Abstract
In this paper, we prove that the number of unimodal sequences of size n is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.
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1 Introduction and statement of results
Nicolas [14] and DeSalvo–Pak [7] proved that p(n), the number of partitions of n, is log-concave for \(n\ge 26\) and even \(n \ge 2\). That is,
This follows for large enough n directly from the Hardy–Ramanujan asymptotic expansion for p(n) (see [7, §6.2]), but log-concavity for \(n\ge 26\) requires a careful argument. DeSalvo and Pak proceeded from Rademacher’s exact formula for p(n), together with work of Lehmer [10, 11]. There has since been a flurry of results studying log-concavity and higher Turán inequalities for partition generating functions and weakly holomorphic modular forms (see for example [6, 8, 9, 15]).
In this paper, we consider unimodal sequences, which distinguish themselves from the other examples by their connection to false theta functions. Let u(n) count the number of unimodal sequences of n,
DeSalvo and Pak [7] mentioned that asymptotic formulas for unimodal sequences (or stacks/convex compositions) are not precise enough to prove log-concavity. We overcome this problem, using recent work of the second author and Nazaroglu, proving an exact formula for u(n). Here, we work out precise asymptotic expansions with explicit error term.
Throughout, we write \(f(n)=O_{\le c}(g(n))\) if \(|f(n)|\le c|g(n)|\) and set \(n_0:=100\,000\).
Theorem 1.1
For \(n\ge n_0\), we have
where the constants A, B, C, D, and E are defined in equation (4.16).
Remark
-
(1)
Presumably both the constant in the error term as well as the number of terms may be improved significantly with our methods. We only stop at the power \(n^{-2}\) to obtain log-concavity.
-
(2)
We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects. Indeed, Mauth in [13] follows this approach to prove log-concavity for so-called partitions without sequences considered by the authors in [2]. Another example arises from irreducible characters of certain vertex operator algebras considered by Cesana [5].
The following asymptotic expansion is a direct consequence of Theorem 1.1.
Corollary 1.2
For \(n\ge n_0\), we have
Combined with a numerical check (see the remark in Section 2), log-concavity follows.
Corollary 1.3
For \(n \in \mathbb {N} \setminus \{1,5,7\}\), we have
For \(n\in \{1,5,7\}\), the above difference is negative.
Remark
Note that a combinatorial proof of the log-concavity for integer partitions is still open. Indeed, as there are so many failures of log-concavity (i.e., for odd \(1\le n\le 25\)), this is likely a difficult problem. The number of exceptions for unimodal sequences by contrast is much more manageable, so perhaps it is reasonable to ask for a combinatorial proof in this case.
The paper is organized as follows. In Sect. 2, we recall the exact formula for u(n) from [4] and state some inequalities for the I-Bessel function. The proof of Theorem 1.1 is carried out in Sects. 3 and 4: In Sect. 3, we bound the contribution to u(n) of the terms for \(k\ge 2\) in Theorem 2.1, in Sect. 4, we use the saddle-point method to prove an asymptotic expansion for the term \(k=1\) in Theorem 2.1, finishing the proof of Theorem 1.1.
2 Preliminaries
Recall that the generating function for unimodal sequences is given by (see [1, equation (3.2) and (3.3)])
Remark
Recall that \((q;q)_\infty ^{-1}\) is the partition generating function. The right-hand side of (2.1) allows for a quick computation of u(n) as a convolution of pairs of partitions and the coefficients \(\{0,\pm 1\}\) in the sparse series. This is especially true with programs like Wolfram Mathematica which have the partitions of large order already hard-coded.
In [4], we found the following exact formula forFootnote 1u(n).
Theorem 2.1
([4], Theorem 1.3, negative of the second term) We have
where \(I_\frac{3}{2}\) is the I-Bessel function of order \(\frac{3}{2}\) and \(K_k(n,r)\) is a certain Kloostermann-type sum (in particular \(|K_k(n,r)|\le k\)).
We require the following bounds for the \(I_\frac{3}{2}\)-Bessel function; the proof uses the integral representation of [12, page 172] and [16, equation (6.25)]. See also [3, Lemma 2.2].
Lemma 2.2
We have
3 The terms \(k\ge 2\)
In this section we bound the contribution from \(k\ge 2\) in Theorem 2.1. We begin by estimating the sum on r.
