Abstract
The zeros of the nth D’Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind \(\eta \)-function. In this paper, we prove that these coefficients are non-vanishing for \(\vert x \vert > \kappa \, (n-1)\) and \(\kappa \approx 9.7225\). Numerical computations imply that 9.72245 is a lower bound for \(\kappa \). This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions.
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1 Introduction
The zero distribution of the D’Arcais polynomials [1, 3, 16]
with initial value \(P_{0}^{\sigma _1 }\left( x\right) =1\), plays an important role in the study of the vanishing properties of the Fourier coefficients of complex powers of the Dedekind \(\eta \)-function [11, 17, 18]. Let \(\sigma _{d}\left( n \right) =\sum _{\ell \mid n}\ell ^{d}\) and \(q=\textrm{e}^{2\pi \textrm{i}z}\), where z is in the upper complex half plane. The identity
encodes that for \(P_{n}^{\sigma _{1}}\left( x_{0}\right) =0\), the nth coefficient of \(q^{ x_{0}/24}\eta \left( z\right) ^{-x_{0}}\) vanishes. The famous conjecture of Lehmer [13] claims that \(x_{0}\) is never equal to \(-24\).
Let \(x_0 \in {\mathbb {Z}} {\setminus } \left\{ 0\right\} \) be even. Then, Serre [18] proved that the sequence \(\{P_n^{\sigma _{1}}(-x_0)\}_n\) is lacunary if and only if \(x_0 \in S_{\mathop {\textrm{even}}}:=\{2,4,6,8,10, 14,26\}\). Based on the records and results (connection to Maeda’s conjecture) provided in [11] we speculate that if \(n \in {\mathbb {N}}\) exists such that \(P_n^{\sigma _1}(-x_0) =0\) for \(x_0 \ne 0\) even, then \(x_0 \in S_{\mathop {\textrm{even}}}\).
Kostant [12], Theorem 4.28, by using techniques from simple complex Lie algebras, proves that \(P_{n}^{\sigma _{1}}\left( 1-m ^{2}\right) \ne 0\) for \(m\ge n\) and \(m >1\). Han [2] deduced from the Nekrasov–Okounkov hook length formula [15] that this already holds true for real \(\left| x\right| \ge n^{2}-1\).
Numerical computations (Fig. 1) indicate that a linear bound of the form \(\left| x\right| \ge \kappa \left( n-1\right) \) is actually in place. Suppose \(\kappa >0\) exists, such that for all complex x we have the following:
Indeed, we have shown [5] that this holds true for \(\kappa \approx 10.82\). On the other hand, Fig. 1 and Table 1 in [8] give the obstruction that \(\kappa \) has to be larger than 9.72245.
In this paper, we prove a theorem on families of polynomials \(\{P_n^{g,h}(x)\}_n\) associated with two arithmetic functions g and h, controlling the growth (Theorem 2.1). As an application, we obtain upper bounds on the possible zeros. This includes Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials related to partitions’ and plane partitions’ numbers, and overpartitions’ numbers. As our most significant application and result we have that (1) holds true for \(\kappa \approx 9.7225\).
2 Statement of the main results
Let \(g:{\mathbb {N}}\rightarrow {\mathbb {C}}\) be normalized with \(g\left( 1\right) =1\) and \(\sum _{n=1}^{\infty } g(n) \, q^{n}\) regular at \( q=0\). Further, let \(h:{\mathbb {N}}\cup \left\{ 0\right\} \rightarrow {\mathbb {C}} \), \( h\left( 0 \right) =0 \) and \( h \left( n\right) \ne 0\) for all \(n\in {\mathbb {N}}\). Let
We define
for \(n\ge 1\) and initial value of \(P_0^{g,h}(x)=1\). Then, \(P_{n}^{\sigma _{1},\mathop {\textrm{id}}}\left( x\right) \) are the D’Arcais polynomials and \( P_{n}^{\sigma _{1},\mathop {\textrm{id}}} \left( x+1\right) \) the Nekrasov–Okounkov polynomials [15]. Further, note that \(\text {p}\left( n\right) =P_n^{\sigma _1, \mathop {\textrm{id}}}(1)\) provides the partition function and \(\mathop {\textrm{pp}}(n)=P_n^{\sigma _2, \mathop {\textrm{id}}}(1)\) the plane partition function (we refer to Sect. 5.2.2). In [10], it has been found that for \(g(n)=n \) and \(h(n) >0\) for \(n\ge 1\), we obtain orthogonal polynomials. The case \(h(n)=1\) leads to Chebyshev polynomials \(U_{n}\left( x\right) \) of the second kind and the case \(h(n)=n\) leads to 1-associated Laguerre polynomials \(L_{n}^{1}\left( x\right) \). We have \(P_{n}^{\mathop {\textrm{id}},\mathop {\textrm{id}}}\left( x\right) =\frac{x}{n}L_{n-1}^{1}\left( -x\right) \) and \(P_{n}^{\mathop {\textrm{id}},1}\left( x\right) =xU_{n-1}\left( \frac{x}{2}+1\right) \) (see also [4, 8]).
