ON DYSON S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS

Transcription

1 ON DYSON S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS KATHRIN BRINGMANN AND JEHANNE DOUSSE Abstract In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the cran generating function We fit this function in a more general family of inverse theta functions which play a ey role in physics Introduction and statement of results Dyson s cran was introduced to explain Ramanujan s famous partition congruences with modulus 5, 7, and Denoting for n N by pn the number of integer partitions of n, Ramanujan [] proved that for n 0 p 5n mod 5, p 7n mod 7, p n mod A ey ingredient of his proof is the modularity of the partition generating function P q := pnq n = = q 4 q; q ητ, n=0 where for j N 0 { } we set a j = a; q j := j l=0 aql, q := e πiτ, and ητ := q 4 n= qn is Dedeind s η-function, a modular form of weight Ramanujan s proof however gives little combinatorial insight into why the above congruences hold In order to provide such an explanation, Dyson [8] famously introduced the ran of a partition, which is defined as its largest part minus the number of its parts He conjectured that the partitions of 5n + 4 resp 7n + 5 form 5 resp 7 groups of equal sie when sorted by their rans modulo 5 resp 7 This Date: July 7, Mathematics Subject Classification 05A7, F0, F0, F50, P55, P8

2 KATHRIN BRINGMANN AND JEHANNE DOUSSE conjecture was proven by Atin and Swinnerton-Dyer [4] Ono and the first author [7] showed that partitions with given ran satisfy also Ramanujan-type congruences Dyson further postulated the existence of another statistic which he called the cran and which should explain all Ramanujan congruences The cran was later found by Andrews and Garvan [, ] If for a partition λ, oλ denotes the number of ones in λ, and µλ is the number of parts strictly larger than oλ, then the cran of λ is defined as { largest part of λ if oλ = 0, cranλ := µλ oλ if oλ > 0 Denote by Mm, n the number of partitions of n with cran m Mahlburg [9] then proved that partitions with fixed cran also satisfy Ramanujan-type congruences In this paper, we solve a longstanding conjecture by Dyson [9] concerning the limiting shape of the cran generating function Conjecture Dyson As n we have with β := π 6n M m, n 4 βsech βm pn Dyson then ased the question about the precise range of m in which this asymptotic holds and about the error term In this paper, we answer all of these questions Theorem The Dyson-Conjecture is true To be more precise, if m π 6 n log n, we have as n Mm, n = β βm 4 sech pn + O β m Remars For fixed m one can directly obtain asymptotic formulas since the generating function is the convolution of a modular form and a partial theta function [6] However, Dyson s conjecture is a bivariate asymptotic Indeed, this fact is the source of the difficulty of this problem We note that Theorem is of a very different nature than nown asymptotics in the literature For example, the partition function can be approximated as pn = Mn + O n α,

3 ON DYSON S CRANK CONJECTURE where Mn is the main term which is a sum of varying length of Kloosterman sums and Bessel functions and α > 0 Rademacher [0] obtained α =, 8 Lehmer improved this to ε and Folsom and Masri [] in their recent wor obtained an impressive error of n δ for some absolute δ > Our result has a very different flavor due to the nonmodularity of the generating function and the bivariate asymptotics In fact we could replace the error by Oβ mα m for any αm such that log n = o αm for all m n 4 π 6 n log n and βmαm 0 as n Here we chose αm = m to avoid complicated expressions in the proof A straightforward calculation shows Corollary Almost all partitions satisfy Dyson s conjecture To be more precise { } n λ n cranλ π 6 log n pn Remars We than Karl Mahlburg for pointing out Corollary to us We can improve and give the sie of the error term Dyson s conjecture follows from a more general result concerning the coefficients M m, n defined for N by C ζ; q = n=0 m= M m, n ζ m q n := q ζq ζ q Note that Mm, n = M m, n Denoting by p n the number of partitions of n allowing colors, we have Theorem 4 For fixed and m 6β log n, we have as n M m, n = β β m 4 sech p n + O β m, with β := π 6n Remars

