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INTEGER PARTITIONS, PROBABILITIES AND QUANTUM MODULAR FORMS HIEU T. NGO AND ROBERT C. RHOADES Abstract. What is the probability that the smallest part of a random integer partition is odd? What is the expected value of the smallest part of a random integer partition? It is straightforward to see that the answers to these questions are both 1, to leading order. We show that the precise asymptotic expansion of each answer is dictated by special values of an L-function and by the asymptotic expansions of a quantum modular form. This paper relates partition probability questions to five interesting quantum modular forms studied by Zagier [83]. Additionally, this work contains new generating function identities for the partition functions and general circle method asymptotics which may be of independent interest.

1. Background Let P(N ) denote the set of all integer partitions of N and write p(N ) = |P(N )| for the number of integer partitions of N . The analytic study of partition statistics started with Hardy–Ramanujan asymptotic formula [49], further strengthened by Rademacher’s exact formula [72]. Hardy–Ramanujan formula gives for any positive integer R the asymptotic  √  ! R−1   exp K N X r R √ pr N − 2 + O N − 2 p(N ) = 4 3N r=0 where K =

2π √ 6

and

s r  X (s + 23 , r − s) −K 2 pr = (−2K) 24 s! s=0
Γ ν + m + 12 (4ν 2 − 1)(4ν 2 − 32 ) . . . (4ν 2 − (2m − 1)2 )
(ν, m) = = m! 4m m! Γ ν − m + 21 −r

for ν ∈ C, m ∈ Z≥0 . For instance, p0 = 1 and p1 = − K1 + p0 = 1 X ps = (−1)j j≥1

X

K . 48

For future reference, set

pr1 · · · prj .

r1 +···rj =s,ri ≥1

It is natural to pose questions about asymptotics of the number of partitions with prescribed properties. An interesting one is: What is the probability that the smallest part of an integer partition is odd? 2000 Mathematics Subject Classification. 05A16, 05A17, 11P81, 11P82, 11P84, 60C05. 1

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HIEU T. NGO AND ROBERT C. RHOADES

The paper establishes a precise asymptotic answer to this question and a collection of similar ones. Theorem 1.1 (probability that the smallest part of an unrestricted partition is odd). Let sm(λ) be the smallest part of the partition λ and set plo (N ) = |{λ ∈ P(N ) : sm(λ) ≡ 1 (mod 2)}| . For any positive integer R, one has  √  exp K N plo (N ) = 1 4 32 N

R−1 X

! Ar N

− r2

+ O(N

−R 2

)

r=0

as N → ∞, where   r X σs (s + 23 , r − s) −K 2 s 1 Ar = 2(−2K)r s=0 s! 24 and where σn is given by (1.5). If PN is the uniform measure on P(N ), then R−1

r

XX r R plo (N ) As pr−s N − 2 + O(N − 2 ). PN (sm(λ) ≡ 1 (mod 2)) = =1+ p(N ) r=1 s=0 The numbers σn arise from the asymptotic expansion of a quantum modular form and are related to special values of a Hecke L-function. Moreover, we show that the asymptotic expansions of many of the most famous quantum modular forms are intimately related to the asymptotics of interesting partition functions. For instance, two of Zagier’s original four examples [83] and an example of Bettin and Conrey [13] arise in our six main theorems. Section 1.2 introduces the quantum modular forms which are needed in the present work. Sect. 1.3 contains four theorems similar to Theorem 1.1, answering the questions: (1) What is the expected value of the smallest part of a random integer partition? (2) What is the expected value of the smallest part of a random integer partition with distinct part sizes? (3) What is the expected value of the largest part of a random integer partition? (4) What is the expected value of the largest part of a random integer partition with distinct part sizes? Section 1.4 contains identities for the generating functions of the relevant counting functions. These identities involve the quantum modular forms introduced in Section 1.2 and are proven in Sect. 2. Section 1.5 contains three general circle method propositions, which are generalizations of Wright’s method [81] and which may be of independent interest. These propositions involve standard facts about Bessel functions gathered in Sect. 3.1 and are proven in Sects. 3.2 and 3.3. To apply the circle method propositions, we need to compute asymptotic expansions near roots of unity of various modular forms and quantum modular forms, and we carry out these computations in Sect. 4. Finally, in Sect. 5 we combine the generating function identities in Section 1.4, the circle method propositions in Sect. 1.5, and the asymptotic expansions in Sect. 4 to derive the combinatorial probability theorems stated in Sect. 1.3.

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

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1.1. A history of partition statistics. There is a long history of the study of partition statistics and questions like those considered in this paper. Perhaps the first systematic study of statistical questions on integer partitions goes back to Erdős and his coauthors, especially Lehner [34] and Turán [35, 36, 37, 38, 39, 40, 41]. Additional results on the limiting distribution of partition statistics were found by Fristedt [44] and Pittel [70, 71]. While those results, and the results of the present paper, deal with the uniform measure on partitions of n, there has also been substantial interest in random partitions with respect to the Plancherel measure. See the works of Vershik and Kerov [76, 77], as well as the work of Oukonkov [68] and the references therein. The work of Erdős, Lehner, Pittel, Turán, Vershik, Kerov, and Oukonkov is largely motivated by the connection between representations of the symmetric groups and integer partitions, see, for instance, the book of MacDonald [63]. There have been many recent works concerned with the rank and crank statistics of integer partitions. These statistics were found by Dyson [32] and Andrews and Garvan [45, 46, 9] and are motivated by Ramanujan’s congruences for the partition function p(N ). Similar to the present work, these statistics have gained a lot of attention because of their connections to exotic modular objects, such as quasimodular forms and mock modular forms [23, 24, 67, 73]. Significant results about the distribution of these statistics appear notably in the works of Bringmann and Mahlburg and their coauthors [15, 21, 31, 30, 69]. There remain many more works dealing with statistics of integer partitions and other probabilistic questions about integer partitions. Notable in those are the works dealing with runs and gaps in parts making up a partition [17, 18, 22, 20, 48, 51, 57, 78, 79]. This paper demonstrates how quantum modular forms and their asymptotic expansions, which involve values of modular L-functions, arise in the resolution of natural integer partition probability problems.

1.2. Quantum modular forms. Zagier [83] introduced the notion of a quantum modular form based on examples from number theory, combinatorics, and quantum invariants of 3manifolds. Let k ∈ 21 Z, Γ be a finite index subgroup of SL2 (Z), and χ : Γ → C× be a finiteorder character. A quantum modular form of weight k for Γ is a function f : P1 (Q) \ S → C,   a b where S is a finite subset of P1 (Q), such that, for every γ = ∈ Γ, the period c d function

−k

hγ (x) := χ(γ)f (x) − (cx + d) f



ax + b cx + d



extends to a function which is C ∞ or real analytic at all but a finite set of points on P1 (R). Furthermore, one usually finds an analytic function fC (z) on C\R whose limit as z approaches x ∈ Q \ S coincides with f .

