Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov-Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.


Introduction and Statement of Results
The estimation of partition functions has a long history. Hardy and Ramanujan [14] initiated this subject by proving the asymptotic formula for the integer partition function p(n). The proof relies on the modularity properties of the Dedekind-eta function, η(τ ) := q 1 24 n≥1 (1 − q n ) with q := e 2πiτ . The partition function is connected to the η-function by the generating function formula: n≥0 p(n)q n = q 1 24 η(τ ) .
Hardy and Ramanujan's proof birthed the Circle Method, which is now an important tool in analytic number theory (see, e.g. [35]); Hardy and Ramanujan also proposed a divergent series for p(n), which Rademacher [31] improved to give an exact formula for p(n). We now know that this was an early example of a Poincaré series, and this has been generalized in many directions [5]. The analytic properties of related functions have frequently been studied. For instance, many people investigated the α-th power η(τ ) α of the Dedekind η-function. For α = 24, one has the famous modular discriminant ∆(τ ). Ramanujan's original conjecture on the growth of the coefficients of ∆(τ ) has been hugely influential in the general theory of L-functions and automorphic forms [33]. It also remained unsolved until it was shown as a consequence of Deligne's proof of the Weil conjectures [9]. More generally, positive powers have been studied in seminal works of Dyson [11] and Serre [34], and encode important Lie-theoretic data thanks to the Macdonald identities [27]. For negative integral powers, one obtains colored partition generating functions. Specifically, for k ∈ N, 1 η(τ ) k =: q − k 24 n≥0 p k (n)q n is the generating function for the number of ways to write the number n as a sum of positive integers using k colors. We consider the coefficients of η(τ ) −α for arbitrary positive real α, although the coefficients no longer have the same combinatorial meaning in counting colored partitions. However, the insertion of a continuous parameter α still gives important information. The most important instance of this is thanks to the famous Nekrasov-Okounkov formula [28] λ∈P q |λ| Here, P is the set of all integer partitions, |λ| denotes the number being partitioned by λ, and H(λ) is the multiset of hook lengths of λ. This formula arose from their study of supersymmetric gauge theory and a corresponding statistical-mechanical partition function, and is related to random partitions.
In several recent papers, Heim, Neuhauser, and others [20,16,17] have studied the analytic properties of the Nekrasov-Okounkov formulas. This led them to make their conjecture below. In order to explain the context of their conjecture further, we now discuss a related chain of partition inequalities which has recently received attention. Independent work by Nicolas [29] and DeSalvo and Pak [10] proved that the partition function p(n) is eventually log-concave, specifically, that p(n) 2 − p(n − 1)p(n + 1) ≥ 0 for all n > 25. This result was vastly generalized to a conjecture for certain higher degree polynomials, arising from so-called Jensen polynomials by Chen, Jia, and Wang [6]. That generalized version was later proven by Griffin, Ono, Zagier, and the third author [13].
Expanding in another direction, Bessenrodt and Ono showed [4] that the partition function satisfies mixed additive and multiplicative properties. Specifically, they showed that for all integers a, b ≥ 2 with a + b > 8, one has Extensions of this result, both rigorous and conjectural, have since been proposed by a number of authors. Alanazi, Gagola, and Munagi [1] gave a combinatorial proof of this result, while Heim and Neuhauser studied the inequality given by replacing the argument a+b by a+b+m−1 [18]. Similar inequalities that mix additive and multiplicative properties for different types of partition statistics have been studied as well [3,8,22].
The first conjecture which we study was made by Chern, Fu, and Tang, who proposed the following analogous conjecture for colored partitions.
Remark. As noted in a paper by Sagan [32], Conjecture 1 is equivalent for k ≥ 3 to the log-concavity of p k (n).
Heim and Neuhauser conjectured a continuous extension.
We study these conjectures with the aim of writing down explicit results which may be of use for the future of related inequalities. To do this, we consider the sign of the general difference of products: p α 1 (n 1 )p α 2 (n 2 ) − p α 3 (n 3 )p α 4 (n 4 ), for any n 1 , n 2 , n 3 , n 4 ∈ N and α 1 , α 2 , α 3 , α 4 ∈ R + . This study leads to our first main result. Theorem 1.1. Fix α 1 , α 2 , α 3 , α 4 ∈ R + , and consider the inequality Without loss of generality, we assume n 1 ≥ n 2 and n 3 ≥ n 4 . If n 3 = o(n 1 ), the inequality is true for n 1 sufficiently large. Conversely, if n 1 = o(n 3 ), the inequality is false for n 3 sufficiently large.
