Exact formula for cubic partitions

We obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng.


Introduction and statement of results
Let n be a non-negative integer.A non-increasing finite sequence of positive integers that sums to n is called a partition of n.We denote by p(n) the number of partitions of n, which can be defined by the coefficients of the following q-series [1] The function p(n) is one of the most well-studied functions troughout number theory and satisfies numerous remarkable identities.One example is the following identity discovered by S. Ramanujan [1], [10].
An immediate consequence is one of Ramanujan's celebrated congruences which asserts that for any non-negative integer p(5n + 4) ≡ 0 (mod 5) Similar to (1.1) we define for non-negative integers the cubic partition function a(n) by H. Chan proved that the function a(n) satisfies a identity similar to the one by Ramanujan given above 4 , a(3n + 2) ≡ 0 (mod 3), see [6].His result is closely related to Ramanujan's continued cubic fraction, see [7].For an introduction to Ramanujan's cubic continued fraction, see [2], [4].Furthermore, from the generating function (1.2) it is obvious that a(n) is the number of partition pairs (λ, µ) such that |λ| + |µ| = n and µ consists of only even numbers.These connections gave rise to the term "cubic partitions".
Another natural question about p(n) is how fast it grows.In the beginnig of the 20th century G.H. Hardy and Ramanujan showed the following asymptotics with their celebrated Circle method [8] Hardy and Ramanujan made extensive use of the fact that the generating function (1.1) is almost a modular form.More precisely, , where η(z) is the Dedekind eta function defined on the upper half-plane The function η(τ ) is a modular form of weight 1 2 and satisfies the transformation law, for γ ∈ SL 2 (Z), Here s(h, k) is the Dedekind sum A proof can be found in [3].A couple years later Rademacher [9] perfected the circle method and obtained the exact formula where A k (n) is the Kloosterman sum given by Note that the series (1.6) converges really fast and that the first term recovers the result by Hardy and Ramanujan (1.3).Here we want to find an exact formula of Rademacher type for the cubic partition function a(n), by using a result of Zuckerman [12], extending Rademacher's work and it can be seen as the apotheosis of the classical Circle Method.Using the Circle Method, Zuckerman computed exact formulas for Fourier coefficients of weakly holomorphic modular forms of arbitrary non-positive weight on subgroups of SL 2 (Z) with finite index in terms of the cusps of the underlying subgroup and the negative coefficients of the form at each cusp.We state the relevant results of Zuckerman in Section 2 and prove the exact formula for a(n) in Section 3. Similar to the case above we will obtain as an immediate consequence the asymptotics , n → ∞.
Finally, we are going to prove a conjecture by Banerjee, Paule, Radu and Zeng [5] which predicts the following asymptotic formula for log a(n) The author wishes to thank Kathrin Bringmann and Walter Bridges for suggesting this problem, William Craig and Andreas Mono for sharing their knowledge on modular forms and helpful suggestions, and Johann Franke for helping me verify the results numerically.The author recieved funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 101001179).