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HIEU T. NGO AND ROBERT C. RHOADES

Many of the examples of quantum modular forms have representations as q-hypergeometric series. Two such series are ∞ X (1.1) σ(q) = 1 + (−1)n q n+1 (1 − q)(1 − q 2 ) · · · (1 − q n ) n=0

(1.2)

= 1 + q − q 2 + 2q 3 − 2q 4 + q 5 + q 7 − 2q 8 + · · · , ∞ X ∗ σ (q) = −2 q n+1 (1 − q 2 )(1 − q 4 ) · · · (1 − q 2n ) n=0

= −2q − 2q 2 − 2q 3 + 2q 7 + 2q 8 + 2q 10 + · · · . With q = e2πiz = e−t , define ( 2πiz 1 e 24 σ(e2πiz ) = q 24 σ(q) z ∈H∪Q f (z) = 1 2πiz −2πiz ∗ −1 ) = −q 24 σ (q ) z ∈ H− ∪ Q. −e 24 σ(e Using the results of Andrews et al. [7], Cohen [29], and Lewis and Zagier [60], Zagier [83] showed that   2πi z h(z) := (2z + 1)f − e 24 f (z) 2z + 1
it is a function which is holomorphic on all of C \ R≤0 . Moreover, the function t 7→ f 2π has an asymptotic expansion near every value t ∈ Q. For instance, near t = 0  n ∞ X t t σn t − 24 −t − 24 ∗ t e σ(e ) = −e σ (e ) ∼ n! 24 n=0    2  3  4 t 5855 t 1538066 t 718108610 t = 2 − 50 (1.3) + − + − ··· 24 2 24 3! 24 4! 24 where the σn are integers given in terms of a Hecke L-function. To describe σn , let T (n) be the number of inequivalent solutions to (1.4)

u2 − 6v 2 = n

such that u + 3v ≡ ±1 (mod 12) minus the number of such solutions with u + 3v√≡ ±5 0 0 0 0 (mod 12). √ rHere two √ solutions (u, v) and (u , v ) to (1.4) are equivalent if (u + v 6) = ±(5 + 2 6) (u + v 6) for some integer r; otherwise they are said to be inequivalent. Define the Dirichlet series ∞ X T (n) D(s) = (Re (s) > 1). ns n=1 Then D(s) is a Hecke L-function and has analytic continuation to the complex s-plane. Set (1.5)

σn := (−1)n D(−n).

See [16] for more on the values D(−n) including their asymptotic properties. Thus Theorem 1.1 shows the asymptotic expansion of this quantum modular form arises in the probability theory of partitions.

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Our work requires two more quantum modular forms. Zagier [82, 83] showed that Kontsevich’s “strange” function F (q) :=

(1.6)

∞ X

(1 − q)(1 − q 2 ) · · · (1 − q n )

n=0

fits into the framework of a quantum modular form. The q-series (1.6) is defined only at roots of unity q. Combining the results of [82] with the results of Bryson, Ono, Pitman and the second author [26] it is possible to construct a function on (C \ R) ∪ Q analogous to f above. Zagier [82] proved  n    2  3 ∞ X t t 1681 t 257543 t Tn t − 24 −t (1.7) e F (e ) ∼ = 1 + 23 + + + ··· n! 24 24 2 24 3! 24 n=0 The Tn , called Glaisher’s numbers, are special values of the Dirichlet L-series L(χ12 , s) where
χ12 = 12· is a quadratic character of conductor 12. More precisely √  n 6 3 12 (1.8) Tn = 2 (2n + 1)! L(χ12 , 2n + 2). π π2 Finally, the third quantum modular form that plays a role in our work is S0 (z) =

∞ X n=1

∞

X qn d(n)q n , with q = e2πiz . = 1 − qn n=1

Bettin and Conrey [13] showed that     4 1 1 1 1 S0 (z) − S0 − = 1− − ψ0 (z) z z 4 z 4 where ψ0 is the period function of an Eisenstein series. Moreover, they established the asymptotic ∞ 2 2n−1 log(t) + γ 1 X B2n t S0 (it/2π) ∼ − + + (t → 0) t 4 n=1 (2n)!(2n) where γ = 0.5772156 . . . is Euler’s constant [58], Bn is the nth Bernoulli number, and B2n =

(−1)n+1 2(2n)! ζ(2n) (2π)2n

where ζ(s) is the Riemann zeta function. See Sect. 4.2 for further discussion on S0 (z). In the remainder we abuse notation and write S0 (q) = S0 (z). For future reference we define the coefficients bn by ! ∞ ∞ 2 2n−1 t γ 1 X B2n t 1X n (1.9) e 24 − + =: bn t . t 4 n=1 (2n)! (2n) t n=0 For instance, b0 = γ, b1 =

γ 24

− 41 , and b2 =

γ 1152

−

1 . 288

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HIEU T. NGO AND ROBERT C. RHOADES

1.3. Asymptotic theorems. This section contains five theorems, each of which relates the asymptotic expansion of a particular partition probability function to the asymptotic expansion of a quantum modular form. Let D(N ) denote the set of all integer partitions of N into distinct parts; set q(N ) = |D(N )|. It follows from [49] or [50] (see also Proposition 1.8) that for any R ≥ 0  √  ! R−1   exp K 0 N X r R q(N ) = qr N − 2 + O N − 2 3 1 4 34 N 4 r=0 where K 0 =

π 1

32

and 0 −r

qr = (−2K )

3 In particular, q0 = 1 and q1 = − 8K 0 −

s r  X K 02 (s + 1, r − s) . 24 s! s=0

K0 . 48

For future reference, define

q0 = 1, X qs = (−1)j

X

qr1 · · · qrj .

r1 +···rj =s,ri ≥1

j≥1

Theorem 1.2 (Expected value of the smallest part in a distinct part partition). If R is a positive integer, then  √  ! R−1 exp K 0 N X X R r esd (N ) := sm(λ) = Br N − 2 + O(N − 2 ) 1 3 4 4 43 N r=0 λ∈D(N ) as N → ∞, where 0 −r

Br = (−2K )

s r  X K 02 σs (s + 1, r − s) . 24 s! s=0

Consequentially, the expected value of the smallest part of a partition with respect to the uniform measure on the distinct part partitions of N is Ed,N (sm(λ)) = 2 +

R−1 X r X

r

R

Bs qr−s N − 2 + O(N − 2 ).

r=1 s=0

Remark. Fristedt [44, Section 9] proved that the probability that j is the smallest of a P part −j −j random distinct part partition is 2 . Consequentially, the expected value is j j2 = 2. Note in our asymptotic that B0 = 2. The following theorem is offered for contrast to the previous theorem. Let `(λ) be the number of parts in λ. Theorem 1.3. One has X λ∈D(N )

(−1)`(λ)+1 sm(λ) =

X d|N

1.