Theorem 1.1 can be made explicit. This is applied below to prove the conjectures of Chern-Fu-Tang and Heim-Neuhauser. Here and throughout the paper, we use the notation f (x) = O ≤ (g(x)) to mean |f (x)| ≤ g(x) for all x in the relevant domain. Theorem 1.2. Fix α ∈ R ≥2 , and let n, ℓ ∈ N ≥2 with n > ℓ + 1. Set N := n − 1 − α 24 and L := ℓ − α 24 , we suppose L ≥ max{2α 11 , 100 α−24 }. Then we have Remark. Theorem 1.2 still holds for L ≥ 2 · 24 11 if α = 24 even though the bound on L is undefined for this value of α.
Because the last expression in parentheses in Theorem 1.2 is always positive, Conjecture 2 is true for ℓ sufficiently large. Note that Conjecture 2 is trivially true if n = ℓ + 1, which is why the theorem is sufficient. Corollary 1.3. Conjecture 2 is true for ℓ ≥ max{2α 11 + α 24 , 100 α−24 + α 24 }. Additionally, for some α ∈ N we are able to numerically verify that the inequality still holds for small values of ℓ and n, giving the following corollary. Corollary 1.4. Conjecture 1 is true. In particular, p 2 (n) is log-concave for n ≥ 6, and p k (n) is log-concave for all n and k ∈ N ≥3 .
Remark. Although Theorem 1.2 turns Conjecture 1 into a finite computer check, the number of cases that must be checked to give Corollary 1.4 is very large. Thus, direct brute force computer checks are not sufficient. Faster methods of verifying such inequalities are described in the proofs below. These may be useful in future partition investigations.
The remainder of the paper is organized as follows. We review basic ingredients needed for the proofs of our theorems in Section 2. These proofs are then carried out in Section 3. In Section 4, we provide lemmas and discussion needed for our computations in order to prove our corollary. We then conclude in Section 5 with some ideas for further work.

Preliminaries
Here, we review the key ingredients for the proof of our results.
2.1. Exact formulas for partitions. In a recently submitted paper, Iskander, Jain, and Talvola [24] gave an exact formula for the fractional partition function in terms of Kloosterman sums and Bessel functions. The α-Kloosterman sum is given by whereh denotes the inverse of h modulo k and s(h, k) is the usual Dedekind sum. The only properties we need of this sum are that A 1,α (n, m) = 1 and |A k,α (n, m)| ≤ k. We have the following result from [24] with β := ⌊ α 24 ⌋. Theorem 2.1. For all α ∈ R + and n > α 24 , we have This provides an exact formula for the numbers we wish to estimate. The difficulty lies in providing precise estimates for the error terms after truncating the series to a finite number of terms in the sum on k. The analysis required for these estimates is continued in the next subsection.

Explicit bounds for Bessel functions.
In order to make the exact formula in Theorem 2.1 useful for our purposes, we need strong estimates on the Bessel functions. Although many Bessel function estimates are standard and a whole asymptotic expansion is known [30, equation 10.40.1], we were unable to find existing bounds suitable for our purposes. Thus, we describe some basic estimates here and sketch our proofs for them. In particular, we prove the following.
(2) We consider first the integral from 0 to 1 which is on the left-hand side of (2.2). Now write, using Taylor's Theorem, where for some ξ ∈ [0, 1], We can bound this by We first consider the contribution from the first 4 terms in (2.3). These are Evaluating the first integral yields the main term. The second integral in (2.4) contributes Using that [30, equation 8.10 one can show that this term overall contributes at most We next estimate the term with C κ (u) in (2.3). Bounding the integral from 0 to 1 against the integral from 0 to ∞, this term can be bounded against Finally, the contribution from the integral from −1 to 0 can be bounded by (estimating the integrand trivially) Overall we obtain By elementary bounds [30, equation 5.6.1], we find Combining the above now easily gives the claim.

Proofs of the theorems
Proof of Theorem 1.1. We use the exact formula from Theorem 2.1 and note that the dominant term comes from m = 0 and k = 1. The claim then follows from Proof of Theorem 1.2. We again use the exact formula from Theorem 2.1 and note that the dominant term comes from m = 0 and k = 1 in each expansion. We see that this main term in p α (n − 1)p α (ℓ + 1) − p α (n)p α (ℓ) is To rewrite the Bessel functions as sums of powers of N and L, we use Lemma 2.2 (2) to obtain where Thus we have As alluded to above, we wish to expand the main term into sums of powers of N's and L's, so we change the (N + 1)'s above into N's. First, note that by using Taylor's Theorem, there exist D * * A,2 , D * A,2 , and D A,2 such that Explicitly bounding in the interval [0, 1], Taylor's Theorem further tells us that for A > 0, Using this, one can prove the bounds In addition to rewriting the powers of N + 1 in (3.4) as powers of N, we also want to replace the √ N + 1 in e π √ 2α 3 (N +1) by a function of √ N instead. This is needed in order to compare the two summands of our main term. To do so, we show that for some D α,3 (N) ∈ R, To prove the second equality and determine a bound for for some ξ ∈ [0, 1 √ N ], so we need to bound G (4) (ξ) on this interval. Using some basic calculus, one finds that for x ∈ [0, 1] Moreover, g ′ (x) > 0 on [0, 1], so we have that Combining these estimates on g(x) and its derivatives, one sees that Assuming that N ≥ 2α 11 , we obtain in this region We now want to write e π √ 2α 3 (N +1) where we need A α,1 , A α,2 , and A α,3 explicitly and a bound on B α (N). To find the A α,j 's, we use (3.5) to rewrite the powers of N + 1 on the left-hand side in terms of powers of N and employ (3.7) to rewrite the exponential term. In doing so and comparing powers of N on each side, one concludes that Below, we need bounds on each of these quantities. By the triangle inequality and the bounds in (3.3), one can find that We next bound the error term B α (N). We can solve for Bα(N ) Similar to finding the A α above, we use (3.7) and (3.5) to expand this as 1 + πα .