Zuckerman's result
The following two sections follow Zuckerman's work [12] closely.Most of the results are completely taken over and translated into more modern language.His method, due to its generality involves numerous technical parameters which appear in the Cirlce Method and do not have obvious meaning without knowing the context they arise in.The interested reader should thus consult [12] for a more detailed account.
Let Γ be a subgroup of SL 2 (Z) of finite index and let F be a weakly holomorphic modular form of weight k = −r, r > 0. Thus, F (τ ) satisfies a transformation equation of the form where ε = ε(a, b, c, d) lies on the unit circle and depends only on the transformation.If c = 0, then c is taken to be positive and we choose the branch of the argument such that Since Γ is of finite index in the modular group we can choose a complete finite system of inequivalent cusps of Γ, which we will denote by P 1 , . . .P s , where In order to treat all our cusps symetrically, we will have to assume that the point at infinity does not belong to our chosen set of inequivalent cusps.This can always be achieved by considering a Γ-equivlant rational point instead.We will obtain a set of Fourier expansions f 1 , • • • , f s corresponding to our set of cusps in the variable (τ − P g ) −1 .Consider now any transformation γ ∈ SL 2 (Z), which must not necessarily belong to our subgroup Γ.We then write where F * (τ ) is now a usual Fourier expansion in τ, obtained from one of the expansions where c g > 0 will be specified below.Note that by choosing the identity matrix in (2.1) we get a usual Fourier expansion F (τ ), which we will later want.Corresponding to each of the cusps P g we can find a transformation in Γ which can be written in the the form We can recover the transformation in the usual form as We then define α g by the equation εe πi 2 = e −2πiαg , where 0 ≤ α g < 1 and ε is taken with respect to τ ′ .We now describe how to find F * (τ ) in terms of the Fourier expansions f g .We therefore choose any transformation γ ∈ SL 2 (Z).Then, a c is a rational point if c = 0 or the point at infinity if c = 0.In any case a c is Γ-equivalent to exactly one of our cusps P g .We can therefore find a transformation which takes P g into a c .We can then express F * (τ ) in terms of the Fourier expansion f g as follows where κ = ±1 is taken as the solution to the following equations p q = κ(ad 1 − cb 1 ), q g = κ(ca 1 − ac 1 ).Furthermore, one obtains This gives the Fourier expansion in spirit of (2.1) at any rational point a c , including the point at infinity.When executing the circle method with these transformation equations there are numerous parameters that appear.First of all consider as usual the Farey fractions h k , gcd(h, k) = 1, k > 0, h ≥ 0.Then, consider the point P = P g − k hcg which is either a rational point or the point at infinity.In any case it is Γ-equivalent to one of our cusps P β , where β = β(h, k, g).There exists a transformation in Γ which takes this point into Associated to this transformation is the parameter σ (g) h,k implicitly defined as the solution of the equations h,k = 0.The remaining technical parameters necessary to state Zuckerman's result are Finally, Zuckerman's result reads now as follows.
Theorem 2.1 (Zuckerman's exact formula).Assume the notation and hypotheses above and denote by I r the Bessel function of order r.If n + α g > 0, then we have

Exact formula for the cubic partition function
We recall that the cubic partition function was defined by the identity which is up to a power of q equal to (η(τ )η(2τ )) −1 .Using the relation we can view η(2τ ) in light of (1.5) as a modular form of weight 1 2 on the subgroup Γ 0 (2) of SL 2 (Z) with multiplier Hence, we find that 2τ) is a weakly modular form of weight −1 with multiplier and thus we can apply Zuckerman's result to find the Fourier coefficients of F (τ ) and therefore a formula of a(n).The group Γ 0 (2) has two inequivalent cusps 0 and 1 2 .We consider the transformations Thus, we find that c 1 = 2 and c 2 = 4. Moreover we can now determine α 1 and α 2 .Since F (τ ) has the multiplier , a short calculation shows that α 1 = α 2 = 7 8 .Since the point at infinity is Γ 0 (2)-equivalent to 1 2 , we obtain that F * (τ )) is given in terms of the Fourier expansion f 2 .To find the exact expression for F * (τ )) we follow Zuckerman's method by choosing a transformation that maps 1  2 to the point at infinity.Obvioulsy Hence, κ = −1 and Furthermore, we consider the point a c = 0 1 , which corresponds to the transformation τ → − 1 τ .In this case the obvious equivalent point is 0 itself and the identity is the corresponding transformation.Thus, in this case We now determine a In the first case we have on the one hand m with ε * = e Comparing coefficients we see that µ −1 = 1 and a −1 = −2e πi 8 .Furthermore, (3.1) implies that the Fourier coefficients are exactly a(n) shifted down by one.Thus, we can recover a(n) through the relation n = m + 1.A similar calculation shows that µ (1) We now continue to determine the remaining parameters only for g = 2 as the Fourier expansion at 1  2 gives us the exact formula.For β(h, k, 2) it is required to find transformations that send P g − k cgh into P β .Let h ′ , h ′′ , h ′′′ be any solutions of for k ≡ 0 (mod 4), for k ≡ 1 (mod 2).Then, the following table gives the remaining parameters for needed to apply Theorem 2.1 Since the series in Theorem 2.1 converges absolutely, the terms can be rearranged and putting everything together, keeping in mind the shift n = m + 1, it follows that where Changing the order of summation one last time we arrive at the following Theorem 3.1.For n ≥ 0 we have the exact formula where l ′ = l if l ≡ 1 (mod 2), l ′ = l if l ≡ 0 (mod 4), l ′ = l 2 if l ≡ 2 (mod 4).It is easy to see that the series converges rapidly and that the term l = 1 dominates the rest of the sum.Thus, we may conclude that For the Bessel function of order ν ∈ N the following asymptotics are known, see [11], where a k (ν) is defined as for some real c i by taking higher order expansions of the involved functions.
We now obtain the following asymptotics.As a final consequence we prove a conjecture by Banerjee, Paule, Radu and Zeng.Proof.We need to shift our approximation from n − 1 8 to n.Thus, write