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

7

Consequentially, as N → ∞ Ed,N (sm(λ) : `(λ) ≡ 1 (mod 2))

 3  √  = Ed,N (sm(λ) : `(λ) ≡ 0 (mod 2)) + o N 4 + exp −K 0 N . P Remark. It is a classical fact that d|N 1 = O (N  ) for any  > 0 (see [64, page 56]). Theorem 1.4 (Expected value of the smallest part of an unrestricted partition). If R is a positive integer, then  √  ! R−1 exp K N X X r R Dr N − 2 + O(N − 2 ) esm (N ) := sm(λ) = 1 2N 4 3 r=0 λ∈P(N ) as N → ∞, where 0 −r

Dr = (−2K )

s r  X Ts (s + 1, r − s) −K 2 24 s! s=0

and Ts is given by (1.8). Consequentially, the expected value of the largest part of an unrestricted partition of N , with respect to the uniform measure, is EN (sm(λ)) = 1 +

R−1 X r X

r

R

Ds pr−s N − 2 + O(N − 2 ).

r=1 s=0

Remark. Again, by [44] a random partition of N almost surely has a part of size 1 and so EN (sm(λ)) ∼ 1. The above theorems concern the smallest part of a partition. We now turn our attention to the largest part of a partition. Theorem 1.5 (Expected value of the largest part of an unrestricted partition). As N → ∞ √   X exp(K N ) K − 21 elg (N ) := lg(λ) = log N + 2γ − 2 log + O(N log N ) 1 1 2 2N 2 4π 2 λ∈P(N ) where γ = 0.5772156 . . . is Euler’s constant. Consequentially, the expected value with respect to the uniform measure on the unrestricted partitions of N is √ √ ! √ 6N 6N 6N EN (lg(λ)) = log + γ + O (log(N )) . π π π Remark. Erdős and Lehner showed in [34] that lim PN

N →∞

π √ lg(λ) − log 6N

! √ ! 6N −v < v = e−e . π

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HIEU T. NGO AND ROBERT C. RHOADES

From this it is straightforward to deduce √ 6N EN (lg(λ)) ∼ log π

√

6N π

!

√ 6N + γ. π

Curiously, in the above the γ occurs from the asymptotic expansion of theR Riemann zeta −v ∞ function ζ(s) near s = 1. It might be interesting to compare with the integral −∞ ve−e −v dv which occurs in the limiting distribution of Erdős and Lehner. Remark. Theorem 1.5, as stated, was previously established by Kessler and Livingston [53] with a slightly worse error term and also by Grabner et al. [48]. The present method makes the structure of the full asymptotic expansion apparent and demonstrates the role of the quantum modular form S0 (z) in determining the expansion. The next theorem has the full asymptotic expansion for the case of partitions with distinct parts. Theorem 1.6 (Expected value of the largest part of a distinct part partition). For any R ≥ 0 as N → ∞ ! √ R−1 X R r exp(K 0 N ) X 00 eld (N ) := lg(λ) = (Fr + Fr0 + Fr log N )N − 2 + O(N − 2 log N ) 1 − 41 4 4π 3 N r=0 λ∈D(N ) where Fr00

0 −r

= 2(−2K )

r X

(−K 02 )s (s, r − s) bs ,

s=0

Fr0

0 −r

= (−2K )

r  X s=0

−K 02 24

s

r−1

1

2s−r a∗r−s−1 (s) (−2) log(K 0 /2) (s, r − s) X (8π) 2 (K 0 ) − , s s! s! 96 s=0

s r  X −K 02 (s, r − s) 0 −r , Fr = (−2K ) 24 s! s=0 where bs is defined in (1.9) in terms of Bernoulli numbers and a∗s is defined in Proposition 1.9 in terms of values of the Gamma function. Consequentially, the expected value of the largest part of a partition into distinct parts of N , with respect to the uniform measure, is  √ n √  Ed,N (lg(λ)) = 2K 0 N log 2K 0 N + γ    0 2   0  √  1 (K ) 1 K 1 13K 0 1 0 +√ + log 2K N + γ + − + 96 16 48 8K 0 96 2K 0 N  o log N +O . N

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

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1.4. Generating function identities. This section discusses the generating functions for the partition problems above. Euler [6] was the first to show the generating function identities P (q) := Q(q) :=

∞ X n=0 ∞ X

n

p(n)q = n

q(n)q =

n=0

∞ Y

1 1 = n 1−q (q; q)∞ n=1 ∞ Y

(1 + q n ) = (−q; q)∞

n=1

Q j where (a; q)n := j=0 (1 − aq j ) and (a; q)∞ := ∞ j=0 (1 − aq ). The following theorem gives results for the generating functions studied in our main theorems. Qn−1

Theorem 1.7. In the notation of Theorems 1.1, 1.2, 1.4, 1.5, and 1.6 ∞ ∞ X X X |λ| N (−1)n ((q; q)n − (q; q)∞ ) q = P (q) · plo (N )q = N =0

n=0

λ∈P sm(λ)≡1 (mod 2)

1 1 = P (q) · σ(q) − 2 2

(1.10) (1.11)

∞ X

esd (N )q N =

N =0

(1.12) (1.13) (1.14)

∞ X

X

sm(λ)q |λ| = − Q(q) · σ ∗ (q) − S0 (q)

λ∈D N

esm (N )q =

N =0 ∞ X N =0 ∞ X

X

sm(λ)q

|λ|

= P (q) ·

X

lg(λ)q |λ| = P (q) ·

N =0

∞ X n=1

λ∈P

eld (N )q N =

(q; q)n − (q; q)∞




n=0

λ∈P

elg (N )q N =

∞ X

X

qn 1 − qn

  1 1 = Q(q) · S0 (q) − + σ(q) 2 2

lg(λ)q |λ|

λ∈D

where ∞ Y

1 , P (q) = (1 − q n ) n=1 σ(q) = 1 +

S0 (q) =

∞ X

(1 + q n ),

n=1

n Y (−1)n q n+1 (1 − q j ),

n=0 ∞ n X n=1

Q(q) =

∞ Y

j=1

σ ∗ (q) = −2

∞ X n=0

q n+1

n Y (1 − q 2j ), j=1

q . 1 − qn

Remark. At all roots of unity, to infinite order, ∞ ∞ X X 1 n (−1) ((q; q)n − (q; q)∞ ) = (−1)n (q; q)n = σ(q) 2 n=1 n=1

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HIEU T. NGO AND ROBERT C. RHOADES

and

∞ X

((q; q)n − (q; q)∞ ) = F (q).

n=1

Thus, these forms behave as quantum modular forms near roots of unity. These identities are discussed in more detail in the proof of Theorem 1.1 and Sect. 4.4. Remark. The identities of Theorem 1.7 are all of the shape Generating function = (P (q) or Q(q)) × Sieving function + Error function.

In the above cases, the sieving functions are quantum modular forms. For example in the first equality the “sieving function” is ∞ X (−1)n ((q; q)n − (q; q)∞ ) n=1

= q + q 3 − q 4 − q 8 + q 10 − q 13 + q 14 + q 15 − q 17 + q 18 − q 19 + · · · Consequentially, plo (n) = p(n − 1) + p(n − 3) − p(n − 4) − p(n − 8) + p(n − 10) + p(n − 13) + · · · This is a typical behavior. See the book of Stanley [75] for discussion of sieving in enumerative combinatorics. See the papers of Andrews [4, 5], Bringmann and Mahlburg [22], Kim and Lovejoy [54], Kim and Jo [52], Kim et al. [55, 56], and Grabner et al. [48] for more examples of this phenomenon. Most commonly, the “sieving function” is a partial theta function, which is known to be a quantum modular form. See [59, 43, 14] for details about the quantum modularity of partial theta functions. In some other cases, the sieving function may be a mock modular form, which may have exponential singularities at some roots of unity, see [4, 12, 74] for examples of this sort. We have chosen the above examples because they have simple partition interpretations and because the quantum modular forms arising are among the most exotic. P∞ nk qn Remark. Define for a natural number k the q-series Mk (q) = n=1 (q;q)n ; here (q; q)n := Qn j n to normalizing by 1/p(n), the kthj=1 (1 − q ). Then the coefficient of q in Mk (q) is, upP moment of the largest part of a partition of n, namely λ lg(λ)k where the sum runs over P nk q n all partitions λ of n. Now define Sk (q) = ∞ n=1 1−q n . One may show that Sk and Mk are related via the recursion  k−1  X k−1 Mk (q) = Sk−1−j (q) Mj (q). j j=0 If k is an odd integer, then Sk is (essentially) an Eisenstein series. If k is an even integer, then Sk is not an Eisenstein series, but it still has pseudo-modularity (see Bettin and Conrey [13]). Thus, the method of this paper can be applied to compute an asymptotic expansion for each of the moments of the largest parts of partitions.

