q := q q q / q q / q / q / fq q / O. Since e π 9z q /9 q /9 = O, the integrand in 4.8 is O as a whole, and is thus uniformly bounded over the sum. In analogy with the bounds for S 6 in Section 4., we can show that S + S = O. For the term S, we identify the portion that contributes exponential growth to the asymptotic expansion. Write 4.0 S = S e + S := h, 6,= ω h, ω h, ω h, ω h, e πihn ν mod ν e πih ν +ν h, ϑ h, e πnz q q / I,ν zdφ q / z e 5π 6z + h, 6,= ω h, ω h, ω h, ω h, e πihn ν mod ν e πih ν +ν h, ϑ h, e πnz z e 5π 6z I,ν zdφ. q Excluding the terms z and I,ν, the remaining portion of the integrand in S e 9/7 is O, and thus the integrand as a whole has magnitude bounded by J 9,,νz. As in 4.5, Lemma 6. part implies that S e = Olog N.

19 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 9 For S, we apply Lemma. part and the bounds of Section 4. to obtain 4. S = h, 6,= = h, 6,= ω h, ω h, ω h, ω h, ω h, ω h, ω h, ω h, h, 6,= e πihn e πihn ν mod ν mod ν e πih ν +ν ν e πih ν +ν h, ϑ h, h, ϑ h, e πnz J 5 6,,νzdΦ e πnz J 5,,νzdΦ + Olog N. 6 roposition. and the bounds from Section 4. give an asymptotic expansion for S in Bessel functions, S = 5π 6 ω h, ω h, ω h, e πihn 6n ν e πih ν +ν 4. ω h, x cosh πiν /6 πx 5 6 ν mod I π 0 n x dx + Olog N Estimation of. Once again, we combine the transformations.5,.9, and. to write = S + S + S := h, 6,= + h, 6,= + h, 6,= ω h, ω h, ω h, ω h, ω h, ω h, ω h, e πihn ω h, ω h, ω h, ω h, e πihn ω h, e πih 4 πihn h, ϑ h, ν mod h, ϑ h, h, ϑ h, q e πnz e π z ν e πih ν ν ze πnz + π 6z e πnz e π z q / q q / q ω q / q / q / q I,ν zdφ. q / dφ q / dφ

20 0 KATHRIN BRINGMANN AND KARL MAHLBURG Following Section 4., both S and S are both O. For S, we imitate 4.0, 4., and 4., using Lemma. and roposition. to find the asymptotic expansion 4. S = h, 6,= = π 6 6n ω h, ω h, ω h, e πihn ω h, h, 6,= ν mod ω h, ω h, ω h, e πihn ω h, ν e πih ν ν ν mod h, ϑ h, ν e πih ν ν e πnz J 6,,νzdΦ + Olog N x π I cosh πiν /6 πx n x dx + Olog N Estimation of. Finally,.6,.9, and. give = S + S + S := + h, 6,= h, 6,= + h, 6,= ω h, ω h, ω h, e πih 4 ω 6h, ω h, ω h, ω 6h, e πihn ωh, ω h, ω h, ω h, e πihn ω 6h, h, ϑ h, ν mod h, ϑ h, πihn h, ϑ h, q e πnz e π 8z ν e πih ν ν e πnz z e π 8z q / q e πnz e π 8z q / q /6 dφ q / q / I,ν zdφ. q /6 q / q /6 ω q / dφ The previous arguments and Lemma. part again imply that S = O. For S, we isolate the main term by writing q q / q /6 = + q q / q /6. q / q / As in 4.9, the total contribution of the second summand to S is O.

21 Thus we are left with S = ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES h, 6,= Using roposition. yields 4.4 S = π 6 n ω h, ω h, ω 6h, e πihn ωh, h, 6,= ω h, ω h, ω 6h, ω h, h, ϑ h, e πihn e πnz + π 8 z dφ + O. π I n + O. Finally, the asymptotic expansion for S also follows as before, and has the form 4.5 S = = h, 6,= π 8 6n ω h, ω h, ω h, e πihn ω 6h, h, 6,= ν mod ω h, ω h, ω h, e πihn ω 6h, ν e πih ν ν ν mod h, ϑ h, ν e πih ν ν e πnz J,,νzdΦ + Olog N 8 x π I n x dx + Olog N. cosh πiν /6 πx 6 Altogether, we have shown that p n = S + S + S + S + Olog N, and the formulas in finish the proof of Theorem.. 5. rincipal part integrals and the proof of Theorem. Using the approximations from Section, we still need to find the asymptotic expansion of the integrals I b,,ν n in order to find the asymptotic expansion for p n. As a first simplification, we will use the Bessel function asymptotic for x see 4..7 in [5] 5. I l x = ex πx + O This implies that 5. I b,,ν n = πbn /4 cosh x /4 πiν /6 πx b e x x. e π bn x + O dx. n x We will address asymptotics in a more general setting, so we first identify the ey properties of the I b,,ν.