4 4 KATHRIN BRINGMANN AND JEHANNE DOUSSE We note that for, the functions C ζ; q are well-nown to be generating functions of Betti numbers of moduli spaces of Hilbert schemes on point blow-ups of the projective plane [] see also [6] and references therein The results of this paper immediately gives the limiting profile of the Betti numbers for large second Chern class of the sheaves Recently, Hausel and Rodrigue-Villegas [6] also determined profiles of Betti numbers for other moduli spaces Note that our method of proof would allow determining further terms in the asymptotic expansion of M m, n Again we could replace the error by Oβ mα m for any α m such that log n = o α n m for all m 4 6β log n and β mα m 0 as n 4 The function C can also be represented as a so-called Lerch sum To be more precise, we have [] m Z n 0 M m, n ζ m q n = ζ q n Z n q nn+ ζq n This representation, which was a ey representation in [6], is not used in this paper 5 The special case = yields the biran of partitions [4] 6 We expect that our methods also apply to show an analogue of for the ran The case of fixed m is considered in upcoming wor by Byungchan Kim, Eunmi Kim, and Jeehyeon Seo [7] This paper is organied as follows In Section, we recall basic facts on modular and Jacobi forms which are the base components of C and collect properties on Euler polynomials In Section, we determine the asymptotic behavior of C In Section 4, we use Wright s version of the Circle Method to finish the proof of Theorem 4 In Section 5, we illustrate Theorem numerically Acnowledgements The research of the first author was supported by the Alfried Krupp Prie for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union s Seventh Framewor Programme FP/007-0 / ERC Grant agreement n 50 - AQSER Most of this research was conducted while the second author was

5 ON DYSON S CRANK CONJECTURE 5 visiting the University of Cologne funded by the Krupp foundation The first author thans Freeman Dyson and Jan Manschot for enlightening conversation Moreover we than Michael Mertens, Karl Mahlburg, and Larry Rolen for their comments on an earlier version of this paper Moreover we than the referee for valuable comments that improved the exposition of this paper Preliminaries Modularity of the generating functions A ey ingredient of our asymptotic results is to employ the modularity of the functions C To be more precise, we write throughout q := e πiτ, ζ := e πiw with τ H, w C i ζ ζ q 4 η τ C ζ; q =, ϑ w; τ where ϑ w; τ := iζ q 8 ητ := q 4 q n, n= q n ζq n ζ q n n= The function η is a modular from, whereas ϑ is a Jacobi form To be more precise, we have the following transformation laws see eg [0] Lemma We have η = iτητ, τ w ϑ τ ; = i iτe πiw τ ϑ w; τ τ Euler polynomials Recall that the Euler polynomials may be defined by their generating function e xt e t + =: E r x tr r! r=0 The following lemma may easily be concluded by differentiating the generating function For the readers convenience we give a proof

6 6 KATHRIN BRINGMANN AND JEHANNE DOUSSE Lemma We have Proof We have Now r=0 t sech = E r+ 0 tr r! = d dt r=0 r=0 E r+ 0 tr r! t r+ E r+ 0 r +! r=0 e t + = E r 0 tr r!, e t + = E r 0 tr r! Taing the difference gives the claim of the lemma since d dt e t + = t e t + sech r=0 We also require an integral representation of Euler polynomials To be more precise, setting for j N 0 w j+ Ej := dw, 0 sinhπw we obtain Lemma We have E j = j+ E j+ 0 Proof We mae the change of variables w w+ i and then use the Residue Theorem to shift the path of integration bac to the real line Using the Binomial Theorem, we may thus write E j = i w + i j+ coshπw dw = i j+ j+ l j + i w l l coshπw dw R l=0 R

7 ON DYSON S CRANK CONJECTURE 7 The last integral is nown to equal i l E l, where E l := l E l denotes the lth Euler number see page 4 of [0] The claim now follows using the well-nown identity see page 4 of [0] j j E j x = x j l E l l l l=0 Asymptotic behavior of the function C Since M m, n = M m, n we from now on assume that m 0 The goal of this section is to study the asymptotic behavior of the generating function of M m, n We define where C m, q := M m, n q n = n=0 = q 4 η τ 0 g w; τ cosπmwdw, i ζ ζ η τ g w; τ := ϑ w; τ C e πiw ; q e πimw dw Here we used that g w; τ = gw; τ In this section we determine the asymptotic behavior of C m, q, when q is near an essential singularity on the unit circle It turns out that the dominant pole lies at q = Throughout the rest of the paper let τ = i π, = β + ixm with x R satisfying x πm β Bounds near the dominant pole In this section we consider the range x We start by determining the asymptotic main term of g Lemma and the definition of ϑ and η immediately imply Lemma For 0 w we have for x as n g w; i π sinπw = π sinh e π w π + O e 4π wre w