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

11

1.5. Wright’s asymptotic expansion and analogues. In this subsection we state generalizations of Wright’s asymptotic expansion using the circle method. This section contains three propositions in a quite general form with the hope that it might be useful for other researchers. All of the propositions concern the asymptotics for the q-coefficients of functions of the form L(q) ξ(q) where ξ(q) has its “main” exponential singularity as q → 1 and L(q) has polynomial or logarithmic behavior as q → 1. Throughout this section let q denote a complex variable with |q| < 1 and q ∈ / R≤0 . Put q = e−t , so t ∈ C satisfies Re t > 0 and |Im t| < π. The setup is as follows. Let L(q) and ξ(q) be two functions meromorphic in |q| < 1 so that L(q)ξ(q) is analytic for |q| < 1. We are interested in the asymptotic expansion of Z 1 L(q)ξ(q) (1.15) V (N ) = dq (N → ∞) 2πi C q N +1 where C is a circle centered at the origin with radius smaller than one and where q describes C counter-clockwise. Integral (1.15) will arise from the combinatorial problems described in Sect. 1. Let 0 < δ < π2 and c > 0 be fixed constants. Set η = √cN . Let C be the circle on the q-plane which is centered at 0 and of radius e−η . Let C1 be the arc of C such that q ∈ C1 precisely when |arg t| < π2 − δ; set C2 = C − C1 . We will make us of four running assumptions: HYPO 1 for every positive integer k, as |t| → 0 in the bounded cone |arg t| < ! k−1 1 X −t s k (1.16) L(e ) = B αs t + Oδ (t ) t s=0

π 2

− δ, either

where αs ∈ C and B is a real constant, in which case we say L(e−t ) has polynomial type near 1, or ! k−1 log t X −t s k (1.17) L(e ) = B αs t + Oδ (t ) t s=0 where αs ∈ C and B is a real constant, in which case we say L(e−t ) has logarithmic type near 1; HYPO 2 as |t| → 0 in the bounded cone |arg t| < π2 − δ

2 (1.18) ξ(e−t ) = tβ ec /t 1 + Oδ e−γ/t where β and γ are real constants such that β ≥ 0 and γ > c2 ; HYPO 3 as |t| → 0 in the bounded cone π2 − δ ≤ |arg t| < π2 (1.19)

L(e−t ) δ t−C

where C = C(δ) is a positive real constant; HYPO 4 as |t| → 0 in the bounded cone π2 − δ ≤ |arg t| < π2 0 −t ξ(e ) δ ξ(|e−t |)e −δt (1.20) where δ 0 = δ 0 (δ) is a positive real constant.

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HIEU T. NGO AND ROBERT C. RHOADES

Remark. The assumptions HYPO 1 and HYPO 2 put asymptotics on L and ξ on the “major arc” C1 near 1, whereas the assumptions HYPO 3 and HYPO 4 require that L and ξ are small on the “minor arc” C2 away from 1. The following result is due to Wright [81, Sect. 6]: Proposition 1.8. (Asymptotic expansion in the polynomial type case) Suppose that the four running assumptions are satisfied. Suppose further that L(q) has polynomial type near 1. Then ! R−1 √ X r 1 R V (N ) = e2c N N 4 (2B−2β−3) pr N − 2 + O(N − 2 ) (N → ∞) r=0

where pr =

r X

αs ws,r−s

s=0

with the αs given by HYPO 1 (1.16) and the ws,r given by
1 r s+β−B+ 12 c − 4c (1.21) ws,r = (s + β − B + 1, r). 1 2π 2 The following is an analogue of Proposition 1.8, now for the case of logarithmic type: Proposition 1.9 (Asymptotic expansion in the logarithmic type with integer-order case). Suppose that the four running assumptions are satisfied. Suppose further that L(q) has logarithmic type near 1 and that B − β = 1 where B is the constant in HYPO 1 (1.17) and β is the constant in HYPO 2 (1.18). Let a∗r (n) be given by (3.11) and set
1 r s− 21 − 4c c ls,r = (s, r) (s, r ∈ Z≥0 ) 1 −4π 2
1 1 r s− 21 log c − 4c c cs−r− 2 ∗ 0 ls,r = (s, r) + δr≥1 r+ 1 ar−1 (s) (s, r ∈ Z≥0 ). 1 2π 2 2 2 One has ! R−1   √ X 1 r R V (N ) = e2c N N − 4 N − 2 (qr0 + qr log N ) + O N − 2 log N (N → ∞) r=0

where qr =

r X

αs ls,r−s

s=0

and

qr0 =

r X

0 αs ls,r−s

s=0

with the αs given by HYPO 1 (1.17). Remark. We will see later from our proof that one can extend Proposition 1.9 to the case B − β ∈ Z (of course with more involved asymptotic formulas). Hence the nature of Proposition 1.9 is of logarithmic type with integer order. The following is another analogue of Proposition 1.8, also for the case of logarithmic type:

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

13

Proposition 1.10 (Asymptotic expansion in the logarithmic type with half-integer-order case). Suppose that the four running assumptions are satisfied. Suppose further that L(q) has logarithmic type near 1 and that B − β = 21 where B is the constant in HYPO 1 (1.17) and β is the constant in HYPO 2 (1.18). Then   √ 2c N − 21 −α0 − 21 V (N ) = e N log N − 2 log c + O(N log N ) (N → ∞). 1 4π 2 Remark. One can write down a full asymptotic expansion for V (N ) in Proposition 1.10 and can extend to the case B − β ∈ 21 + Z (of course with more involved asymptotic formulas). Hence the nature of Proposition 1.10 is of logarithmic type with half-integer-order. The main idea of the proofs of Propositions 1.8, 1.9 and 1.10 is the circle method. The main technical ingredients are various asymptotic expansions of Bessel functions. 1.6. Acknowledgements. This work is a sequence to previous work [61, 62], which started from a project group led by Ken Ono in the Arizona Winter School 2013. Without the excellent working environment provided by the group leader and the winter school organizers, this work would not have been realized. The authors are grateful to Jeffrey C. Lagarias for his support and encouragement. We would like to thank Yingkun Li for discussions about early aspects of this work. We would like to thank H. M. Bui for his literature help. 2. Proof of generating function identities This section contains the proof of the generating function identities of Theorem 1.7. The proof uses the following theorems of Andrews and Freistas. Theorem 2.1 (Theorem 4.1 of [8]). Let g be a function defined by the series g(x) = P∞ n n=0 gn x . Then   ∞ ∞ X (t)n (t)∞ (t)∞ X (a/t)n n n gn − = g(q )t . (a)n (a)∞ (a)∞ n=1 (q)n n=0 Theorem 2.2 (Theorem 4.4 of [8]). One has   ∞  ∞  X (a)∞ (b)∞ X (q/b)n bn (c/a)n an 1 (a)n (b)n (a)∞ (b)∞ − = + . n (c) (q) (c) (q) (c) (q) (q) (b) 1 − q n n ∞ ∞ ∞ ∞ n n n=0 n=1 Proof of Theorem 1.7. (1) Recall that plo (N ) is the number of partitions of N for which sm(λ) ≡ 1 (mod 2). We begin by establishing ∞ ∞ X X X qn N |λ| (2.1) plo (N )q = q = . 2n ) (q) (1 − q n−1 n=1 N =1 λ:sm(λ)≡1(mod 2)