22 KATHRIN BRINGMANN AND KARL MAHLBURG roposition 5.. Let gx := x /4 for a > 0, b R. Then hx = gx + g x is a coshai + bx monotonically decreasing real function for x [0, ]. roof. First, x is monotonically decreasing as x increases, so we need only address the denominators. Basic algebra gives the simplification coshai + bx + coshai bx + coshai + bx 5. = coshai bx coshai + bx coshai bx cosa coshbx = cosa + coshbx. Thus we need only show that is coshbx is monotonic. The derivative of this function cosa + coshbx b sinhbxcosa + coshbx coshbx b sinhbx. cosa + coshbx When x = 0, this is zero, and for x > 0, it is not difficult to show that the numerator is always negative. Thus the function is monotonically decreasing, as claimed. Note that hx as in the preceding proposition has a Taylor series in a neighborhood of x = 0. Furthermore, the function hx x is also monotonically decreasing in a neighborhood of 0, which will be helpful later in bounding error terms. To set further notation, let 5.4 I f,an := 0 fx e aπ n x dx. roposition 5.. Suppose that fx is a positive function defined on [0, ] that is bounded, differentiable, and monotonically decreasing, and that also has a Taylor series on some neighborhood of 0. If a > 0, then we have for all 0 < c < as n 8 If,an = a f0n roof. To finish the proof it is enough to show that 4 e πa n + O 5.5 I an + J an I f,an I an with and Ian := a f0n 4 e πa n J an n 4 c e πan. n 4 c e πa n

23 We first show the upper bound. We have ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES I f,an f0e πa n Using the upper bound x x gives I f,an f0e πa n f0e πan n 4 We next consider the lower bound. We have e πa n x dx. e πa nx dx = f0e πan n 4 I f,an = e πan n 4 R e πan n 4 e πax dx = n 4 0 n c 0 y f n 4 y f n 4 n 4 n a f0eπa n 4 e πa n y n dy e πa n for any 0 < c <. For x < 4, the Taylor series expansion x = n! n n!n! xn easily implies the bound 5.7 x x n n x n 0 y n x x4. The estimate 5.7 and the monotonicity of fx applied to 5.6 then give n If,an e πan n 4 f n c 4 e c πan4c 5.8 e πay dy. 0 e πax dx = I a n. We will now use the Taylor series expansion of fx to write fn c /4 = f0 E n, where E n = On c /4, and we also write the exponential function e πan4c / = E n, where E n = On 4c /. Note that both E and E are positive functions. Furthermore, we use the following bound for the error function: n c lugging in to 5.8, we obtain 5.9 e πay dy n c e πay πan dy = c πa e. I f,an e πan n 4 f0 E n E n dy a πa e πan c

24 4 KATHRIN BRINGMANN AND KARL MAHLBURG In other words, recalling 5.5 we have shown that for any 0 < c < 8. J an n 4 c e πa n Remar. Since f is uniformly bounded, roposition 5. could also be proven through a reformulation in terms of distributions. In particular, similar arguments as those used in the proof also imply that as N, 5.0 µ N := N 4 e aπ N x dx a δ 0, where δ 0 is the Dirac delta measure at x = 0. roof of Theorem.. We prove the expansion for p n by determining the main terms in Theorem., and estimating the remainder. First, consider the last sum. The contribution coming from the term = is given by π 6 n I π n = 4 n 4 e π n + O e π n by 5.. Next we estimate, again using 5., the terms coming from > up to a constant against e π n e π 5 n, n 4 5 N 0 h< which is exponentially smaller than the main term. The first and second sums in Theorem. are estimated in exactly the same way, and also contribute only to the error term. This leaves the third sum. Again we start with the term =, which by 5. can be estimated against 5. π 8 6n I 8,,n = 6 n 4 x /4 e π cosh πi πx 6 6 n 5 4 n x + O n x We first consider the main term in this expression. Using ropositions 5. and 5. this yields the bound 6n cosh e π n e π n + O = πi n +c 6 8 n e π n e π n + O. n +c The big-o term in 5. can similarly be estimated against n e π n. dx.