8 8 KATHRIN BRINGMANN AND JEHANNE DOUSSE In view of Lemma it is therefore natural to define Thus G m, := 4π G m, := 0 0 sinπw sinh π w g w; i π e π w cosπmwdw, π sinπw sinh π w e π w cosπmwdw The dominant contribution comes from G m, C m, q = q 4 η τ G m, + G m, Lemma Assume that x and m 6β log n Then we have as n G m, = β m 4 sech + O β m sech β m Proof Inserting the Taylor expansion of sin, exp, and cos, we get sinπwe π w cosπmw = j+ν π j +!ν!r! πj+ πm ν r w j+ν+r+ j,ν,r 0 This yields that G m, = 4π j,ν,r 0 j+ν j +!ν!r! πj+ πm ν π r I j+ν+r where for l N 0 we define I l := 0 w l+ sinh π w dw We next relate I l to E l defined in For this, we note that I l = 0 w l+ sinh dw I π w l

9 with I l := w l+ sinh π w ON DYSON S CRANK CONJECTURE 9 dw w l+ e π wre dw Re l Γ l + ; π Re Here Γα; x := x e w w α dw denotes the incomplete gamma function and throughout gx fx means that gx = O fx Using that as x Γ l; x x l e x thus yields that I l Re e π Re e π Re In the first summand in we mae the change of variables w w and then shift π the path of integration bac to the real line by the Residue Theorem Thus we obtain that w l+ l+ l+ 0 sinh l+ E l+ 0 dw = El =, π w π π where for the last equality we used Lemma Thus G m, = = ν=0 j,ν,r 0 m ν ν! r+ j+r+ j +!ν!r! mν j+ν+r+ E j+ν+r+ 0 + O j ν r e π Re E ν+0 + O = 4 sech m where for the last equality we used Lemma approximate sech m and coshm We have coshm = cosh β m + iβ m x = coshβ m cos β m x = coshβ m + O β m + O coshm, To finish the proof we have to + i sinhβ m sin β m x

10 0 KATHRIN BRINGMANN AND JEHANNE DOUSSE This implies that m sech = yielding Thus we obtain cosh = m cosh β m + O β m m sech = sech β m + O β m + O β m + O β +, coshβ m G m, = β m 4 sech m = β m 4 sech + O β m sech β m + O β coshβ m We may now easily finish the proof distinguishing the cases on whether β m is bounded or goes to We next turn to bounding G m, Lemma Assume that x Then we have as n G m, q e 5π 4β β Proof By Lemma we obtain that G m, sinπw Re w +w 0 e 4π w eπ dw It is not hard to see that sinπw e 4π w Moreover, Re = β + m x β, β

11 ON DYSON S CRANK CONJECTURE The claim now follows, using that the maximum of w + w on [0, ] is obtained for w = Combining the above yields Proposition 4 Assume that x Then we have as n C m, q = + sech β m e π 4π 6 + O β + m sech β m e π n 6 Proof Recall from that Lemma easily gives that C m, q = q 4 η τ G m, + G m, q 4 η τ = e π 6 + Oβ π The functions G m, and G m, are now approximated using Lemma and Lemma, respectively It is not hard to see that the main error term arises from approximation G m, We thus obtain C m, q = + 4π e π 6 sech The claim follows now using that β m β, Re = β + O β m sech 6n π β m e π Re 6 Bounds away from the dominant pole We next investigate the behavior of C m, away from the dominant cusp q = To be more precise, we consider the range x πm β as in [] Let us start with the following lemma, which proof uses the same idea