Indeed, the first equality is the definition of plo (N ). Let p0lo (N ) denote the number of partitions of N in which the largest part, indexed P∞ by0 n say,N occurs an odd number of times. Then the third member of (2.1) is precisely N =1 plo (N )q . that we can express a partition λ of N as a decreasing sequence (λ1 , . . . , λr ) with PRecall r i=1 λi = N by a Young diagram, which by definition is a rectangular array of N boxes,

14

HIEU T. NGO AND ROBERT C. RHOADES

with r rows where the ith-row is of length λi . The conjugate partition λ0 of λ is the partition whose Young diagram is obtained by reflecting the Young diagram of λ about the diagonal so that rows become columns and columns become rows. For example, the conjugate partition of the partition (4, 3, 2, 2, 1) is the partition (5, 4, 2, 1). Taking the conjugate partitions gives a bijection between the partitions of N in which the largest part occurs an odd number of times with the partitions of N with the smallest part being odd. In particular we have plo (N ) = p0lo (N ); this shows the second equality of (2.1). The desired identity is an immediate consequence of the theorem of Andrews and Freistas, Theorem 2.1 (taking gn = (−1)n , a = 0, t = q). (2) Recall that esd (N ) is the sum of the smallest parts of the partitions of N with distinct parts. By definition it is clear that ∞ X

N

esd (N )q =

X

N =0

sm(λ)q

|λ|

=

∞ X

nq n (−q n+1 ; q)∞ .

n=0

λ

The desired identity follows from previous work of the authors and Li [62, Section 6.3]. (3) Recall that esm (N ) is the sum of the smallest parts of the partitions of N . We show that (2.2)

∞ X

esm (N )q N =

N =0

X

sm(λ)q |λ| =

∞ X n=1

λ

qn . (1 − q n )(q)n

The first equality is the definition of esm (N ). Let clg (λ) denote the number of occurrences of the largest part  of the partition λ. Then the third member of (2.2) is equal to P P∞ N λ∈P(N ) clg (λ) q . On the other hand, by looking at Young diagrams of partiN =1 tions P and the conjugate transformation described above, it is easy to see that esm (N ) = λ∈P(N ) clg (λ). Thus we deduce the second equality of (2.2). In Theorem 2.2, we put a = b = q and c = 0 to deduce ∞ X n=0

((q)n − (q)∞ ) = (q)∞

∞ X n=1

qn , (1 − q n )(q)n

which is precisely (1.12). (4) Recall that elg (N ) is the sum of the largest parts of the partitions of N . By standard arguments ∞ ∞ X X X nq n elg (N )q N = lg(λ)q |λ| = . (q) n n=1 N =0 λ∈P In previous work [62, Section 6.3] the authors and Li established ∞ ∞ X nq n 1 X qn = . (q)n (q)∞ n=1 1 − q n n=1

Equation (1.13) follows.

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

15

(5) Recall that eld (N ) is the sum of the largest parts of the partitions of N with distinct parts. It is clear, by definition, that ∞ X

N

eld (N )q =

N =1

X

lg(λ)q

|λ|

=

∞ X

nq n (−q)n−1 .

n=1

λ∈D

Equation (1.14) follows from an identity established in previous work of the authors and Li [62, Section 6.4] ∞ X

nq n (−q)n−1 = (−q)∞

n=1

∞ X n=1

qn 1 − n 1−q 2

!

1 + σ(q). 2 

3. Circle method This section contains the proofs of the Propositions of Sect. 1.5. 3.1. Preliminaries: a recollection on Bessel functions. We recall some standard facts about Bessel functions. For references, see [80, Chapters 6 and 7], [66, Section 10], [65, 25]. The I-Bessel function is defined by Schläfli’s integral (3.1)

Iν (z) =


ν 1 z 2 2πi

Z

(0+) −ν−1

t

 exp

−∞

 z2 + t dt 4t

(z ∈ C − R≤0 ; ν ∈ C).

In (3.1), the contour runs from −∞ below the negative real axis to 0 and circles around 0 counter-clockwise and comes back to −∞ above the negative real axis, and arg z is taken to be −π below the negative real axis and π above the negative real axis (see [80, Section 6.22]). The K-Bessel function is defined by π (I−ν (z) − Iν (z)) 2 sin(πν) Km (z) = lim Kν (z) Kν (z) =

ν→m,ν ∈Z /

(z ∈ C − R≤0 ; ν ∈ C − Z) (z ∈ C − R≤0 ; m ∈ Z).

For any ν ∈ C, the pair {Iν (z), Kν (z)} forms a basis of linearly independent solutions of the Bessel’s differential equation   y0 ν2 y + − 1 + 2 y = 0. z z 00

16

HIEU T. NGO AND ROBERT C. RHOADES

Theorem 3.1. ([80, Section 7.23], [66, Section 10.40]) Suppose that ν ∈ C and that p is a positive integer which satisfies p ≥ Re ν − 12 . Then ( p−1 ) p−1 X (−1)m (ν, m) e−z+(ν+ 12 )πi X ez (ν, m) (3.2) Iν (z) = + + Oν,p (z −p ) 1 1 m m (2z) (2z) (2πz) 2 m=0 (2πz) 2 m=0  π  π − < arg z < , |z| → ∞ , 2 ( p−1 2 )  π  12 X (ν, m) Kν (z) = e−z (3.3) + Oν,p (z −p ) m 2z (2z) m=0  π  π − < arg z < , |z| → ∞ . 2 2 For brevity, we rewrite Theorem 3.1 as follows: Theorem 3.2. If 0 < δ < 12 π and ν ∈ C, then I±ν (z) =

(3.4)

(3.5)

K±ν (z) =

R−1 X

ez z e

1 2

z −r ar (ν) + O(z −R )

(| arg z|
0). sin πs 2

Lemma 4.3. Let M be a positive integer. As |t| → 0 in the right half-plane Re t > 0, one 1 has AM (t) M t2M + 2 . Proof. Using the functional equation of the Riemann zeta function ζ(s) together with the 1 π identities Γ(s)Γ(s + 21 ) = π 2 21−2s Γ(2s) and Γ(s)Γ(1 − s) = sin(πs) , we see that ζ 2 (s)

Γ(s) (2π)2s−1 −s 2 −s (it) = πs Γ(1 − s)ζ (1 − s)(it) . sin πs cos 2 2

Therefore (2π)−2s−1 2 −s ds (Re t > 0). πs Γ(1 + s)ζ (1 + s)(it) 1 cos ( 2 +2M ) 2 R R∞ Write s = 21 + 2M + it0 . We break the integral ( 1 +2M ) ds = i −∞ dt0 into three parts: 2 (a) |t0 | ≥ 220M , (b) 1 ≤ |t0 | < 220M , and (c) |t0 | < 1 and hence write A = Aa + Ab + Ac accordingly. On part (a), i.e., |t0 | ≥ 220M and σ = 12 + 2M , we have the following estimates 1 AM (t) := πi