Theorem 2.1
Let \(c\in {\mathbb {C}}\) and let \(R>0\) be the radius of convergence of
Let \(q_{1}=\sup \left\{ 0\le q<R:G_{c}\left( q\right) <1\right\} \). For \(0<q<q_{1 }\le R\) let
Then for all \(n \in {\mathbb {N}}\) and \(x\in {\mathbb {C}}\) with \(\left| x \right| \ge {\kappa } \, H\left( n-1\right) \), we have:
An application of the triangle inequality shows the following:
Corollary 2.2
Assumptions as in Theorem 2.1. Then
if \(\left| x \right| \ge {\kappa } H\left( n-1\right) \), \(n\ge 1\).
From Corollary 2.2 we obtain:
Corollary 2.3
Assumptions as in Theorem 2.1. We have \(P_{n}^{g,h}\left( x\right) \ne 0\) for \(\left| x\right| \ge \kappa \, H \left( n-1\right) \), \(n\ge 1\).
Proof
We have \(P_{1 }^{g,h}\left( x\right) = \frac{x}{h\left( 1\right) }\ne 0\) for \(x\ne 0\). By induction, and also as we have shown for \(\left| x\right| \ge \kappa H \left( n-1\right) \) in the preceding
and \(\left( 1-G_{c}\left( q\right) \right) \left| x\right| -\left| c \right| H \left( n-1\right) \ge \frac{H \left( n-1\right) }{q }>0\) for \(\left| x\right| \ge \kappa H \left( n-1\right) \), \(n\ge 2\). \(\square \)
Corollary 2.4
Let g, h be normalized non-negative arithmetic functions with h positive. Suppose that \(\sum _{n=1}^{\infty } g\left( n\right) \, q ^{n}\) is regular at \(q =0\). Suppose that \(\frac{1}{x} P_n^{g,h}(x)\) is a Hurwitz polynomial and let \(\kappa ^{g}\) be given (Corollary 2.3). Then, there exists a zero \(\alpha _n^{g,h}\) of \(P_n^{g,h}(x)\) such that
A polynomial with real and non-negative coefficients is called Hurwitz polynomial if all zeros have negative real part. D’Arcais polynomials, up to the zero \(x=0\), possess this property for \(n \le 1500\) (we refer to [7], Section 5.3).
Proof
It follows from (2) that the sum of all zeros is equal to
Therefore, there exists a zero of \(P_n^{g,h}(x)\), due to the Hurwitz property and Corollary 2.3, satisfying (3). \(\square \)
If the D’Arcais polynomial \(P_n^{\sigma _1}(x)\) is Hurwitz this implies there exists a zero \(\alpha _n\) with
For the value of \(\kappa ^{\sigma _1}\) we refer to Corollary 4.1 and Corollary 4.6.
Remark 2.5
Note that for fixed c the function
is convex on \(\left( 0,q_{1}\right) \), as
where F is a power series in q whose coefficients are not negative. This implies that the second derivative \(\left( \frac{\partial }{\partial q}\right) ^{2}\kappa =\frac{2}{q^{3}}+ F^{\prime \prime }\left( q\right) >0\) on \(\left( 0,q_{1}\right) \).
Before providing the proof of Theorem 2.1, we illustrate how to utilize our main result towards the finding of upper bounds for the zeros with the largest absolute value.
