On the complex conjugate zeros of the partial theta function

TL;DR

This paper characterizes the location of complex conjugate zeros of the partial theta function for q in (-1,1), providing bounds on their real and imaginary parts for different q ranges.

Contribution

It establishes precise bounds on the complex conjugate zeros of the partial theta function depending on the parameter q, extending understanding of its zero distribution.

Findings

Zeros for q in (0,1) lie within specified bounds in the complex plane.

Zeros for q in (-1,0) are confined within a particular rectangle.

The results provide explicit regions where zeros can be located.

Abstract

We prove that 1) for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ belong to the set $\{$~Re\,$x\in (-5792.7,0),$~$|$Im\,$x|<132~\}$ $\cup$ $\{ ~|x|<18~\}$ and 2) for any $q\in (-1,0)$, they belong to the rectangle $\{$~$|$Re\,$x|< 364.2,$~$|$Im\,$x|<132~\}$.

Equations

$${\Theta }^{\ast }(q,x)=QPR,\mathrm{where}Q:=\prod _{m=1}^{\infty }(1-q^m),P:=\prod _{m=1}^{\infty }(1+x{q}^m)\mathrm{and}R:=\prod _{m=1}^{\infty }(1+q^{m-1}/x).$$

$$|G|\le \sum _{j=0}^{\infty }|q{|}^{j(j-1)/2}|x{|}^{-j-1}\le \sum _{j=0}^{\infty }|x{|}^{-j-1}=1/(|x|-1).$$

$$|{\Theta }^{\ast }(q,-q^{-\nu -1/2})|>|G(q,-q^{-\nu -1/2})|,$$

$$P^{\bullet }\ge P^{♭}{P}^{†},\mathrm{where}{P}^{♭}:=(1-q^{-7/2})(1-q^{-5/2})(1-q^{-3/2})(1-q^{-1/2})$$

$$|{\Theta }^{\ast }(q,-q^{-\nu -1/2})|>4.68\cdot 0.288^2=0.388\dots .$$

$$P^{△}:=\prod _{m=1}^{\nu }(1-q^{-\nu -1/2+m})=q^{-{\nu }^2/2}(-1{)}^{\nu }{P}^{♯},P^{♯}:=\prod _{m=1}^{\nu }(1-q^{\nu +1/2-m}).$$

$$\begin{array}[]{c}\tau=((1/(n+1))/(1+h^{1/2}))^{2}>1/4(n+1)^{2}~{}~{}~{}~{}\,{\rm and}\\ \\ |\Theta^{*}|=|QPR|>Q^{2}|P|=Q^{2}|P^{\triangle}P^{\dagger}|>Q^{2}(P^{\dagger})^{2}q^{-\nu^{2}/2}>Q^{4}(1-\sqrt{q})^{2}q^{-\nu^{2}/2}\\ \\ >Q^{4}\tau e^{8(n+1)}>e^{-4(\pi^{2}/6)n}\tau e^{8(n+1)}=e^{(8-2\pi^{2}/3)n+8}\tau>e^{n+8}/4(n+1)^{2}~{}.\end{array}$$

$$\begin{array}[]{ccccccc}|G|&\leq&1/(q^{-\nu-1/2}-1)&\leq&1/(h^{-\nu-1/2}-1)&\leq&1/(h^{-4(n+1)-1/2}-1)\\ \\ &\leq&1/(e^{4}h^{-1/2}-1)&\leq&1/(e^{4}-1)&<&e^{10}/36~{}.\end{array}$$

Full text

Key words: partial theta function, Jacobi theta function, Jacobi triple product

AMS classification: 26A06

1 Introduction

For any fixed $q\in\mathbb{C}$ ($|q|<1$), the sum of the bivariate series $\theta(q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^{j}$ is an entire function in $x$ called the partial theta function. We remind that the Jacobi theta function equals $\Theta(q,x):=\sum_{j=-\infty}^{\infty}q^{j^{2}}x^{j}$ and that $\theta(q^{2},x/q)=\sum_{j=0}^{\infty}q^{j^{2}}x^{j}$. We consider the case when $q$ is real. In the present note we prove the following theorem:

Theorem 1**.**

(1) For any $q\in(0,1)$, all complex conjugate pairs of zeros of $\theta(q,.)$ belong to the set $\{$ Re $x\in(-5792.7,0),$ $|$Im $x|<132~{}\}$ $\cup$ $\{~{}|x|<18~{}\}$.