Z

(2π)−2s−1 ∼ (2π)−4M −2 1 − 12 πt0 πs  e cos 2 1

Γ(1 + s)  t1+2M e− 2 πt0 0

(by Stirling’s formula)

ζ 2 (1 + s) M 1 1

1

1

(it)−s = e 2 πis ts ∼ e− 2 πt0 t 2 +2M ; here all estimates except for the second line are apparent, whereas for the second line we note 1 1 1 1 1 1 1 πs = e 2 πis + e− 2 πis = e 2 πiσ e− 2 πt0 + e− 2 πiσ e 2 πt0  e 2 πt0 . 2 cos 2 R ∞ 1+2M − 3 πt 1 1 +2M e 2 0 dt0 M t 2 +2M . Similar arguments show that Ab M Hence Aa M t 2 t M 0 1 1 t 2 +2M and that Ac M t 2 +2M . This concludes the lemma.  Corollary 4.4. Let M be a positive integer. One has   M 2 X 1 it B2n = −i t2n−1 + O(t2M + 2 ) g0 2π (2n)! n n=1   M 2 X 1 it 4 log t 2πi − 4γ B2n ψ0 = + −2 t2n−1 + O(t2M + 2 ) 2π t t (2n)! n n=1

(Re t > 0, |t| → 0). (Re t > 0, |t| → 0).

26

HIEU T. NGO AND ROBERT C. RHOADES

From the relation between the divisor generating function S0 (z) and the associated period function ψ0 (z), namely     1 1 1 4 1 S0 (z) = 1− − ψ0 (z) + S0 − , 4 z 4 z z
it we are now ready to see that the function S0 2π satisfies HYPO 1 and HYPO 3. Corollary 4.5. (1) Let M be a positive integer. As |t| → 0 in the bounded cone |arg t| < π − δ, one has 2  (4.5)

S0

it 2π

M

 =−

2 1 log t γ 1 X B2n + + + t2n−1 + O(t2M + 2 ). t t 4 n=1 (2n)! (2n) π 2

− δ ≤ |arg t| < π2 , one has   3 it (4.6) S0 δ t− 2 . 2π
it The corollary shows that the function S0 2π is a sum of a series of
polynomial type and it a series of logarithmic type in the sense of HYPO 1, and that S0 2π satisfies HYPO 3. (2) As |t| → 0 in the bounded cone

Proof.

(1) This is an immediate consequence of Corollary 4.4
and the observation that in
1 the given bounded cone, the function S0 − z = S0 2πi decays exponentially with t 1 . respect to |t| (2) We have  X    ∞ 4π 2 n 1 2πi d(n)e− t S0 − = S0 = z t n=1 

∞ X

2

4π nRe t  − |t|2

ne

≤

n=1

∞ X

2

e

n  log n− 2π1/2 N

.

n=1

P P P 2/3 We then break this latter sum as ∞ , we n=1 = n 1 be any (fixed) constant. We shall work out an integral representation for R (z) which is analogous to Theorem 4.2. Our starting observation is that R (z) is an inverse Mellin transform of a certain
Hecke L-function. We will look at the period function of type ψ (z) = R (z) − cz1 R − cz1 where c > 0 is a normalizing constant. Define  s 2 X T (n) + T (−n) s −s 2π L+ (s) = ; Λ (s) = (1152) Γ L+ (s) + ns 2 n≥1  2 X T (n) − T (−n) s+1 s+1 −(s+1) 2 L− (s) = ; Λ− (s) = (1152) π Γ L− (s). ns 2 n≥1 We collect the relevant results from [7, 29, 16] in the following Theorem 4.6. [7, 29, 16] (1) (2) (3) (4)

T (n) = O(log n) for  ∈ {+, −} and n ∈ Z>0 ; L (s) converges absolutely for Re s > 1; Λ (s) extends to an entire function of order 1 and satisfies Λ (s) = Λ (1 − s); Outside the critical strip 0 < Re s < 1, L+ (s) has trivial zeros (of order 2) at s ∈ 2Z≤0 = {0, −2, −4, −6, . . .}. Outside the critical strip 0 < Re s < 1, L− (s) has trivial zeros (of order 2) at s ∈ −1 + 2Z≤0 = {−1, −3, −5, −7, . . .}.

The following is an analogue of Theorem 4.2: Theorem 4.7. For z ∈ H one has R+ (z) − (4.7)

(4.8)

1 1

 R+ −

1 1152z Z



1152 2 z M X i 1 −s = r+,n + Γ(s) L+ (s) ds πs (2πz) 1 2πi sin − −2M ( ) 2 n=0 2   1 1 R− (z) + − 1 R− 1152z 1152 2 z Z M X 1 1 −s = r−,n + Γ(s) L− (s) ds πs (2πz) 1 2πi cos − −2M ( 2 ) 2 n=0

28

HIEU T. NGO AND ROBERT C. RHOADES

where the r,n are given by r+,0 = 0 and (−1)n+1 i (2πz)2n−1 L+ (1 − 2n) (2n − 1)! (−1)n = (2πz)2n L− (−2n) (2n)!

(4.9)

r+,n =

(n ∈ Z>0 )

(4.10)

r−,n

(n ∈ Z≥0 ).

Proof. We have X X T (n) + T (−n) Z 2πinz (−2πinz)−s Γ(s) ds R (z) = (T (n) + T (−n)) e = 2πi (A) n>0 n>0 Z 1 1 = (4.11) e 2 iπs (2πz)−s Γ(s) L (s) ds. 2πi (A)
Now let  = + and move the line of integration to the left, say to the line − 12 − 2M for 1 any positive integer M . The poles of the integrand in (4.11), namely e 2 iπs (2πz)−s Γ(s) L+ (s), are simple poles at s ∈ −1 + 2Z≤0 . For n ∈ Z>0 we have  1  (−1)n+1 i iπs −s 2 Ress=1−2n e (2πz)2n−1 L+ (1 − 2n) = r+,n . (2πz) Γ(s) L+ (s) = (2n − 1)! One can use the functional equation of L+ to rewrite this expression of r+,n . The same thing can be said for  = −. In this case, the poles of the integrand in (4.11), namely 1 e 2 iπs (2πz)−s Γ(s) L− (s), are simple poles at s ∈ 2Z≤0 . For n ∈ Z≥0 we have  1  (−1)n Ress=−2n e 2 iπs (2πz)−s Γ(s) L− (s) = (2πz)2n L− (−2n) = r−,n . (2n)! One can use the functional equation of L− to rewrite this expression of r−,n . Since r+,0 = 0, we see that Z M X 1 1 (4.12) R (z) = r,n + e 2 iπs (2πz)−s Γ(s) L (s) ds. 2πi (− 12 −2M ) n=0 Hence M X

