Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $\zeta_{\mathfrak{so}{(5)}}$, which is of independent interest.


Introduction and statement of results
1.1.The Circle Method.In analytic number theory and combinatorics, one uses complex analysis to better understand properties of sequences.Suppose that a sequence (c(n)) n∈N 0 has moderate growth and the generating function is holomorphic in the unit disk with radius of convergence 1. Via Cauchy's integral formula one can then recover the coefficients from the generating function for any closed curve C contained in the unit disk that surrounds the origin exactly once counterclockwise.The so-called Circle Method uses the analytic behavior of f (q) near the boundary of the unit circle to obtain asymptotic information about c(n).For instance, if the c(n) are positive and monotonically increasing, it is expected that the part close to q = 1 provides the dominant contribution to (1.1).These parts of the curve are the major arcs and the complement are the minor arcs.To obtain an asymptotic expansion for c(n), one then evaluates the major arc to some degree of accuracy and bounds the minor arcs.Depending on the function f (q), both of these tasks present a variety of difficulties.
In the present paper, we are interested in infinite product generating functions of the form Such generating functions are important in the theory of partitions, but also arise, for example, in representation theory.If a(n) is a "simple" sequence of nonnegative integers and f is "bounded" away from q = 1, then Meinardus [28] proved an asymptotic expression for c(n).Debruyne and Tenenbaum [15] eliminated the technical growth conditions on f by adding a few more assumptions on the a(n), which made their result more applicable.Our main results, Theorems 1.4 and 4.4, yield asymptotic expansions given mild assumptions on a(n) and have a variety of new applications.
1.2.The classical partition function.Let n ∈ N. A weakly decreasing sequence of positive integers that sum to n is called a partition of n.The number of partitions is denoted by p(n).If λ 1 + . . .+ λ r = n, then the λ j are called the parts of the partition.The partition function has no elementary closed formula, nor does it satisfy any finite order recurrence.However, setting p(0) := 1, its generating function has the following product expansion where |q| < 1.In [21], Hardy and Ramanujan used (1.2) to show the asymptotic formula which gave birth of the Circle Method.With Theorem 1.4 we find, for certain constants B j and arbitrarily N ∈ N, Similarly, one can treat the cases for k-th powers (in arithmetic progressions), see [15].
1.3.Plane partitions.Another application is an asymptotic formula for plane partitions.A plane partition of size n is a two-dimensional array of non-negative integers π j,k for which j,k π j,k = n, such that π j,k ≥ π j,k+1 and π j,k ≥ π j+1,k for all j, k ∈ N. We denote the number of plane partitions of n by pp(n).MacMahon [23] proved that Using Theorem 1.4, we recover Wright's asymptotic formula [35]  with ζ the Riemann zeta function.
1.4.Partitions into polygonal numbers.The n-th k-gonal number is given by (k ∈ N ≥3 ) The study of representations of integers as sums of polygonal numbers has a long history.Fermat conjectured in 1638 that every n ∈ N may be written as the sum of at most k k-gonal numbers which was finally proved by Cauchy.Let p k (n) denotes the number of partitions of n into k-gonal numbers.We have the generating function such that r k counts the number of k in (1.4).As a result, the number of representations equals p(n).It is natural to ask whether this can be generalized.The next case is the unitary group su(3), whose irreducible representations W j,k indexed by pairs of positive integers.Note that (see Chapter 5 of [20]) dim(W j,k ) = 1  2 jk(j + k).Like in the case of su(2), a general n-dimensional representation decomposes into a sum of these W j,k , again each with some multiplicity.So analogous to (1.2), the numbers , again with r su(3) (0) := 1.In [31], Romik proved that, as n → ∞, , with explicit constants 3 C 0 , A 1 , . . ., A 4 expressible in terms of zeta and gamma values.Two of the authors [7] improved this to an analogue of formula (1.3), namely, for any N ∈ N 0 , we have as n → ∞, where the constants C j do not depend on N and n and can be calculated explicitly.