(2) For any $q\in(-1,0)$, all complex conjugate pairs of zeros of $\theta(q,.)$ belong to the set $\{$ $|$Re $x|<364.2,$ $|$Im $x|<132~{}\}$.

One of the most recent applications of $\theta$ is connected with a problem on section-hyperbolic polynomials, i.e. real univariate polynomials with all roots real and such that all their truncations of positive degree have also all their roots real. Inspired by classical results of Hardy, Petrovitch and Hutchinson (see [6], [16] and [7]), the study of the problem was continued in [15], [8] and [14]. Other areas of application of $\theta$ are modularity and asymptotics of regularized characters and quantum dimensions of the $(1,p)$-singlet algebra modules (see [3] and [5]), the theory of (mock) modular forms (see [4]), Ramanujan-type $q$-series (see [18]), asymptotic analysis (see [2]) and statistical physics and combinatorics (see [17]); see also [1].

The following properties of $\theta$ are proved in [10]: For each $q\in(0,1)$, the function $\theta(q,.)$ has infinitely-many negative and no nonnegative zeros. There exists a sequence of values of $q$ (tending to $1^{-}$) $0=:\tilde{q}_{0}<\tilde{q}_{1}=0.3092\ldots<\tilde{q}_{2}=0.5169\ldots<\cdots<1$ for which $\theta$ has exactly one multiple zero. This is the rightmost of its (negative) zeros; it is a double one. For $q\in(\tilde{q}_{j},\tilde{q}_{j+1}]$, $\theta(q,.)$ has exactly $j$ complex conjugate pairs of zeros counted with multiplicity.

Analogous properties for $q\in(-1,0)$ read (see [11]): For each $q\in(-1,0)$, the function $\theta(q,.)$ has infinitely-many negative and infinitely-many positive zeros. There exists a sequence of values $\bar{q}_{j}$ of $q$ (tending to $-1^{+}$, the largest of which is $\bar{q}_{1}=-0.727133\ldots$) for which $\theta$ has exactly one multiple zero; it is a double one. This is the rightmost of its negative zeros for $j$ odd and the second from the left of its positive zeros for $j$ even. For $q\in[\bar{q}_{1},0)$, all zeros of $\theta$ are real.

2 Proof of Theorem 1

Proof of part (1).

We write “CCP” for “complex conjugate pair (of zeros of $\theta(q,.)$)”. We use a result of [12]: For any $q\in(0,1)$, all zeros of $\theta(q,.)$ belong to the domain \{{\rm Re}~{}x<0,|{\rm Im}~{}x|<132\}$$\cup$$\{{\rm Re}~{}x\geq 0,|x|<18\}. To prove part (1) of Theorem 1 means to show that the real parts of the CCPs are $>-5792.7$. The proof is based on a comparison between the functions $\theta$, $\Theta^{*}:=\sum_{j=-\infty}^{\infty}q^{j(j+1)/2}x^{j}$ and $G:=\sum_{j=-\infty}^{-1}q^{j(j+1)/2}x^{j}$. Obviously, $\theta=\Theta^{*}-G$. For the function $\Theta^{*}$ we use the following formula (derived from the Jacobi triple product, see [9]):

[TABLE]

Consider the functions $\theta$, $\Theta^{*}$ and $G$ restricted to the vertical line in the $x$-plane $\mathcal{L}_{\nu}(q)~{}:~{}$Re $x=-q^{-\nu-1/2}$, $\nu\in\mathbb{N}$ which avoids the zeros of $\Theta^{*}$. The product $|Q|\cdot|P|\cdot|R|$ is minimal for $x=-q^{-\nu-1/2}$ (because for $g:=q^{m-1}$, $u<-1$ and $v\in\mathbb{R}^{*}$, one has $|1+g/(u+iv)|^{2}-|1+g/u|^{2}=-(2u+g)v^{2}g/(u^{2}(u^{2}+v^{2}))>0$ and $|1+(u+iv)q^{m}|>|1+uq^{m}|$). Obviously,

