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On Dyson's Crank Conjecture and the Uniform Asymptotic

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On Dyson's Crank Conjecture and the Uniform Asymptotic 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 crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics. 1. Introduction and statement of results Dyson’s crank was introduced to explain Ramanujan’s famous partition congru- ences with modulus 5, 7, and 11. Denoting for n 2 N by p(n) the number of integer partitions of n, Ramanujan [22] proved that for n ≥ 0 p (5n + 4) ≡ 0 (mod 5); p (7n + 5) ≡ 0 (mod 7); p (11n + 6) ≡ 0 (mod 11): A key ingredient of his proof is the modularity of the partition generating function 1 1 X 1 q 24 P (q) := p(n)qn = = ; (q; q) η(τ) n=0 1 Qj−1 ` 2πiτ where for j 2 N0 [ f1g we set (a)j = (a; q)j := `=0(1 − aq ), q := e , and 1 1 1 24 Q n η(τ) := q n=1(1 − q ) is Dedekind’s η-function, a modular form of weight 2 . Ramanujan’s proof however gives little combinatorial insight into why the above congruences hold. In order to provide such an explanation, Dyson [8] famously intro- duced the rank 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 size when sorted by their ranks modulo 5 (resp. 7). This Date: July 7, 2014. 2010 Mathematics Subject Classification. 05A17, 11F03, 11F30, 11F50, 11P55, 11P82. 1 2 KATHRIN BRINGMANN AND JEHANNE DOUSSE conjecture was proven by Atkin and Swinnerton-Dyer [4]. Ono and the first author [7] showed that partitions with given rank satisfy also Ramanujan-type congruences. Dyson further postulated the existence of another statistic which he called the “crank” and which should explain all Ramanujan congruences. The crank was later found by Andrews and Garvan [1, 12]. If for a partition λ, o(λ) denotes the number of ones in λ, and µ(λ) is the number of parts strictly larger than o(λ), then the crank of λ is defined as largest part of λ if o(λ) = 0, crank(λ) := µ(λ) − o(λ) if o(λ) > 0. Denote by M(m; n) the number of partitions of n with crank m. Mahlburg [19] then proved that partitions with fixed crank also satisfy Ramanujan-type congruences. In this paper, we solve a longstanding conjecture by Dyson [9] concerning the limiting shape of the crank generating function. Conjecture 1.1 (Dyson). As n ! 1 we have 1 1 M (m; n) ∼ βsech2 βm p(n) 4 2 with β := pπ . 6n Dyson then asked the question about the precise range of m in which this asymp- totic holds and about the error term. In this paper, we answer all of these questions. p Theorem 1.2. The Dyson-Conjecture is true. To be more precise, if jmj ≤ p1 n log n, π 6 we have as n ! 1 β 2 βm 1 1 M(m; n) = sech p(n) 1 + O β 2 jmj 3 : (1.1) 4 2 Remarks. 1. 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. 2. We note that Theorem 1.2 is of a very different nature than known asymp- totics in the literature. For example, the partition function can be approxi- mated as p(n) = M(n) + O n−α ; ON DYSON’S CRANK CONJECTURE 3 where M(n) is the main term which is a sum of varying length of Kloosterman 3 sums and Bessel functions and α > 0. Rademacher [20] obtained α = 8 , 1 Lehmer improved this to 2 − " and Folsom and Masri [11] in their recent −δ 1 work obtained an impressive error of n for some absolute δ > 2 . Our result has a very different flavor due to the nonmodularity of the generating function and the bivariate asymptotics. 1 2 3. In fact we could replace the error by O(β 2 mα (m)) for any α(m) such that log n 1 p 1 = o (α(m)) for all jmj ≤ p n log n and βmα(m) ! 0 as n ! 1. n 4 π 6 − 1 Here we chose α(m) = jmj 3 to avoid complicated expressions in the proof. A straightforward calculation shows Corollary 1.3. Almost all partitions satisfy Dyson’s conjecture. To be more precise p n ] λ ` njcrank(λ)j ≤ p log n ∼ p(n): (1.2) π 6 Remarks. 