The expansion (1.5) with improved error term O N (n − N+1 10 ) and explicit values for A j (1 ≤ j ≤ 4) and C 0 , can also be obtained using Theorem 4.4.This framework generalizes to other groups.For example, one can investigate the Witten zeta function for so (5), which is (for more background to this function, see [25] and [26]) where the ϕ are running through the finite-dimensional irreducible representations of so (5).We prove the following; for the more precise statement see Theorem 5.14.
1 Note that asymptotics for polynomial partitions were investigated in a more general setting by Dunn and Robles in [17]. 2 Explicit asymptotic formulas for p3(n), p4(n), and p5(n) are given in Corollary 5.4.
3 Note that Romik used different signs for the constants in the exponential.
We prove the following.
Theorem 1.3.As n → ∞, we have, for any N ∈ N, exp A 1 n where C, A 1 , A 2 , A 3 , and A 4 are given in (5.17)- (5.19) and the B j can be calculated explicitly.
1.6.Statement of results.The main goal of this paper is to prove asymptotic formulas for a general class of partition functions.To state it, let f : N → N 0 , set Λ := N \ f −1 ({0}), and for q = e −z (z ∈ C with Re(z) > 0), define We require the following key properties of these objects.(P1) Let α > 0 be the largest pole of L f .There exists L ∈ N, such that for all primes p, we have The series L f (s) converges for some s ∈ C, has a meromorphic continuation to {s ∈ C : Re(s) ≥ −R}, and is holomorphic on the line {s ∈ C : Re(s) = −R}.The function L * f (s) := Γ(s)ζ(s + 1)L f (s) has only real poles 0 < α := γ 1 > γ 2 > . . .that are all simple, except the possible pole at s = 0, that may be double.(P3) For some a < π 2 , in every strip σ 1 ≤ σ ≤ σ 2 in the domain of holomorphicity, we uniformly have, for Note that (P1) implies that |Λ \ (bN ∩ Λ)| ≥ L > α 2 for all b ≥ 2. Theorem 1.4.Assume (P1) for L ∈ N, (P2) for R > 0, and (P3).Then, for some M, N ∈ N, are given by4 L (defined in (1.8)), and 0 < β 2 < β 3 < . . .are given by M + N , where M and N are defined in (1.9) and (1.10), respectively.The coefficients A j and B j can be calculated explicitly; the constants A 1 , C, and b are provided in (1.11) and (1.12).Moreover, if α is the only positive pole of L f , then we have M = 1.

Remarks.
(1) Debruyne and Tenenbaum proved Theorem 1.4 in the special case that f is the indicator function of a subset Λ of N.They also assumed that the associated L-function can be analytically continued except for one pole in 0 < α ≤ 1.Our refined assumption (P1) on the set Λ is necessary to bound minor arcs in this more general setup.
(2) The complexity of the exponential term depends on the number and positions of the positive poles of L f .Theorem 4.4 is more explicit and covers the case of exactly two positive poles.This case has importance for representation numbers of su(3) and so (5).
In Section 2, we collect some analytic tools, properties of special functions and useful properties of asymptotic expansions that are heavily used throughout the paper.In Section 3, we apply the Circle Method and calculate asymptotic expansions for the saddle point ̺ n and the value of the generating function G f (̺ n ).In Section 4, we complete the proof of Theorem 1.4, and we also state and prove a more explicit version of Theorem 1.4 in the case that L f has two positive poles (Theorem 4.4).The proofs of Theorems 1.1, 1.2, and 1.3 are given in Section 5; this includes a detailed study of the Witten zeta function ζ so(5) which is of independent interest.