Lemma 3.1
For \(|x|\le 1\) and \(k \ge 2\), we have
Proof
We use for \(0\le y \le \pi \)
to bound
A short calculation shows that for \(|x|\le 1\),
Thus, using \(|K_k(n,r)|\le k\), we have
The claim now follows from the integral comparison criterion. \(\square \)
Now we bound the sum over from all \(k\ge 2\) in Theorem 2.1 as follows.
Lemma 3.2
For \(n\ge n_0\), we have
Proof
By Lemma 3.1, we can bound the left-hand side by
To estimate the integral of the \(I_\frac{3}{2}\)-Bessel function, we apply Lemma 2.2.
First we consider the range
Applying Lemma 2.2 and extending the range of integration, we have
Using the evaluation
this part contributes overall
Next we consider the range
If \(k\ge \frac{\pi \sqrt{24n+1}}{3\sqrt{2}}\), then this range is empty and we have no contribution. Otherwise, Lemma 2.2 gives that the corresponding contribution to (3.1) can be bound against
We now trivially bound this against the exponential
where in the last step we use \(n\ge n_0\). Combining (3.2) and (3.3) proves the lemma. \(\square \)
4 The main term and the proof of Theorem 1.1
The term from \(k=1\) in Theorem 2.1 equals
Using
we obtain that (4.1) equals
We bound the second term in (4.2) for \(n\ge n_0\) as
We split the integral for the first term in (4.2) as
We bound the contribution from the second range as
The rest of this section is devoted to obtaining an asymptotic expansion for
of the form in Theorem 1.1. That is, our error term needs to take the shape
In keeping with the saddle-point method, we eventually let \(x\mapsto cn^{-\frac{1}{4}}x\) for some constant c, so \(dx\mapsto cn^{-\frac{1}{4}}dx\), and thus we obtain the factor \(O(n^{-\frac{5}{4}})\) outside of the integral. Hence, we need to expand the integrand itself up to \(O(n^{-\frac{5}{2}})\). To that end, we begin with the following lemma. Set
Lemma 4.1
For \(|x|\le n^{-\frac{1}{8}}\), we have
Proof
By expanding the Taylor series for \(\sqrt{1-x^2}\) and noting that \(|x|\le n^{-\frac{1}{8}}\le \frac{1}{2}\), we have
Now using that \(e^u\le 1+2|u|\) for \(u\in [-1,1]\) and \(|x|\le n^{-\frac{1}{8}}\), we have
Next, we note that
so that in particular \(|y|\le 1.3\) for \(|x|\le n^{-\frac{1}{8}}\). Hence,
The lemma follows. \(\square \)
Next, we require the Taylor expansions of the other functions in the integrand in (4.4), namely
where \(P_{\text {odd}}(x)\) is an odd polynomial, so does not contribute to the integral, and also
Thus, by Lemma 4.1 and equations (4.5) and (4.6), we see that (4.4) equals
Set \(\lambda _n:=(\frac{\pi }{6\sqrt{2}}\sqrt{24n+1})^{\frac{1}{2}}\). It is not hard to see that
Next we make the change of variables \(x\mapsto \frac{x}{\lambda _n}\) in (4.7) to get
Now note that
where for \(|X|\le 0.5\)
Thus \(2F(\frac{x}{\lambda _n})=O_{\le 0.06}(\frac{x^{10}}{\lambda _n^{10}})\).
We now plug (4.10) into (4.9) and simplify (using a computer algebra system as aide) to get
where \(E_n(x)\) contains only powers \(\lambda _n^{-m}\) with \(m \ge 10\). We bound the integral of \(E_n\) explicitly using a computer algebra system to carry out the integral and arrive at
Now we use that for m even with \(0\le m\le 16\),
We now use for \(w\in \mathbb {R}_{\ge 1}\) and m even with \(0\le m\le 16\),
Define
With (4.8), we have
We use (4.13) in (4.11), simplify and combine with (4.12). Thus (4.11) equals
Finally, we apply to (4.14) the Taylor expansions for \(n\ge n_0\)
Thus (4.14) equals
Define
We now combine (4.15) with Lemma 3.2 and (4.3). From
we conclude Theorem 1.1.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We are grateful to Lukas Mauth for helping us verify log-concavity for the remaining cases \(7\le n\le n_0\) with Wolfram Mathematica. We thank the referee for providing feedback which improved the exposition. The authors have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001179).
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Bridges, W., Bringmann, K. Log concavity for unimodal sequences. Res. number theory 10, 6 (2024). https://doi.org/10.1007/s40993-023-00490-6
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DOI: https://doi.org/10.1007/s40993-023-00490-6