From here, we can simply expand out this product. Because all terms with power greater than N −2 are already subtracted, we can factor this out of everything remaining and bound the absolute value of what is left using (3.6), (3.3), (3.8), and the fact that N ≥ 2α 11 to obtain a bound on B α (N), namely Now, using (3.2) and (3.9), (3.1) becomes We write the expression in the outer parentheses as for some function f (N, L), where the equality follows from (3.10). We wish to bound . Note that f (N, L) can be easily calculated from (3.13) by simply expanding the products. We do not write out every term but instead explain how to bound just a couple of the terms. For example, the next largest term (asymptotically) that arises when computing f (N, L) is We can bound the first product above using (3.10) and (3.3) as where the inequality follows from basic calculus. Then when f (N, L) is multiplied by , this term can be bounded in absolute value using (3.11) and (3.3) by (using All of the exact terms that arise in f (N, L) (i.e., those not involving the error terms D α,1 (N), D α,1 (L), B α (N), or B α (L)) can be bounded in this way. As for the remaining terms of f (N, L), one can use that N is decreasing as a function of N to bound it above by where the second inequality holds for L ≥ 2α 11 ≥ 2 12 by calculus. This allows us to bound all of the other terms of f (N, L). For example, one of the remaining terms is Hence, when f (N, L) is multiplied by N , this term can be bounded utilizing (3.15), (3.12), and (3.3) by (using that L ≥ 2α 11 ) All of the other terms are bounded in a similar manner (using the fact that N ≥ L).
Combining these bounds and using that α ≥ 2, we obtain Combining (3.13), (3.14), and (3.16), the main term can be written as We now need to bound the remaining terms in the expansion of p α (n−1)p α (ℓ+1)−p α (n)p α (ℓ) coming from Theorem 2.1. To estimate the contribution from k ≥ 2, we bound, for X ∈ R + F κ (X) := k≥2 I κ X k .
Note that F κ (X) is monotonically increasing because I κ is. We estimate the first ⌊X⌋ − 1 terms using Lemma 2.2 (1) For the second bound, we are using that √ ke To bound the remaining terms of F κ (X), we use Lemma 2.2 (3) to conclude that Combining (3.18) and (3.19) and using basic calculus and the fact that κ ≥ 2, we determine that (3.20) Using this, we may bound the non-main terms as follows. We first bound the non-main terms corresponding to p α (n − 1)p α (ℓ + 1). For this, we consider terms with (i) k 1 , k 2 ≥ 2, (ii) terms with k 1 ≥ 2 and k 2 = 1, (iii) terms with k 1 = 1 and k 2 ≥ 2, (iv) terms with k 1 = k 2 = 1 and m 1 ≥ 1, and finally (v) terms with k 1 = k 2 = 1, m 1 = 0, and m 2 ≥ 1. As is done above, we do not write out all of these sums but instead just illustrate how to bound the terms corresponding to (ii). We get an upper bound of where the inequality follows from the monotonicity of F and I. Using (3.20) and Lemma 2.2 (1), we can further bound (3.21) by Now, we claim that p α (n) ≤ e π √ 2αn 3 . This follows from virtually the same proof as in the α = 1 case; see [2] for details. Using this bound on p α (n), the fact that √ L ≤ √ L + 1 ≤ √ L + 1, and factoring out the terms outside of parentheses in (3.17), we see that (3.22) is at most It is easy to check that this is decreasing in N for N ≥ 2α 11 ≥ 2 12 , so utilizing that N ≥ L, we get an upper bound on (3.23) of 804 25 Similarly, this term is decreasing in L for L ≥ 2α 11 ≥ 2 12 , so we can plug in L = 2α 11 to obtain a bound of 3216 25 One can bound the exponent by − 17 10 α 6 , and estimate β + 1 ≤ 13α 24 . The resulting term is 11323 300 √ 2α 24 e − 17 10 α 6 .