Z 1 1 R(z) = R+ (z) + R− (z) = (r+,n + r−,n ) + e 2 iπs (2πz)−s Γ(s) L(s) ds. 1 2πi (− 2 −2M ) n=0
Now let’s look at z1 R − z1 for  ∈ {+, −}. We have    −s Z 1 1 1 1 1 −2π iπs R − = e2 Γ(s) L (s) ds z z z 2πi (A) z Z 1 1 = e− 2 iπs (2π)−s z s−1 Γ(s) L (s) ds 2πi (A) Z 1 1 = e− 2 iπ(1−s) (2π)−(1−s) z −s Γ(1 − s) L (1 − s) ds. 2πi (1−A)

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

29

One applies the functional equation of Λ (s) (Theorem 4.6) to see that   1 1 R+ − = z z   1 1 R− − = z z

1 2πi

Z

1 2πi

Z

(−i) cot

1 1 πs 1152s− 2 e 2 iπs (2πz)−s Γ(s) L+ (s) ds 2

(−i) tan

1 1 πs 1152s− 2 e 2 iπs (2πz)−s Γ(s) L− (s) ds. 2

(1−A)

(1−A)

Since the current integrands do not have poles when Re s ≤ 0, we are free to move the line of integration to the left and deduce that   1 1 R+ − = z z   1 1 R− − = z z

(4.13) (4.14)

Z 1 1 1 πs (−i) cot 1152s− 2 e 2 iπs (2πz)−s Γ(s) L+ (s) ds 2πi (− 12 −2M ) 2 Z 1 1 πs 1 (−i) tan 1152s− 2 e 2 iπs (2πz)−s Γ(s) L− (s) ds. 1 2πi (− 2 −2M ) 2

We now note two identities: i πs = 2 sin πs 2 1 1 πs 1 e 2 iπs − ie 2 iπs tan = . 2 cos πs 2 1

1

e 2 iπs + ie 2 iπs cot

(4.15) (4.16)

We add (4.12) and (4.13) (with z replaced by 1152z) and use (4.15) to conclude (4.7). We add (4.12) and (4.14) (with z replaced by 1152z) and use (4.16) to conclude (4.8).  Proceeding in the same fashion as Corollaries 4.4 and 4.5, with the aid of Theorem 4.6 we arrive at: Corollary 4.8. Let M be a positive integer. One has 



M X −L+ (1 − 2n)

2n−1

2M + 21

2πi



2πi )− R+ = t + O(t 1 R+ (2n − 1)! 1152t 1152 2 t n=1   X   M it L− (−2n) 2n−1 2πi 2πi 2M + 12 R− = t + O(t )+ 1 R− 2π (2n)! 1152t 1152 2 t n=0 it 2π

 (Re t > 0, |t| → 0). (Re t > 0, |t| → 0)


1 1 it We are now ready to see that each of the functions q 24 σ(q) = R 48π and q − 24 σ ∗ (q) =
it R∗ 48π satisfies HYPO 1 and HYPO 3. Recall R = 12 (R+ + R− ) and R∗ = 12 (R+ − R− ).

30

HIEU T. NGO AND ROBERT C. RHOADES

Corollary 4.9. (1) Let M be a positive integer. Let q = e2πiz = e−t . As |t| → 0 in the bounded cone |arg t| < π2 − δ, one has (4.17) e

t − 24

 2n−1  2n M X 1 L+ (1 − 2n) t L− (−2n) t −t σ(e ) = L− (0) + − + 2 (2n − 1)! 2 24 (2n)! 2 24 n=1

! 1

+ O(t2M + 2 ),

(4.18)  2n−1  2n M X 1 L+ (1 − 2n) t L− (−2n) t ∗ −t e σ (e ) = − L− (0) + − − 2 (2n − 1)! 2 24 (2n)! 2 24 n=1

!

t 24

(2) As |t| → 0 in the bounded cone C for which t

π 2

e− 24 σ(e−t ) δ t−C

1

+ O(t2M + 2 ).

− δ ≤ |arg t| < π2 , there is a positive real constant t

and e 24 σ ∗ (e−t ) δ t−C .

Let us simplify the notation of Corollary 4.9 by the following Definition 4.10. Let the coefficients σn and σn∗ be given by  n ∞ X t σn t −t − 24 e σ(e ) =: n! 24 n=0  n ∞ X t σn∗ t ∗ −t e 24 σ (e ) =: n! 24 n=0

(Re t > 0) (Re t > 0).

In particular, σn∗ = (−1)n+1 σn . 4.4. The Kontsevich–Zagier function F (q). Kontsevich defined the series F (q) = 1 +

∞ X

(q)n .

n=1

This series does not converge in any open subset of C but it makes sense when q is a root of unity. Zagier [82] investigated this series in great details. An interesting aspect of his work is a “strange identity”   ∞ ∞ ∞ X X n2 −1 1X 12 n (4.19) (q)n = 1 + (1 − q) . . . (1 − q ) = − n q 24 . 2 n=1 n n=0 n=1 To quote Zagier: “The meaning of the equality is that the function on the left agrees at roots of unity with the radial limit of the function on the right, and similarly for the derivatives of all orders. Equation (4.19), of which  n ∞ ∞ X X t Tn t − 24 −t −nt e (1 − e ) · · · (1 − e ) = n! 24 n=0 n=0 is a consequence, is related to the Dedekind η-function and the theory of periods of modular forms.”

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

31

To be precise, Zagier defined for |q| < 1 the following functions F1 (q) = F2 (q) =

∞ X n=1 ∞ X

n

n(q)n−1 q = lim

N →∞

N X

n(q)n−1 q n

n=1

[(q)n − (q)∞ ]

n=0

  ∞ X n2 −1 12 H(q) = n q 24 = 1 − 5q − 7q 2 + 11q 5 + 13q 7 − · · · n n=1 E(q) =

∞ X n=1

∞

X qn = d(n)q n = q + 2q 2 + 2q 3 + 3q 4 + · · · 1 − qn n=1

Theorem 4.11. (Zagier [82])
  (1) One has F1 (q) = F2 (q) = − 21 H(q) + 21 − E(q) (q)∞ as an identity in Z 21 [[q]]. Note however that F1 = F2 ∈ Z[[q]]. (2) As t ∈ R+ , t → 0 one has   ∞ ∞ X t 12 − n2 t X n −t − 24 n γn t e H(e ) = e 24 ∼ n n=1 n=0
(−1)n 12 where γn = 24n n! L(−2n − 1, · ). One can then use the same arguments as Sect. 4.3 to deduce the following: Corollary 4.12. (1) Let M be a positive integer. As |t| → 0 in the bounded cone |arg t| < π − δ, one has 2 M

t

t

e− 24 F1 (e−t ) = e− 24 F2 (e−t ) = −

1X n γn t + O(tM +1 ) 2 n=1


n 12 where γn = (−1) L(−2n − 1, ) and − γ2n = 24Tnn n! . n 24 n! · (2) As |t| → 0 in the bounded cone π2 − δ ≤ |arg t| < π2 , there is a positive real constant C for which t t e− 24 F1 (e−t ) = e− 24 F2 (e−t ) δ t−C . 5. Proofs of combinatorial probability theorems In this section we prove the main theorems stated in Sect. 1. Proof of Theorem 1.1. Recall that plo (N ) is the number of partitions of N for which sm(λ) ≡ P 1 (mod 2). The first identity of Theorem 1.7 expresses the generating function N ≥0 plo (N )q N as the product of a modular form and a function with quantum modular behavior. We show the additional q-identity ! ∞ ∞ X X 1 1 (5.1) (−1)n (q)n = 1+ (−1)n q n+1 (q)n = σ(q). 2 2 n=0 n=0

32

HIEU T. NGO AND ROBERT C. RHOADES

Consider the basic hypergeometric q-series F (a, b; t) = F (a, b; t : q) :=

∞ X (aq; q)n n=0

(bq; q)n

tn .