[TABLE]

When $x\in\mathcal{L}_{\nu}(q)$, the right-hand side is maximal for $x=-q^{-\nu-1/2}$. If for given $q$ and $\nu$ one has

[TABLE]

then the inequality $|\Theta^{*}|>|G|$, hence $\Theta^{*}-G\neq 0$, holds true along $\mathcal{L}_{\nu}(q)$ and the function $\theta$ has no zeros on the line $\mathcal{L}_{\nu}(q)$. We set $[\tilde{q}_{1},1)=\cup_{n=1}^{\infty}K_{n}$, where $K_{n}:=[\tilde{q}_{1},1-1/(n+1)]$. We set also $\ell_{1}:=4$ and $\ell_{n}:=4(n+1)$, if $n\geq 2$.

Lemma 2**.**

For $q\in K_{n}$, $\nu\geq\ell_{n}$ and $x\in\mathcal{L}_{\nu}(q)$, one has $|\Theta^{*}|>|G|$ (hence $\theta\neq 0$).

Proof.

For $n=1$, one has $Q\geq\prod_{j=1}^{\infty}(1-2^{-j})=0.288\ldots$ and for $\nu\in\mathbb{N}$, $\nu\geq\ell_{1}$, $R|_{x=-q^{-\nu-1/2}}>Q$. Denote by $P^{\bullet}$ the product $P|_{x=-q^{-\nu-1/2}}$ and set $P^{\dagger}:=\prod_{j=1}^{\infty}(1-q^{j-1/2})$. Thus

[TABLE]

with equality only for $\nu=\ell_{1}=4$. Both factors $P^{\flat}$ and $P^{\dagger}$ can be minorized by their respective values $36.3\ldots$ and $0.129\ldots$ for $q=1/2$, so $P^{\bullet}\geq P^{\bullet}|_{\nu=4,q=1/2}>36.3\cdot 0.129=4.68\ldots$ and

[TABLE]

At the same time, for $x=-q^{-\nu-1/2}\leq-q^{-9/2}$ and for $q\in[\tilde{q}_{1},1/2]$, the majoration $1/(|x|-1)$ of $|G|$ (see (1)) is maximal for $q=1/2$, $\nu=4$, in which case it equals $0.046\ldots<0.388\ldots$.

For $n\geq 2$, we use the inequality $Q\geq e^{-(\pi^{2}/6)n}$ which holds true for $q\in K_{n}$, see Lemma 4 in [9]. As in the case $q\in K_{1}$, one obtains $R|_{x=-q^{-\nu-1/2}}>Q$, $\mathbb{N}\ni\nu\geq\ell_{n}$. The product $P$ is represented in the form $P=P^{\triangle}P^{\dagger}$ (with $P^{\dagger}$ as above), where

[TABLE]

It is clear that $P^{\sharp}>P^{\dagger}>(1-\sqrt{q})Q$; the rightmost inequality follows from $1-q^{j-1/2}<1-q^{j}$, $j\in\mathbb{N}$. Set $h:=1-1/(n+1)$, $f^{*}:=(1-1/x)^{-x}$, and $\tau:=(1-h^{1/2})^{2}$. The factor $q^{-\nu^{2}/2}$ is minorized by $h^{-8(n+1)^{2}}=((f^{*})(n+1))^{8(n+1)}$. The function $f^{*}$ is decreasing on $[2,\infty)$, with $f^{*}(2)=4$ and $\lim_{x\rightarrow\infty}f^{*}(x)=e$, therefore $q^{-\nu^{2}/2}\geq((f^{*})(n+1))^{8(n+1)}>e^{8(n+1)}$. Hence

[TABLE]

The function $e^{x+8}/4(x+1)^{2}$ is increasing on $[2,\infty)$, so $|\Theta^{*}|>e^{10}/36$. For $|G|$ one obtains

[TABLE]

∎

Lemma 3**.**

For $j\in\mathbb{N}$, the double zero of $\theta(\tilde{q}_{j},.)$ belongs to the interval $I:=[-1226,0]$.

Proof.