Using that this expression is decreasing in α, we get an upper bound by plugging in α = 2 yielding a numerical answer of 51 · 10 −40 . One can similarly bound all of the other error terms; the only significant departure occurs when bounding terms corresponding to (iv) and (v), where a term ( α 24 − 1) − 1 4 occurs. It is here that we need to use the bound L ≥ 100 to ensure that our argument of L is large enough. The proof in this case is still similar in nature and is omitted. The largest of the errors that arise from these cases is 10 −4 , which when combined with the error of (3.17) gives the statement of the theorem.

Proofs of the corollaries
As alluded to after the statement of Theorem 1.2, all of the terms in the expansion of p α (n − 1)p α (ℓ + 1) − p α (n)p α (ℓ) are positive, so Corollary 1.3 follows. Thus, only the proof of Corollary 1.4 remains. In order to prove this for a fixed value of k, we only need to compute the ratios p k (ℓ+1) p k (ℓ) up to ℓ ≥ ⌈2k 11 + k 24 ⌉ (note that for k ∈ N, max{2k 11 , 100 k−24 } = 2k 11 ) and see that they are decreasing, except for k = 2. For k ∈ {2, 3}, we can do this directly, but for k ∈ {4, 5}, we need to find a way to make the computation more efficient and store less memory; we provide the necessary details to do so in the following subsection. At the end of the section, we describe how proving the result for k ∈ {2, 3, 4, 5} is sufficient to prove Corollary 1.4. Namely, in Proposition 4.3, we show that p k (n) is log-concave for k ∈ {3, 4} and point out that convolution of log-concave sequences is log-concave, which shows that the same property also holds for k ∈ N ≥3 . 4.1. Tools needed for the proof of Corollary 1.4. To verify the initial cases of the conjecture, of course a direct approach using Rademacher sums, recursive formulas, or by convoluting the partition generating function can be used. However, due to the large number of cases that have to be checked (for example, for p 4 (n) we need to compute all values with n ≤ 2 · 4 11 + 6 ), these direct methods are not sufficient. An approach with lower time and memory requirements is thus essential in practice. As a result, we begin by defining sequences that approximate our partition numbers well enough to prove the lemma and which also require less memory and speed to compute. To do this, let d = (d j ) ∞ j=1 be a sequence of positive integers d j ≤ j, and for n ∈ N recursively define We also set the negative values to be zero: p ± k,d (n) = 0 for n ≤ −1 Lemma 4.1. For n ∈ N, we have p − k,d (n) ≤ p k (n) ≤ p + k,d (n).
Proof. Using (3) of [19], we find that for n ≥ 1, we have p k (n) = k n n ℓ=1 σ(ℓ)p k (n − ℓ). (4.1) We prove the claimed inequalities by induction. The base case, n = 0, is trivial as p k (0) = 1. Assume inductively that for every 0 ≤ m < n the claim holds. Note that both p k (n) and σ(n) are non-negative for all n ∈ N. Hence the inequality d n ≤ n and the inductive hypothesis p k (n − ℓ) ≥ p − k,d (n − ℓ) imply that This gives the first inequality.
To obtain the upper bound, we note that p k (n) is increasing, and hence Therefore p k (n) ≤ k n dn ℓ=1 σ(ℓ)p k (n − ℓ) + k n p k (n − d n − 1) n 2 .
Remark. In the special case d n = n, one has p − k,d (n) = p k (n) = p + k,d (n) by (4.1). In order to compute p ± k,d (n) for every 1 ≤ n ≤ N, the number of steps required is O ( 1≤n≤N d n ). Thus the number of steps to compute p k (n) directly (i.e., d n = n) is O(n 2 ). If d n ≪ n δ , then the number of steps to compute the lower and upper bounds p ± k,d (n) is ≪ N 1+δ . Moreover, in order to compute p + k,d (n) with a computer one only needs to keep d n = O(n δ ) numbers in memory (this is O(n) in the special case d n = n). Hence computing the sequences p ± k,d (n) is better than p k (n) both in the speed of the calculation and in the memory requirement.
These numbers grow very quickly. Thus, if d is chosen appropriately so that n is small in comparison with an exponential of the shape e 2πc k ( √ n− √ n−dn−1) , (c k > 0) Remark. Since the sequences p ± k,d (n) are generally faster to compute than p k (n) and have a smaller memory requirement, in practice Lemma 4.2 gives us an easier and faster criterion to check to numerically verify the log-concavity of p k (n) for 1 ≤ n ≤ N for some fixed N. similar to how we have done here, which may show when exactly the inequalities begin to hold (see for example [12,23]).