Example 2.6
Let \(b>0\), \(g\left( n\right) =b^{n-1}\), and \(h\left( n\right) =1\) for all n (except 0). We choose \(c=b\) and for arbitrary \(q>0 \) we obtain
for \(\left| x\right| \ge \kappa =\kappa _{q }\) where \(G_{b}\left( q\right) =0\) and \(\kappa =\frac{1}{1-G_{b}\left( q\right) }\left( \frac{1}{q }+c \right) =\frac{1}{1-0 }\left( \frac{1}{q }+b\right) \).
Therefore, q can be chosen arbitrarily large and \(\kappa \rightarrow b \). As \(P_{n}^{g,1}\left( x\right) =\left( x+b\right) ^{n-1}x\) for all \(n\ge 1\) equality (4) certainly holds always true.
From the theorem we then also find that \(P_{n}^{g,1}\left( x\right) \ne 0\) for all \(\left| x\right| \ge \kappa _{q}\) without knowing the explicit form of \(P_{n}^{g,1}\left( x\right) \). Therefore, in this case, the estimate is sharp as \(\kappa _{q} \rightarrow b\) for \(q \rightarrow \infty \).
Example 2.7
Let \(g\left( k\right) =3-2\delta _{1,k}\). We choose \(c=1\). Therefore, \(G_{1}\left( q\right) =2q\) and \(\kappa =\frac{1}{1-2q}\left( \frac{1}{q}+1\right) \) for any \(0<q<1/2=q_{1}\). Then
if and only if \(0=2q^{2}+4q-1\). Thus, \(q=\frac{-4+\sqrt{16+8 }}{4 }=-1 +\sqrt{3/2 }\) and
Note, that this value is actually the best possible for \(h\left( n\right) =1\), compare [8], Corollary 4.3 and Remark 4.4.
Example 2.8
Let \(g\left( k\right) =k\). For positive \(h\left( n\right) >0\), the sequence of polynomials \(P^{\mathop {\textrm{id}},h}\left( x\right) \) is related to orthogonal polynomials [10]. Then for \(c=\frac{3}{2}\) we obtain
Therefore, \(\kappa = \frac{\left( 3q+2 \right) \left( q-1\right) ^{3}}{-\left( 2q^{3} - 4q^{2} + 5q - 2\right) q}\) and
The relevant zero of the numerator is located at \(q\approx 0.37609\) and yields a minimal value of \(\kappa \approx 5.5928\).
Remark 2.9
To verify our speculation and to determine also all other integer zeros of \(P_n^{\sigma _{1}}(x)\) the results of Kostant [12] and Han [2] demand to check the values of \(P_n^{\sigma _{1}}(-m)\) for \(1 \le m \le n^2\). The results from [5] and this paper reduce the amount of values to \(1 \le m \le 9.7225 \, (n-1)\).
3 Proof of Theorem 2.1
We provide the proof by induction on n. The base case \(n=1\) holds true:
for \(\left| x\right| \ge \kappa H \left( 0\right) \). Let now \(n\ge 2\). Then
The basic idea for the induction step is to use the inequality
To estimate the sum, we apply the induction hypothesis for \(1\le j\le n-1\) to show that
for \(\left| x\right| \ge \kappa H\left( j-1\right) \). Note, that for \(\left| x\right| \ge \kappa H \left( n-1\right) \) we have
Therefore,
Iterating this inequality leads to
for \(\left| x\right| \ge \kappa H\left( n-1\right) \ge \kappa H\left( n-k\right) \) for all \(k=2,\ldots ,n\) by definition. We estimate the sum by
and obtain
By estimating the sum using the assumption from the theorem, we obtain
since
which is equivalent to \(\frac{\left( 1-G_{ c}\left( q\right) \right) \left| x\right| }{H \left( n-1\right) }-\left| c \right| \ge \frac{1}{q }\) and \(G_{c}\) is increasing on \(\left[ 0,R\right) \) as \(\left| g\left( k+1\right) -c g\left( k\right) \right| \ge 0\) for all \(k\in {\mathbb {N}}\). Finally, this proves the theorem.
4 D’Arcais polynomials
One of the main applications of Theorem 2.1 provides the following comparison of the growth of \(P_n^{\sigma _1, h}\left( x\right) \) and \(P_{n-1}^{\sigma _1, h}\left( x\right) \). Here, the function h can be considered as a deformation.
