A Rademacher Type Formula For Partitions and Overpartitions

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of $n$ and the Zuckerman formula for the Fourier coefficients of $\vartheta_4(0\vert \tau)^{-1}$ is presented.

Euler [12] was the first to systematically study partitions. He showed that Euler also showed that and since the exponents appearing on the right side of (1.2) are the pentagonal numbers, Eq. (1.2) is often called "Euler's pentagonal number theorem." Although Euler's results can all be treated from the point of view of formal power series, the series and infinite products above (and indeed all the series and infinite products mentioned in this paper) converge absolutely when |x| < 1, which is important for analytic study of these series and products.
Recently, Bringmann and Ono [4] have given exact formulas for the coeffcients of all harmonic Maass forms of weight ≦ 1 2 . The generating functions considered herein are weakly holomorphic modular forms of weight − 1 2 , and thus they are harmonic Maass forms of weight ≦ 1 2 . Accordingly, the results of this present paper could be derived from the general theorem in [4]. However, here we opt to derive the results via classical method of Rademacher.
Letp(n) denote the number of overpartitions of n and letf (x) denote the generating function ∞ n=0p (n)x n ofp(n). Elementary techniques are sufficient to show thatf .
Note that 1 via an identity of Gauss [1, p. 23, Eq. (2.2.12)], so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a perfect square, just as the reciprocal of the generating function for partitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a pentagonal number. Hardy and Ramanujan, writing more than 80 years before the coining of the term "overpartition," stated [28, p. 109-110] that the function which we are callinḡ p(n) "has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple ϑ-funciton, of particular interest." They went on to state that (1.5) In fact, (1.5) was improved to the following Rademacher-type convergent series by Zuckerman [55, p. 321, Eq. (8.53)]: A simplified proof of Eq. (1.6) was given by L. Goldberg in this Ph.D. thesis [17].

1.3.
Partitions where no odd part is repeated. Let pod(n) denote the number of partitions of n where no odd part appears more than once. Let g(x) denote the generating function of pod(n), so we have Via another identity of Gauss [1, p. 23, Eq. (2.2.13)], it turns out that so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the exponents which are triangular (or equivalently, hexagonal) numbers. The analogous Rademacher-type formula for pod(n) is as follows.
Eq. (1.7) is the case r = 2 of Theorem 2.1 to be presented in the next section.

A common generalization
Let us define where r is a nonnegative integer. Thus, Notice that f r (x) can be represented by several forms of equivalent infinite products, each of which has a natural combinatorial interpretation: Thus, p r (n) equals each of the following: • the number of overpartitions of n where nonoverlined parts are multiples of 2 r (by (2.5)); • the number of partitions of n where all parts are either odd or multiples of 2 r (by (2.6)), provided r ≧ 1; • the number of partitions of n where where nonmultiples of 2 r−1 are distinct (by (2.7)), provided r ≧ 1.
Of fundamental importance is the path of integration to be used. In [47], Rademacher improved upon his original proof of (1.4) given in [46], by altering his path of integration from a carefully chosen circle to a more complicated path based on Ford circles, which in turn led to considerable simplifications later in the proof.
3.1. Farey fractions. The sequence F N of proper Farey fractions of order N is the set of all h/k with (h, k) = 1 and 0 ≦ h/k < 1, arranged in increasing order. Thus, e.g., 3 4 . For a given N , let h p , h s , k p , and k s be such that hp kp is the immediate predecessor of h k and hs ks is the immediate successor of h k in F N . It will be convenient to view each F N cyclically, i.e. to view 0 1 as the immediate successor of N −1 N .

Ford circles and the Rademacher path. Let h and k be integers with
Note that we have Every Ford circle is in the upper half plane. For h1 k1 , h2 k2 ∈ F N , C(h 1 , k 1 ) and C(h 2 , k 2 ) are either tangent or do not intersect.
The Rademacher path P (N ) of order N is the path in the upper half of the τ -plane from i to i + 1 consisting of traversed left to right and clockwise. In particular, we consider the left half of the Ford circle C(0, 1) and the corresponding upper arc γ(0, 1) to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over P (N ).

3.3.
Set up the integral. Let n and r be fixed, with n > (2 r − 1)/24. Since Cauchy's residue theorem implies that where C is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable so that the unit disk |x| ≦ 1 in the x-plane maps to the infinitely tall, unit-wide strip in the τ -plane where 0 ≦ ℜτ ≦ 1 and ℑτ ≧ 0. The contour C is then taken to be the preimage of P (N ) under the map x → e 2πiτ . Better yet, let us replace x with e 2πiτ in (3.4) to express the integration in the τ -plane: 3.4. Another change of variable. Next, we change variables again, taking Thus C(h, k) (in the τ -plane) maps to the clockwise-oriented circle K (−) k (in the z-plane) centered at 1/2k with radius 1/2k.
So we now have where √ z is the principal branch, (h, k) = 1, and H is a solution to the congruence hH ≡ −1 (mod k).
where H j is divisible by 2 r−j and is a solution to the congruence hH j ≡ −1 (mod k), and is the Kronecker δ-function.
Notice that in particular, for ⌊ r 2 ⌋ ≦ j ≦ r, (3.11) simplifies to Since the r = 0 case was established by Zuckerman, and the r = 1 case by Rademacher, we will proceed with the assumption that r > 1.
Apply (3.11) to (3.7) to obtain 3.6. Normalization. Next, introduce a normalization ζ = zk. (This is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature.) Let us now rewrite (3.14) as

3.7.
Estimation. It will turn out that as N → ∞, only I j,2 for j = 0 and ⌊r/2⌋ < j ≦ r ultimately make a contribution (provided r < 5). Note that all the integrations in the ζ-plane occur on arcs and chords of the circle K of radius 1 2 centered at the point 1 2 . So, inside and on K, 0 < ℜζ ≦ 1 and ℜ 1 ζ ≧ 1.
3.7.1. Estimation of I j,2 for 1 ≦ j ≦ ⌊r/2⌋. The regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I (h, k) and ζ T (h, k) to the directed segment. By [2,p. 104,Thm. 5.9], the length of the path of integration does not exceed 2 √ 2k/N , and on the segment connecting ζ I (h, k) to ζ T (h, k), |ζ| < √ 2k/N . Thus, the absolute value of the integrand, Thus, for 1 ≦ j ≦ ⌊r/2⌋, for a constant C ′ j (recalling that n and r are fixed).

3.7.2.
Estimation of I j,1 for 1 ≦ j ≦ ⌊r/2⌋. We have the absolute value of the integrand: for a constant c j . So, for 1 ≦ j ≦ ⌊r/2⌋, Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I (h, k) and ζ T (h, k) to the directed segment.
With this in mind, we estimate the absolute value of the integrand: for a constant c 0 . So, for a constant C 0 .
3.7.4. Estimation of I j,1 for 1 + ⌊r/2⌋ ≦ j ≦ r ≦ 4. Again, the regularity of the integrand allows us to alter the path of integration from the arc connecting ζ I (h, k) and ζ T (h, k) to the directed segment.
With this in mind, for a constant c j . So, for a constant C j , when 1 + ⌊r/2⌋ ≦ j ≦ r.