Notation
For β ∈ R, we denote by {β} := β − ⌊β⌋ the fractional part of β.As usual, we set H := {τ ∈ C : Im(τ ) > 0} and E := {z ∈ C : |z| < 1}.For δ > 0, we define where Arg uses the principal branch of the complex argument.For r > 0 and z ∈ C, we set For a, b ∈ R, we let R a,b;K be the rectangle with vertices a ± iK and b ± iK, and we let ∂R a,b;K be the path along the boundary of R a,b;K , surrounded once counterclockwise.For −∞ ≤ a < b ≤ ∞, we denote S a,b := {z ∈ C : a < Re(z) < b}.We also set, for real σ 1 ≤ σ 2 and δ > 0, For k ∈ N and s ∈ C, the falling factorial is (s . For f : N → N 0 , we let P be the set of poles of L * f , and for R > 0 we denote by P R the union of the poles of L * f greater than −R with {0}.We define ) (1.10) We set, with ω α := Res s=α L f (s), (1.12)

Preliminaries
In this section, we collect and prove some tools used in this paper.
2.1.Tools from complex analysis.We require the following results from complex analysis.The first theorem describes Taylor coefficients of the inverse of a biholomorphic function; for a proof, see Corollary 11.2 on p. 437 of [10].
Proposition 2.1.Let φ : B r (0) → D be a holomorphic function such that φ(0) = 0 and φ ′ (0) = 0, with φ(z) =: n≥1 a n z n .Then φ is locally biholomorphic and its local inverse of φ has a power series expansion φ −1 (w) =: k≥1 b k w k , where To deal with certain zeros of holomorphic functions, we require the following result from complex analysis, the proof of which is quickly obtained from Exercise 7.29 (i) in [9].Proposition 2.2.Let r > 0 and let φ n : B r (0) → C be a sequence of holomorphic functions that converges uniformly on compact sets to a holomorphic function φ : B r (0) → C, with φ ′ (0) = 0. Then there exist r > κ 1 > 0 and κ 2 > 0 such that, for all n sufficiently large, the restrictions ) are biholomorphic functions.2.2.Asymptotic expansions.We require two classes of asymptotic expansions.
(2) Let φ be holomorphic on the right half-plane.Then φ ∈ H(R) if there are real numbers If there is no risk of confusion, then we write N , ν j , and a j in the above.The R-dependence of the error only matters if R varies, for instance, if we can choose it to be arbitrarily large.
Note that any sequence g(n) with can be extended to a function g in K(R).Conversely, each function in K(R) can be restricted to a sequence g(n) n∈N satisfying (2.2).In addition, we include functions in K(R) that have asymptotic expansion as in ( 1), but are initially defined only on intervals (r, ∞) for some large r > 0. The reason for this is that it does not matter how the function is defined up to r, and therefore it can always be continued to (0, ∞).If g ∈ K(R) for all R > 0, then we write We use the same abbreviation if φ ∈ H(R) for all R > 0. In this case we write g ∈ K(∞) and φ ∈ H(∞), respectively.In some situations, we write for R ∈ R ∪ {∞} where R might depend on the choice of the function g.If R = ∞, then one may ignore the error O R (x −R ) and use the notation (2.3) instead.We have the following useful lemmas, that can be obtained by a straightforward calculation.
, and h ∈ K(R 2 ).Then we have the following: (1) We have λg ∈ K(R 1 ) and g + h ∈ K(min{R 1 , R 2 }).The exponents ν g+h,j run through ( ).The exponents ν gh,j run through We next deal with compositions of asymptotic expansions with holomorphic functions.
is defined for all x > 0 sufficiently large, and we have h We need a similar result for general asymptotic expansions.

Special functions.
The following theorem collects some facts about the Gamma function.
(1) The gamma function Γ is holomorphic on C \ (−N 0 ) with simple poles in −N 0 .For n ∈ N 0 we have Res s=−n Γ(s) = (−1) n n! . ( The bound also holds for compact intervals sin(πs) .For s, z ∈ C with s / ∈ −N, the generalized Binomial coefficient is defined by .
We require the following properties of the Riemann zeta function.
(1) The ζ-function has a meromorphic continuation to C with only a simple pole at s = 1 with residue 1.For s ∈ C we have (as identity between meromorphic functions) ( (3) Near s = 1, we have the Laurent series expansion . For the Saddle Point Method we need the following estimate.Finally, we require the following in our study of the Witten zeta function ζ so(5) .
Lemma 2.9.Let n ∈ N 0 .The function g : R → R defined as g(u The lemma follows, since g is an even function.