Corollary 4.1
Let \(g=\sigma _1\). Suppose \(\left| x \right| \ge \frac{107}{11} H \left( n-1 \right) \) for all \(n\ge 1\). Then
In particular, \(P_{n}^{\sigma _{1},h}\left( x\right) \ne 0\) for \(\left| x\right| \ge \frac{107 }{11}H \left( n-1\right) \), \(n\ge 2\).
For the proof, the following lemma will be useful. Note, that we want to apply our Theorem 2.1 for \(c=4/3\).
Lemma 4.2
\(\left| \frac{4}{3}\sigma _{1}\left( k\right) -\sigma _{1}\left( k+1\right) \right| \le \frac{1}{2}\left( {\begin{array}{c}k+2\\ 2\end{array}}\right) \) for all \(k\ge 2\).
Proof
It can be checked that \(\left| \frac{4}{3}\sigma _{1}\left( k\right) -\sigma _{1}\left( k+1\right) \right| \le \frac{1}{2}\left( {\begin{array}{c}k+2\\ 2\end{array}}\right) \) for all \(2\le k\le 42\) (see Table 1 for \(k\le 1 3\)). For \(k\ge 43\) holds \(1+\ln \left( k+1\right) \le \frac{k+1}{44 }+\ln \left( 44\right) \le \frac{3}{7}\frac{k+2}{4}\) as \(\exp \left( 3\cdot 45/28-1\right) >44\). Therefore,
\(\square \)
Remark 4.3
We can now easily demonstrate that \(\kappa <10\). We have
for \(0<q <1\). For \(q=\frac{4}{19}<q_{1}\), we obtain
Remark 4.4
In the proof of Corollary 4.1, we will use the following notation. For \(K\in {\mathbb {N}}\), let \(\gamma \left( k\right) =\left| \sigma _{1}\left( k+1\right) -4\sigma _{1}\left( k\right) /3\right| \) for \(1\le k\le K-1\) and \(\gamma \left( k\right) =\frac{1}{2}\left( {\begin{array}{c}k+2\\ 2\end{array}}\right) \) for \(k\ge K\). (Note that \(\gamma \left( 2\right) =0\) see Table 1.)
Then, \(\left| \sigma _{1}\left( k+1\right) -\frac{4}{3}\sigma _{1}\left( k\right) \right| \le \gamma \left( k\right) \) for all \(k\ge 1\) by Lemma 4.2. Let \(M_{K} \left( q\right) =\sum _{k=1}^{\infty } \gamma \left( k\right) q^{k}\). The power series \(M_{K}\) is almost (except for the first K terms) a multiple of the second derivative of the geometric series in q.
Proof of Corollary 4.1
We have
We use the notation from Remark 4.4 with \(K=9\) and estimate
for \(0\le q<q_{0}=\sup \left\{ 0\le q<R:M_{9}\left( q\right) <1\right\} \le q_{1}\le R\). Therefore,
For \(q= \frac{11 }{50 }\) we obtain
and
The claim now follows from Theorem 2.1. \(\square \)
Remark 4.5
Let us choose \(c=4/3\) and \( K=10 \). Then, \(q=0.22\) leads to \(\kappa \approx 9.723\) by applying the same method.
Corollary 4.6
(of Theorem 2.1) Let \(c=4/3\) and \( K=13 \). Then, \(q=0.22\) leads to \(\kappa \approx 9.7225\).
We leave the related calculations to the interested reader. Note, that the values of \(\kappa \) apply to all admissible functions h, in particular to \(h\left( n\right) =1\), compare [8], Table 1. Therefore, we can conclude that \(\kappa \) is very close to the best possible value.
Remark 4.7
Consider the original problem with the sum of divisors function \(\sigma _{1}=g\) and h the identity. It is worth noting that this allows to improve Corollary 3 of [5] from \(\kappa =\frac{119}{11} \) to \(\kappa =\frac{107 }{11}<10 \) by Corollary 2.3 and by Corollary 4.6 to \(\kappa \approx 9.7225\).
5 Final remarks
5.1 The geometric case \(h(n)=1\)
A special case of Theorem 2.1 leads to the following non-vanishing result:
Corollary 5.1
Let g be a normalized arithmetic function. Suppose that \(\textrm{G}\left( q \right) := \sum _{n=1}^{\infty } g\left( n\right) \, q^{n}\) is regular at \(q =0\) with radius of convergence \(\textrm{R} \). Then,
Let \(c \in {\mathbb {C}}\) and \(0<q < R\) with \(G_{c}\left( q\right) <1\). Then, \(P_{n}^{g,1}\left( x\right) \ne 0\) for all \(n\ge 1\) and complex numbers x with \(\vert x \vert \ge \kappa _{q}\), where
The identity (5) already appeared in [6].