12 KATHRIN BRINGMANN AND JEHANNE DOUSSE Lemma 5 Assume that τ = u + iv H with Mv u for u > 0 and v 0, we have that P q [ π v exp v ] π + M Proof We rewrite logp q = log q n = n= n= m= q nm m = m= q m m q m Therefore we may estimate q m logp q m q m q q q q + q m m q m m= m= = logp q q q q We now split u into ranges If Mv u, then we have cosπu cosπmv 4 Therefore q = e πv cosπu + e 4πv e πv cosπmv + e 4πv Taylor expanding around v = 0 we find that For 4 u Hence, for all Mv u, Furthermore we have By Lemma, we have q πv + M + O v 4 we have cosπu 0 Therefore q > πv + M q πv + M + O v 5 q = e πv = πv + O v 6 P q = e πv 4 ηiv = ve π v + Ov

13 Thus ON DYSON S CRANK CONJECTURE logp q = π v + logv + Ov 7 Combining 5, 6, and 7, we obtain logp q π v + logv + Ov + O πv + M = π v + logv + O π + M Exponentiating yields the desired result We are now able to bound C m, q away from q = Proposition 6 Assume that x πm β Proof We have by definition C m, q n 4 exp Then we have, as n, n 6n π 6 8π m Note that by C m, q = P q gw; τ cosπmwdw 0 g w; τ = + ζ n We thus may bound g w; τ n n q n +n + ζ ζq n q n +n q n q n n n q n +n ζ q n e β n β n 4 Thus C m, q P q n 4

14 4 KATHRIN BRINGMANN AND JEHANNE DOUSSE Using Lemma 5 with v = β, u = β m x, and M = m, yields for x π π πm β, Therefore [ P q n π π 4 exp β π [ C m, q n π 4 exp n 4 exp n 4 exp β π π 6n [ n π 6 π n 6n π 6 8π + m ] + m ] ] + m m 4 The Circle Method In this section we use Wright s variant of the Circle Method and complete the proof of Theorem 4 and thus the proof of Dyson s conjecture We start by using Cauchy s Theorem to express M as an integral of its generating function C m, : M m, n = πi C C m, q q n+ dq, 4 where the contour is the counterclocwise transversal of the circle C := {q C ; q = e β } Recall that = β + ixm Changing variables we may write M m, n = β C πm m, e e n dx We split this integral into two pieces x πm β M m, n = M + E

15 with ON DYSON S CRANK CONJECTURE 5 M := E := β πm β πm x x πm β C m, e e n dx, C m, e e n dx In the following we show that M contributes to the asymptotic main term whereas E is part of the error term 4 Approximating the main term The goal of this section is to determine the asymptotic behavior of M We show Proposition 4 We have M = β 4 sech β m m p n + O n 4 A ey step for proving this proposition is the investigation of P s, := πi +im im v s e π n 6 v+ v dv for s > 0 These integrals may be related to Bessel functions Denoting by I s the usual I-Bessel function of order s, we have Lemma 4 As n n n P s, = I s π + O exp π m Proof We use the following loop integral representation for the I-Bessel function [4] x > 0 I l x = t l e xt+ t dt, 4 πi Γ where the contour Γ starts in the lower half plane at, surrounds the origin counterclocwise and then returns to in the upper half-plane We choose for Γ

16 6 KATHRIN BRINGMANN AND JEHANNE DOUSSE the piecewise linear path that consists of the line segments γ 4 : i, i, γ m m : i γ : i, i, γ : i, + i m m m m m, i, which are then followed by the corresponding mirror images γ, γ, and γ 4 Note that P s, = γ Thus, to finish the proof, we have to bound the integrals along γ 4, γ, and γ the corresponding mirror images follow in the same way First γ γ 4 n exp π 6 e π n 6 t t + im using Next m n exp π 6 Finally γ t im s dt + t s e π n 6 t dt n s+ Γ s + ; π exp where we used that t + proof exp π π n t 6 t rm + n m t t + + m t im m t im, s dt n n e π n 6, 6 + im t s dt e π n 6 t t im t + m s dt, obtains its maximum at t = This finishes the