Minor and major arcs
3.1.The minor arcs.For z ∈ C with Re(z) > 0, we define, with G f given in (1.7), Note that we assume throughout, that the function f grows polynomially, which is implicitly part of (P2).We apply Cauchy's Theorem, writing where ̺ n → 0 + is determined in Subsection 3.3.We split the integral into two parts, the major and minor arcs, for any β ≥ 1 The first integral provides the main terms in the asymptotic expansion for p f (n), the second integral is negligible, as the following lemma shows.
Lemma 3.1.Let 1 < β < 1 + α 2 and assume that f satisfies the conditions of Theorem 1.4.Then Sketch of proof.The proof may be adapted from [15,Lemma 3.1].That is, we estimate the quotient, where ||x|| is the distance from x to the nearest integer.We then throw away m-th factors depending on the location of t ∈ [ ̺ β n 2π , 1  2 ].The proof follows [15, Lemma 3.1] mutatis mutandis; key facts are hypothesis (P3) of Theorem 1.4 and that (which follows from [32, Theorem 7.28 (1)]) 3.2.Inverse Mellin transforms for generating functions.We start this subsection with a lemma on the asymptotic behavior of the function Φ f near z = 0.
Lemma 3.2.Let f : N → N 0 satisfy (P2) with R > 0 and (P3).Fix some 0 < δ < π 2 − a. Then we have, as z → 0 in C δ , For the k-th derivative (k ∈ N), we have Here we use (P2), giving that there are no poles of J f (s; z) on the path of integration.By Proposition 2.7 (2), [7, Theorem.2.1 (3)], and (P3), we find a constant c(R, κ) such that, as |v| → ∞, For the second integral in (3.2), applying the Residue Theorem gives since s = 0 is a double pole of J f (s; z).For the last two integrals in (3.2) we have, for some m(I) ∈ N 0 , depending on which vanishes as K → ∞ and thus the claim follows by distinguishing κ = 0 and κ ∈ N.

3.3.
Approximation of saddle points.We now approximately solve the saddle point equations 3) The following proposition provides an asymptotic formula for certain functions.
Then we have the following: (1) There exists a positive sequence (̺ n ) n∈N , such that for all n sufficiently large, φ(̺ n ) = n holds. ( , and the corresponding exponent set In particular, we have ̺ n → 0 + .
Proof.In the proof we abbreviate ν n := ν φ,n and a n := a φ,n .
(1) For n ∈ N, set As φ is holomorphic on the right-half plane by assumption, so are the ψ n .Using (2.1), write where the error satisfies We next show that, for all n sufficiently large, the ψ n only have one zero near w = 1.We argue with Rouché's Theorem.First, we find that, for n sufficiently large, the inequality holds on the entire boundary of B κ(ν 1 ) (1), with 0 < κ(ν 1 ) < 1 2 sufficiently small such that w → 1 − w ν 1 only has one zero in B κ(ν 1 ) (1).By Rouché's Theorem and (3.5), for n sufficiently large ψ n also has exactly one zero in B κ(ν 1 ) (1).We denote this zero of ψ n by w n .It is real as φ is real-valued on the positive real line and a holomorphic function.One can show that ̺ n = ( n a 1 ) 5 Recall that we can consider the sequence ̺n as a function on R + .
(2) We first give an expansion for w n .By Proposition 2.2, there exists κ > 0, such that for all n sufficiently large and all z ∈ B κ (0), the inverse functions ψ −1 n of ψ n are defined and holomorphic in B κ (1).Using this, we can calculate w n , satisfying ψ n (w n ) = 0.For this, let h n (w) := ψ n (w + 1) − ψ n (1).