1. We thank Karl Mahlburg for pointing out Corollary 1.3 to us. 2. We can improve (1.2) and give the size of the error term. Dyson’s conjecture follows from a more general result concerning the coefficients Mk(m; n) defined for k 2 N by 1 1 X X (q)2−k C (ζ; q) = M (m; n) ζmqn := 1 : k k (ζq) (ζ−1q) n=0 m=−∞ 1 1 Note that M(m; n) = M1(m; n). Denoting by pk(n) the number of partitions of n allowing k colors, we have. Theorem 1.4. For k fixed and jmj ≤ 1 log n, we have as n ! 1 6βk 1 βk 2 βkm 1 M (m; n) = sech p (n) 1 + O β 2 jmj 3 ; k 4 2 k k q k with βk := π 6n . Remarks. 4 KATHRIN BRINGMANN AND JEHANNE DOUSSE 1. We note that for k ≥ 3, the functions Ck(ζ; q) are well-known to be generating functions of Betti numbers of moduli spaces of Hilbert schemes on (k − 3)−point blow-ups of the projective plane [13] (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 Rodriguez-Villegas [16] also determined profiles of Betti numbers for other moduli spaces. 2. Note that our method of proof would allow determining further terms in the asymptotic expansion of Mk(m; n). 1 2 2 3. Again we could replace the error by O(βk mαk(m)) for any αk(m) such that log n 1 1 = o (αk(m)) for all jmj ≤ log n and βkmαk(m) ! 0 as n ! 1. (kn) 4 6βk 4. The function Ck can also be represented as a so-called Lerch sum. To be more precise, we have [1] n n(n+1) X m n 1 − ζ X (−1) q 2 M (m; n) ζ q = n : (1.3) (q)1 1 − ζq m2Z n2Z n≥0 This representation, which was a key representation in [6], is not used in this paper. 5. The special case k = 2 yields the birank of partitions [14]. 6. We expect that our methods also apply to show an analogue of (1.1) for the rank. The case of fixed m is considered in upcoming work by Byungchan Kim, Eunmi Kim, and Jeehyeon Seo [17]. This paper is organized as follows. In Section 2, we recall basic facts on modular and Jacobi forms which are the base components of Ck and collect properties on Euler polynomials. In Section 3, we determine the asymptotic behavior of Ck. In Section 4, we use Wright’s version of the Circle Method to finish the proof of Theorem 1.4. In Section 5, we illustrate Theorem 1.2 numerically. Acknowledgements The research of the first author was supported by the Alfried Krupp Prize 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 Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. Most of this research was conducted while the second author was ON DYSON’S CRANK CONJECTURE 5 visiting the University of Cologne funded by the Krupp foundation. The first author thanks Freeman Dyson and Jan Manschot for enlightening conversation. Moreover we thank Michael Mertens, Karl Mahlburg, and Larry Rolen for their comments on an earlier version of this paper. Moreover we thank the referee for valuable comments that improved the exposition of this paper. 2. Preliminaries 2.1. Modularity of the generating functions. A key ingredient of our asymptotic results is to employ the modularity of the functions Ck. To be more precise, we write (throughout q := e2πiτ , ζ := e2πiw with τ 2 H; w 2 C) 1 − 1 k 3−k i ζ 2 − ζ 2 q 24 η (τ) C (ζ; q) = ; (2.1) k # (w; τ) where 1 1 Y n η(τ) := q 24 (1 − q ) ; n=1 1 1 1 Y n n −1 n−1 # (w; τ) := iζ 2 q 8 (1 − q ) (1 − ζq ) 1 − ζ q : n=1 The function η is a modular from, whereas # is a Jacobi form. To be more precise, we have the following transformation laws (see e.g. [20]). Lemma 2.1. We have 1 p η − = −iτη(τ); τ w 1 p πiw2 # ; − = −i −iτe τ # (w; τ) : τ τ 2.2. Euler polynomials. Recall that the Euler polynomials may be defined by their generating function 1 2ext X tr =: E (x) : (2.2) et + 1 r r! r=0 The following lemma may easily be concluded by differentiating the generating func- tion (2.2).
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