5.1.1 Sign changes and reciprocal of Eisenstein series
In [6] we studied properties of the q-expansion of reciprocals of Eisenstein series \(E_m\) of weight m. We have determined for \(g_m(n):= \sigma _{m-1}(n)\) the associated \(\kappa _{m}\), m even, (here we follow the notation given in [6]) to examine if
where \(B_m\) is the mth Bernoulli number. Let (6) be satisfied, then all the coefficients of \(1/E_m\) are non-vanishing. If \(B_m\) is negative, then the coefficients of \(1/E_m\) have alternating signs. For example, this is the case for \(1/E_4\).
With the method provided in this paper, we can improve the \(\kappa _m\). These are recorded in Table 2.
5.1.2 Chebyshev polynomials
Let \(U_n(x)\) be the Chebyshev polynomials of the second kind. Let \(g(1)=g(2)=1\) and let \(g(n)=0\) for \(n \ge 3\). Then [9], Section 3.4 shows that
Then \(G_{c}\left( q\right) =\left| 1-c\right| q+\left| c\right| q^{2}\) and \(\kappa =\frac{1}{1-G_{c}\left( q\right) }\left( \frac{1}{q}+\left| c\right| \right) \). Assume \(0\le c\le 1\) then
which attains its minimum at \(q=\frac{1}{2}\). For this q, we obtain \(\kappa =4\). This implies that \(U_n(x/2) \ne 0\) for \(\left| -x^2 \right| \ge 4\) which implies, that \(U_n(x) \ne 0\) if \(\vert x \vert \ge 1\). This shows that \(\kappa =4\) is the best possible.
5.2 The exponential case \(h(n)=n\)
5.2.1 Hermite polynomials
Let \(H_0(x)=1\), \(H_1(x)=2x\), \( H_2(x) = 4x^2 -2\), \(H_3(x) = 8x^3-12x\), and more generally,
denote the Hermite polynomials. Let \(g(1)=g(2)=1\) and \(g(n)=0\) for \(n \ge 3\). Then, \(P_n^{g,\mathop {\textrm{id}}}(-2x^2) = (-x)^n \, H_n(x)/(n!).\) This follows from [9], Section 3.4. We have already obtained \(\kappa =4\), which is optimal. This implies that
5.2.2 Plane partitions and overpartitions
Let \(g(n)=\sigma _2(n)\). Then, the polynomials \(P_n^{\sigma _2,\mathop {\textrm{id}}}(x)\) interpolate the plane partition function \(\mathop {\textrm{pp}}(n)\). We have \(P_n^{\sigma _2,\mathop {\textrm{id}}}(1)=\mathop {\textrm{pp}}(n)\). With the method presented in this paper, we can demonstrate that \(P_n^{\sigma _2, \mathop {\textrm{id}}}(x) \ne 0\) for \(\left| x \right| \ge \kappa \, (n-1)\), where \(\kappa = 16.022\).
Let \(\bar{\textrm{p}}\left( n\right) \) be the number of overpartitions of n (we refer to [14] for recent results). Let \(g(n) = \sigma _1(n) - \sigma _1(n/2)\), where \(\sigma _1(y)=0\) for \(y \not \in {\mathbb {N}}\). Then, \(P_n^{g, \mathop {\textrm{id}}}(1) = \bar{\textrm{p}}\left( n\right) \). With the method of this paper, we obtain \(P_n^{g, \mathop {\textrm{id}}}(x) \ne 0\) for \(\vert x \vert \ge \kappa \, (n-1)\), where \(\kappa =6. 2133\).
Data Availability
The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.
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Heim, B., Neuhauser, M. Estimate for the largest zeros of the D’Arcais polynomials. Res Math Sci 11, 1 (2024). https://doi.org/10.1007/s40687-023-00412-z
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DOI: https://doi.org/10.1007/s40687-023-00412-z
Keywords
- Arithmetic functions
- Dedekind eta function
- Fourier coefficients
- Location of zeros
- Polynomials
- Recurrence relations