We have h n (0) = 0, and we find, with Proposition 2.1, where the b m can be explicitly calculated.First, ψ n (1) m (m ∈ N 0 ) have expansions in n by (3.4) and Lemma 2.4.They have exponent set 2≤j≤N φ (1 − . We find, for k ∈ N, Again by Lemma 2.4, and (3.6), ψ . By Lemma 2.4 we have the following expansion in n . By the formula in Proposition 2.1, the b m (n) are essentially sums and products of terms ψ ′ n (1) −1 and ψ for n → ∞, one has, for M sufficiently large and not depending on n, Now, as w n ∼ 1, we conclude the theorem recalling that ̺ n = ( n a 1 ) We next apply Proposition 3.3 to −Φ ′ f .For the proof one may use Lemma 3.2 with k = 1.
Corollary 3.4.Let ̺ n solve (3.3).Assume that f : N → N 0 satisfies the conditions of Theorem 3.4.The major arcs.In this subsection we approximate, for some 1 + α 3 < β < 1 + α 2 , where α is the largest positive pole of L f .The following lemma can be shown using [15, §4].Then we have where The following lemma shows that the first term in Lemma 3.5 dominates the others; its proof follows with Lemma 2.5, Lemma 3.2, and Corollary 3.4 by a straightforward calculation.Lemma 3.6.Let k ≥ 2 and assume the conditions as in Lemma 3.5.Then we have where the η j run through We next use Lemma 2.5 and Corollary 3.4 to give an asymptotic expansion for G f (̺ n ).
Lemma 3.7.Assume that f : N → N 0 satisfies the conditions of Theorem 1.4.Then, we have where ν F,j run through (the inclusion follows by Corollary 3.4) Note that, again by Lemma 2.5 and Lemma 3.2, we obtain We split the sum in (3.7) into two parts: one with nonpositive ν F,1 , . . ., ν F,M , say, and the one with positive ν F,j < R α+1 .Note that M is bounded and independent of R. Exponentiating (3.7) yields Note that the positive ν F,j run through (3.8) with −∞ replaced by 0. By Lemma 2.4, we have exp for some K ∈ N and with exponents ε j running through N .Recall that, by Corollary 3.4, we have . By Lemma 2.3 (2), we obtain, for some N ∈ N, B j ∈ C, and δ j running through M + N , Setting C j := a F,j for 1 ≤ j ≤ M , the lemma follows.
Another important step for the proof of our main theorem is the following lemma.
Hence, for some 1 ≤ M ≤ N ̺ , we obtain As α ∈ P R , this is a subset of N , so the above exponents are given by N , proving the lemma.
The following corollary is very helpful to prove our main theorem.Corollary 3.9.Let f : N → N 0 satisfy the conditions of Theorem 1.4.Then we have  ) , and the ν j run through In particular, we have ν 1 = α+2 2(α+1) .We prove the following lemma.Lemma 4.2.Assume that f satisfies the conditions of Theorem 1.4 and that L f has only one positive pole α.Then we have where To show the lemma, we need an expansion for ̺ n .We have, by (3.3) and again by Lemma 3.2, By Corollary 3.4, we have an expansion for ̺ n with an error o(1).We iteratively find the first terms.By Corollary 3.4 we have ̺ n ∼ a as small as possible.One finds that and hence Plugging (4.3) into Φ f leads, by (4.1), to As a result, using (4.3), we conclude the claim.
We are now ready to prove Theorem 1.4.
4.2.The case of two positive poles of L f .If α > 0 is the only positive pole of L f , then we can calculate the single term in the exponential in the asymptotic of p f (n) explicitly, by Theorem 1.4.
In this subsection we assume that L f has exactly two positive simple poles, α and β.In this case, Lemma 3.2 with k = 1 gives In the next lemma, we approximate the saddle point in this special situation.
Lemma 4.3.Let f satisfy the conditions of Theorem 1.4.Additionally assume that L f has exactly two positive poles α and β that satisfy ℓ+1 ℓ β < α ≤ ℓ ℓ−1 β for some ℓ ∈ N, where we treat the case ℓ = 1 simply as 2β < α.Then there exists 0 < r ≤ R α+1 such that for some constants K j independent of n and c 3 as in (4.4).In particular, we have Proof.By Corollary 3.4, the exponents of ̺ n that are at most 1 are given by combinations with j ∈ N and m ∈ N 0 .A straightforward calculation shows that ℓ+1 ℓ β < α ≤ ℓ ℓ−1 β if and only if for all 1 ≤ j ≤ ℓ + 1 but not for j > ℓ + 1. Together with the error term induced by Corollary 3.4, (4.5) follows.Assuming ℓ ≥ 5, K 1 to K 5 and the term c 3 (α+1)n can be determined iteratively.