17 We now turn to the proof of Proposition 4 ON DYSON S CRANK CONJECTURE 7 Proof of Proposition 4 Using Proposition 4 and maing a change of variables, we obtain by Lemma 4 M = + O β + 4π exp π sech β m P +, + O β + m sech n 6 = β + 4π + sech β m + m I π β m n + O β + m sech Using the Bessel function asymptotic see 47 in [] I l x = ex e x + O πx yields M = β + n sech β m eπ 4π π n n + O exp π m 4 + O x e π n n 4 + O β + m sech e π n β m β m It is not hard to see that the last error term is the dominant one Thus e π n M = Using that [5, ] β + 4π p n = now easily gives the claim sech β m π n 4 + O m n n + 4 e π n + O n e π n e π n

18 8 KATHRIN BRINGMANN AND JEHANNE DOUSSE 4 The error arc We finally bound E and show that it is exponentially smaller than M The following proposition then immediately implies Theorem 4 Proposition 4 As n E n n 6n 4 exp π 8π m Proof Using Proposition 6, we may bound E β n n 6n m x πm 4 exp π 6 8π β n n 6n 4 exp π 8π m m e β n dx 5 Numerical data We illustrate our results in tables n M 0, n M 0, n M0,n M0,n n M, n M, n M,n M,n where we set Mm, n := β 4 sech βm pn References [] G Andrews and F Garvan, Dyson s cran of a partition, Bull Amer Math Soc 8 988, 67 7 [] A Atin and H Swinnerton-Dyer, Some properties of partitions, Proc Lond Math Soc 4 954, 84 06

19 ON DYSON S CRANK CONJECTURE 9 [] G Andrews, R Asey, and RRoy, Special functions, Encyclopedia Math Appl 7, Cambridge University Press, Cambridge 999 [4] G Aren, Modified Bessel functions, Mathematical Methods for Physicists, rd ed, Orlando, FL: Academic Press 985, [5] K Bringmann, K Mahlburg, and R Rhoades, Asymptotics for ran and cran moments, Bull Lond Math Soc 4 0, [6] K Bringmann and J Manschot, Asymptotic formulas for coefficients of inverse theta functions, submitted for publication [7] K Bringmann and K Ono, Dyson s rans and Maass forms, Ann of Math 7 00, [8] F Dyson, Some guesses in the theory of partitions, Eurea Cambridge 8 944, 0 5 [9] F Dyson, Mappings and symmetries of partitions, J Combin Theory, Ser A 5 989, [0] A Erdélyi, W Magnus, F Oberhettinger, and F Tricomi, Higher transcendental functions, Vol I, Robert E Krieger Publishing Co, Inc, Melbourne, Fla 98 [] A Folsom and R Masri, Equidistribution of Heegner points and the partition function, Math Ann 48 0, [] F Garvan, New combinatorial interpretations of Ramanujan s partition congruences mod 5, 7, and, Trans Amer Math Soc , [] L Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math Ann , 9 07 [4] P Hammond and R Lewis, Congruences in ordered pairs of partitions, Int J Math , [5] G Hardy and S Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc Lond Math Soc 6 98, [6] T Hausel and Rodrigue-Villegas Cohomology of large semiprojective hyperaehler varieties, preprint [7] B Kim, E Kim, and J Seo, Asymptotics for q-expansions involving partial theta functions, in preparation [8] D Lehmer, On the remainders and convergence of the series for the partition function, Trans Amer Math Soc 46 99, 6 7 [9] K Mahlburg, Partition congruences and the Andrews-Garvan-Dyson cran, Proc Natl Acad Sci 0 005, [0] H Rademacher, Topics in analytic number theory, Die Grundlagen der math Wiss, Band 69, Springer Verlag, Berlin 97 [] H Rademacher and H Zucerman, On the Fourier coeffcients of certain modular forms of positive dimension, Ann of Math 9 98, 4 46 [] S Ramanujan, Congruence properties of partitions, Math Z 9 9, 47 5 [] E Wright, Asymptotic partition formulae II Weighted partitions, Proc Lond Math Soc 6 9, 7 4 [4] E Wright, Stacs II, Q J Math 97, 07 6

20 0 KATHRIN BRINGMANN AND JEHANNE DOUSSE Mathematical Institute, University of Cologne, Weyertal 86-90, 509 Cologne, Germany address: LIAFA, Universite Denis Diderot - Paris 7, 7505 Paris Cedex, FRANCE address: jehannedousse@liafauniv-paris-diderotfr