25 ASYMTOTICS FOR ARTITIONS WITHOUT SEQUENCES 5 hx x This follows by recalling that is initially decreasing, as well as the fact that 5. is a principal part integral. Finally we treat the > terms from the third sum of Theorem.. These may be estimated against up to a constant 5. n 5 <N 0 h< max ν I 8,,νn n 4 5 <N 0 h< max ν x /4 πiν cosh 6 πx 6 e π n x dx For a given, roposition 5. again bounds the integrals up to a constant that is uniform in ν by n π 4 e n n π 4 e 5 n. πiν cosh 6 This yields that 5. may be estimated against n 4 e π 5 n, which is again an exponential error, and thus the proof is complete. References [] G. Andrews, On the theorems of Watson and Dragonette for Ramanujan s moc theta functions, Amer. J. Math. 88 No. 966, [] G. Andrews, The theory of partitions, Cambridge University ress, Cambridge, 998. [] G. Andrews, artitions with short sequences and moc theta functions, roc. Nat. Acad. Sci. 0 No. 005, [4] G. Andrews, A survey of multipartitions: Congruences and identities, Surveys in Number Theory, K. Alladi, ed., Developments in Mathematics 7, pp. 9, Springer, 008. [5] G. Andrews, R. Asey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 7. Cambridge University ress, Cambridge, 999. [6] G. Andrews, H. Erisson, F. etrov, and D. Romi, Integrals, partitions and MacMahon s theorem, J. Comb. Theory A 4 007, [7] K. Bringmann, On the explicit construction of higher deformations of partition statistics, Due Math. J , 95. [8] K. Bringmann, Asymptotics for ran partition functions, Transactions of the AMS, accepted for publication. [9] K. Bringmann, F. Garvan and K. Mahlburg, artition statistics and quasiwea Maass forms, Int. Math. Res. Not. 008, rmn 4. [0] K. Bringmann and K. Ono, The fq moc theta function conjecture and partition rans, Invent. Math ,

26 6 KATHRIN BRINGMANN AND KARL MAHLBURG [] K. Bringmann and K. Ono, Coefficients of harmonic wea Maass forms, to appear in roc. of the 008 Univ. of Florida Conference on partitions, q-series, and modular forms. [] K. Bringmann and K. Ono, Dyson s rans and Maass forms, Annals of Math. 7 00, [] J. Bruinier and J. Fune, On two geometric theta lifts, Due Math. Journal 5 004, [4] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc , 6 5. [5] L. Dragonette, Some asymptotic formulae for the moc theta series of Ramanujan, Trans. Amer. Math. Soc. 7 95, [6] G.H. Hardy, Ramanujan, Cambridge University ress, 940. [7] G.H. Hardy and S. Ramanujan, Asmptotic formulae for the distribution of integers of various types, roc. London Math. Soc. Series, 6 98,. [8] A. Holroyd, T. Liggett and D. Romi, Integrals, partitions, and cellular automata, Trans. Amer. Math. Soc , [9] A. Ingham, Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc , [0] K. Ono, The Web of Modularity, CBMS Regional Conference Series in Mathematics 0 American Mathematical Society, 004. [] H. Rademacher, On the expansion of the partition function in a series, Ann. of Math. 44, 94, [] H. Rademacher and H. S. Zucerman, On the Fourier coefficients of certain modular forms of positive dimension, Ann. of Math. 9 No. 98, [] G. Watson, The final problem: An account of the moc theta functions, J. London Math. Soc. 96, [4] D. Zagier, Ramanujan moc theta functions and their applications [d aprés Zwegers and Bringmann-Ono] 006, Séminaire Bourbai, No [5] S. Zwegers, Moc theta-functions and real analytic modular forms, q-series with applications to combinatorics, number theory, and physics Urbana, IL, 000, 69 77, Contemp. Math. 9, Amer. Math. Soc., rovidence, RI, 00. [6] S. Zwegers, Moc theta functions, h.d. Thesis, Universiteit Utrecht, 00. Mathematical Institute, University of Cologne, Weyertal 86-90, 509 Cologne, Germany address: bringma@math.uni-oeln.de Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 09, U.S.A. address: mahlburg@math.mit.edu Department of Mathematics, rinceton University, rinceton, NJ 08544, U.S.A. address: mahlburg@math.princeton.edu