We are now ready to prove asymptotic formulas if L f has exactly two positive poles.
Theorem 4.4.Assume that f : N → N 0 satisfies the conditions of Theorem 1.4 and that L f has exactly two positive poles α > β, such that ℓ+1 ℓ β < α ≤ ℓ ℓ−1 β for some ℓ ∈ N. Then we have with and for all k ≥ 3 Here, C and b are defined in (1.11) and (1.12), the ν j run through M + N , the K j are given in Lemma 4.3, and c 1 , c 2 , and c 3 run through (4.4).
Proof.Assume that g : N → C has an asymptotic expansion as n → ∞ and denote by [g(n)] * the part with nonnegative exponents.With Lemmas 3.2 and 4.1 we obtain, using that L f has exactly two positive poles in α and β, with the δ j running through M. With the Binomial Theorem and Lemma 4.3, we find Applying the Multinomial Theorem to (4.7) gives Similarly, we have −β m 0≤j 1 ,j 2 ,...,j ℓ ≤m Finally, we obtain, with Lemma 4.3, Combining (4.8), (4.9), and (4.10), we find that where 1 .
Note that we have by definition of c 1 , c 2 (see (4.4)), K 1 , and K 2 (see Lemma 4.3), , which gives (4.6).Hence we indeed obtain, as n → ∞, for suitable where the ν j run, as in Theorem 1.4, through M + N .This proves the theorem.
5. Proofs of Theorems 1.1, 1.2, and 1.3 We require the zeta function associated to a polynomial P , Z (s) := n≥1 1 P (n) s with P (n) > 0 for n ∈ N. In particular, we consider P = P k , where The following lemma ensures that all the P k satisfy (P1) with L arbitrary large.
Lemma 5.1.Let k ≥ 3 be an integer and let For every prime p, we have We next show that (P2) and (P3) hold.
(1) The function Z P k has a meromorphic continuation to C with at most simple poles in 1 2 − N 0 .The positive pole lies in s = 1  2 .
(2) We have Proof.(1) The meromorphic continuation of Z P k to C follows by [27,Theorem B].By [27, Theorem A (ii)] the only possible poles (of order at most one) are located at This proves the existence of a pole in s = ( Proof. (1) Since the roots of P k are not in R ≥1 , we may use [27,Theorem D] to obtain that For the derivative, one applies [27, Theorem E] yielding ( , the result follows as the sum has residue 1 2 at s = 1 2 by equation ( 16) of [27].
The previous three lemmas are used to prove Theorem 1.1.
Proof of Theorem 1.1.We may apply Theorem 1.4 as Lemma 5.1 and Proposition 5. We consider some special cases of Theorem 1.1.
the constants B j are explicitly computable,

Lemma 2 . 8 .
Let µ n be an increasing unbounded sequence of positive real numbers, B > 0, and P a polynomial of degree m ∈ N 0 .Then we have µn −µn P (x)e −Bx 2 dx = ∞ −∞ P (x)e −Bx 2 dx + O B,P µ

] 3 (
* is the part of the expansion of c 1 α̺ α n involving nonnegative powers of n, i.e., for m ≥ 2 in the sum on the right of (4.7) we can ignore the term c

1 3
2 ensure that conditions (P1)-(P3) are satisfied.Hence, one obtains an asymptotic formula for p k (n).The constants occurring in Theorem 1.4 are computed using (1.11), (1.12), and Proposition 5.3.That the exponential consists only of the term A 1 n follows by Theorem 1.4, since Z P k (s) has exactly one positive pole, lying in s =1  2 .Note that we are allowed to choose L and R arbitrarily large due to Lemma 5.1 and Proposition 5.2 (1).