# Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

## Abstract

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.

## 1. Introduction and statement of results

The estimation of partition functions has a long history. Hardy and Ramanujan Reference 14 initiated this subject by proving the asymptotic formula

for the integer partition function The proof relies on the modularity properties of the .*Dedekind-eta function*, with The partition function is connected to the . by the generating function formula: -function

Hardy and Ramanujan’s proof birthed the Circle Method, which is now an important tool in analytic number theory (see, e.g. Reference 37); Hardy and Ramanujan also proposed a divergent series for which Rademacher ,Reference 33 improved to give an exact formula for We now know that this was an early example of a Poincaré series, and this has been generalized in many directions .Reference 5.

The analytic properties of related functions have frequently been studied. For instance, many people investigated the power -th of the Dedekind For -function. one has the famous modular discriminant , Ramanujan’s original conjecture on the growth of the coefficients of . has been hugely influential in the general theory of and automorphic forms -functionsReference 35. It also remained unsolved until it was shown as a consequence of Deligne’s proof of the Weil conjectures Reference 9. More generally, positive powers have been studied in seminal works of Dyson Reference 11 and Macdonald Reference 28, and encode important Lie-theoretic data thanks to the Macdonald identities Reference 28. For negative integral powers, one obtains colored partition generating functions. Specifically, for ,

is the generating function for the number of ways to write the number as a sum of positive integers using colors. We consider the coefficients of for arbitrary positive real although the coefficients no longer have the same combinatorial meaning in counting colored partitions. However, the insertion of a continuous parameter , still gives important information. The most important instance of this is thanks to the famous *Nekrasov–Okounkov formula* Reference 29

Here, is the set of all integer partitions, denotes the number being partitioned by and , is the multiset of hook lengths of 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 Reference 16Reference 17Reference 20 have studied the analytic properties of the Nekrasov–Okounkov formulas. For a fixed the , Fourier coefficient of -thEquation 1.1 is a polynomial in which Heim and Neuhauser conjectured to be unimodal. Partial progress towards this result was recently given by Hong and Zhang ,Reference 22. On the other hand, considering all of the coefficients of Equation 1.1 for a fixed led Heim and Neuhauser 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 Reference 30 and DeSalvo and Pak Reference 10 proved that the partition function is eventually log-concave, specifically, that

for all This result was vastly generalized to a conjecture for certain higher degree polynomials, arising from so-called Jensen polynomials by Chen, Jia, and Wang .Reference 6. That generalized version was later proven by Griffin, Ono, Zagier, and the third author Reference 13.

Expanding in another direction, Bessenrodt and Ono Reference 4 showed that the partition function satisfies mixed additive and multiplicative properties. Specifically, they showed that for all integers with one has ,

Extensions of this result, both rigorous and conjectural, have since been proposed by a number of authors. Alanazi, Gagola, and Munagi Reference 1 gave a combinatorial proof of this result, while Heim and Neuhauser studied the inequality given by replacing the argument by Reference 18. Similar inequalities that mix additive and multiplicative properties for different types of partition statistics have been studied as well Reference 3Reference 8Reference 23.

The first conjecture which we study was made by Chern, Fu, and Tang, who proposed the following analogous conjecture for colored partitions.

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:

for any and This study leads to our first main result. .

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 to mean for a positive function and for all in the domain in which the functions are defined.

Because the last expression in parentheses in Theorem 1.2 is always positive, Conjecture 2 is true for sufficiently large. Note that Conjecture 2 is trivially true if which is why the theorem is sufficient. ,

Additionally, for some we are able to numerically verify that the inequality still holds for small values of and giving the following corollary. ,

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.

## 2. 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 Reference 25 gave an exact formula for the fractional partition function in terms of Kloosterman sums and Bessel functions. The * sum -Kloosterman* is given by

where denotes the inverse of modulo and is the usual Dedekind sum. The only properties we need of this sum are that and We have the following result from .Reference 25.

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 The analysis required for these estimates is continued in the next subsection. .

### 2.2. 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 Reference 31, 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.