When |t| = 1, but t 6= 1 and |q| < 1 the series converges via the recurrence 1 − b b − atq + F (a, b; tq), 1−t 1−t which appears as (2.4) in [42]. Applying (5.2) with t = −1, a = 1, b = 0, we obtain (5.1). Using the notation of Sects. 1.5 and 3, it follows from (1.10) that Z 1 Lξ plo (N ) = V (N ) := dq 2πi C q N +1 1 1 P (2π) 2 n (−1) ((q) − (q) ) and ξ(q) = where L(q) = 1 1 q 24 ∞ . Corollary 4.9 shows that n ∞ n=0 η(q) F (a, b; t) =

(5.2)

(2π) 2

L(e−t ) verifies HYPO 1 and HYPO 3, whereas Theorem 4.1 shows that ξ(e−t ) verifies HYPO 2 and HYPO 4. In the notation of Sect. 1.5, we have 1 σs B = 0, αs = · , 1 (2π) 2 2 s! 24s π2 c = , 6

1 β= , 2

2

γ = 4π 2 .

Proposition 1.8 then yields V (N ) = e2π

√N 6

R−1 X

N −1

! Ar N

− r2

+ O(N

−R 2

)

r=0

where Ar =

Pr

s=0

αs ws,r−s and ws,r =


−1 r 4



π √ 6

2π

s+1−r 3 (s + , r). 2

1 2



This concludes Theorem 1.1. Proof of Theorem 1.2. It follows from (1.11) that Z 1 Lξ esd (N ) = V (N ) := dq − d(N ) 2πi C q N +1 1

1

1

where L(q) = − 11 q − 24 σ ∗ (q) and ξ(q) = 2 2 q 24 (−q)∞ . Corollary 4.9 shows that L(e−t ) 22 verifies HYPO 1 and HYPO 3, whereas Theorem 4.1 shows that ξ(e−t ) verifies HYPO 2 and HYPO 4. In the notation of Section 1.5, we have 1 σs B = 0, αs = √ · , 2 s! (−24)s π2 β = 0, c2 = , γ = 2π 2 . 12

PARTITIONS, PROBABILITY, AND QUANTUM MODULAR FORMS

33

Proposition 1.8 then yields V (N ) = eπ

√N

N −1

3

R−1 X

! Br N

− r2

+ O(N

−R 2

)

r=0

where Br =

Pr

s=0

αs ws,r−s and ws,r−s

1 s−r = √ (−2K 0 ) 2 π



K0 2

s+1/2 (s + 1, r − s). 

This concludes Theorem 1.2. Proof of Theorem 1.3. By definition we have ∞ X X `(λ)+1 |λ| nq n (q n+1 ; q)∞ . (−1) sm(λ)q =

(5.3)

n=0

λ

Theorem 1.3 is now an immediate consequence of (5.3) and an identity which was proved in [62, Sect. 6.3]: ∞ ∞ X X qn n n+1 . nq (q ; q)∞ = 1 − qn n=1 n=1 

See also (1.13) and its proof in Section 2. Proof of Theorem 1.4. Using the notation of Sect. 3, it follows from (1.12) that Z 1 Lξ esm (N ) = V (N ) := dq 2πi C q N +1 1

where L(q) =

1 (2π) 2

1

1

q 24 F (q) and ξ(q) =

(2π) 2 η(q)

. Corollary 4.12 shows that L(e−t ) verifies

HYPO 1 and HYPO 3, whereas Theorem 4.1 shows that ξ(e−t ) verifies HYPO 2 and HYPO 4. In the notation of Sect. 1.5, we have B = 0,

αs =

Ts

=

1 2

1 2

(2π) 2 (2π) s! 24s π2 c2 = , γ = 4π 2 . 6

1 β= , 2 Proposition 1.8 then yields V (N ) = e2π

−γs

√N 6

N −1

R−1 X

,

! Dr N

− r2

+ O(N

−R 2

)

r=0

where Dr =

Pr

s=0

αs ws,r−s and ws,r−s

1 = √ (−2K)s−r 2 π

This concludes Theorem 1.4.



K 2

s+1

3 (s + , r − s) 2 

34

HIEU T. NGO AND ROBERT C. RHOADES

Proof of Theorem 1.5. Using the notation of Section 3, it follows from (1.13) that Z 1 Lξ elg (N ) = V (N ) := dq; 2πi C q N +1 1 1 P (2π) 2 qn −t where L(q) = 1 1 q 24 ∞ n=1 1−q n and ξ(q) = η(q) . Corollary 4.5 shows that L(e ) verifies (2π) 2

HYPO 1 and HYPO 3, whereas Theorem 4.1 shows that ξ(e−t ) verifies HYPO 2 and HYPO 4. Moreover, we write ! ∞ 2 X t log t B 1 γ 1 2n − 24 − L(e−t ) = + + + t2n−1 1 e t t 4 (2n)!(2n) 2 (2π) n=1 2 P B2n 2n−1 and splits into two parts: (a) − logt t , (b) γt − 41 + ∞ . In the notation of n=1 (2n)!(2n) t Sect. 1.5, we now have a combination of the logarithmic type case with half-integer-order (part (a)) and of the polynomial type case (part (b)). Proposition 1.10 applies to part (a), whereas Proposition 1.8 applies to part (b). For example, the parameters in the logarithmic type case are

B = 1, 1 β= , 2 Applying Propositions 1.8 and 1.10,

α0 =

−1

1 , (2π) 2 π2 c2 = , γ = 4π 2 . 6 we conclude Theorem 1.5.



Proof of Theorem 1.6. Using the notation of Sect. 3, it follows from (1.14) that Z 1 Lξ eld (N ) = V (N ) := dq 2πi C q N +1   P 1 1 1 1 qn where L(q) = 2− 2 q − 24 − 12 + ∞ and ξ(q) = 2 2 q 24 (−q)∞ . Corollary 4.5 shows that n n=1 1−q L(e−t ) verifies HYPO 1 and HYPO 3, whereas Theorem 4.1 shows that ξ(e−t ) verifies HYPO 2 and HYPO 4. Moreover, we write ! ∞ 2 t log t γ 1 X B2n −t − 21 24 2n−1 L(e ) = 2 e − + − + t t t 4 n=1 (2n)!(2n) 2 P B2n 2n−1 and splits into two parts: (a) − logt t , (b) γt − 41 + ∞ . In the notation of n=1 (2n)!(2n) t Sect. 1.5, we now have a combination of the logarithmic type case with integer-order (part (a)) and of the polynomial type case (part (b)). Proposition 1.9 applies to part (a), whereas Proposition 1.8 applies to part (b). For example, the parameters in the logarithmic type case are π2 B = 1, β = 0, c2 = , γ = 2π 2 . 12 Applying Propositions 1.8 and 1.9, we conclude Theorem 1.6. 

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[83] D. Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc. 11, 659–675 (2010) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA E-mail address: [email protected] Center for Communications Research, Princeton, NJ 08540, USA E-mail address: [email protected]