We use a result of [13]: Consider the function $\theta$ with $q$, $x\in\mathbb{C}$, $|q|<1$. Set $\alpha_{0}:=\sqrt{3}/(2\pi)=0.27\ldots$. Then for $k\geq n\geq 5$ and for $|q|\leq 1-1/(\alpha_{0}n)$, there exists exactly one (and simple) zero $\xi_{k}$ of $\theta(q,.)$ satisfying the conditions $|q|^{-k+1/2}<|\xi_{k}|<|q|^{-k-1/2}$, no zero is of modulus $|q|^{-k\pm 1/2}$, and exactly $n-1$ zeros counted with multiplicity are of modulus $<|q|^{-n+1/2}$.

Hence for $|q|\in[1-1/(\alpha_{0}(n-1)),1-1/(\alpha_{0}n)]$, for any multiple zero $\xi$ of $\theta$ one has $|\xi|\leq\gamma_{n}:=(1-1/(\alpha_{0}(n-1)))^{-n+1/2}$. Indeed, the quantity $|q|^{-n+1/2}$ is maximal for $|q|=1-1/(\alpha_{0}(n-1))$ and no multiple zero of $\theta$ is of modulus $\geq|q|^{-n+1/2}$. For $n\geq 6$, the sequence $\gamma_{n}$ is decreasing, $\gamma_{6}=1225.1\ldots$ and $1-1/(5\alpha_{0})=0.27\ldots<\tilde{q}_{1}$, so for $q\in(0,1)$, any double zero of $\theta$ is in $I$. ∎

Simple zeros of $\theta$ (real or complex) depend continuously on $q$. Two simple real zeros of $\theta(\tilde{q}_{j}^{-},.)$ coalesce for $q=\tilde{q}_{j}$ to become a complex pair for $q=\tilde{q}_{j}^{+}$ (as we say, a pair born from the double zero of $\theta(\tilde{q}_{j},.)$), see [10]. For each $q$ fixed, the line $\mathcal{L}_{\nu}(q)$ is to the right of $\mathcal{L}_{\nu+1}(q)$. As $q$ increases, the number $-q^{-\nu-1/2}$ increases and $\mathcal{L}_{\nu}(q)$ moves from left to right. For any $q\in K_{1}$, the line $\mathcal{L}_{\nu}(q)$, $\nu\geq\ell_{1}=4$, is to the left of the double zero $-7.5\ldots$ of $\theta(\tilde{q}_{1},.)$, see [14] or [10]; one has $\mathcal{L}_{4}(\tilde{q}_{1})~{}:~{}$Re $x=-196.7\ldots$ and $\mathcal{L}_{4}(1/2)~{}:~{}$Re $x=-22.6\ldots$. Recall that $\tilde{q}_{2}=0.51\ldots\not\in K_{1}$. Hence for $q\in K_{1}\setminus\tilde{q}_{1}$, there is exactly one CCP, the one born from the double zero of $\theta(\tilde{q}_{1},.)$. For $q\in K_{1}$, there are no zeros of $\theta$ on $\mathcal{L}_{4}(q)$, so as $q$ increases from $\tilde{q}_{1}$ to $1/2$, the line $\mathcal{L}_{4}(q)$ does not encounter the only CCP of $\theta$ and thus there are no CCPs on and to the left of $\mathcal{L}_{4}(q)$.

Thus there are no CCPs on and to the left of $\mathcal{L}_{\ell_{2}}(1/2)=\mathcal{L}_{12}(1/2)$. One has $\mathcal{L}_{12}(1/2)~{}:~{}$Re $x=-5792.6\ldots$, $\mathcal{L}_{12}(0.56)~{}:~{}$Re $x=-1404.9\ldots$ and $\mathcal{L}_{14}(0.56)~{}:~{}$Re $x=-4479.9\ldots$ (here and below the values $0.56$, $0.6$ and $0.63$ of $q$ are chosen for the convenience of computation). As $q$ increases from $1/2$ to $0.56$, the line $\mathcal{L}_{12}(q)$ moves from Re $x=-5792.6\ldots$ to Re $x=-1404.9\ldots$ and encounters no zeros of $\theta$ on its way. As CCPs can be born only on the interval $I$, for $q\in[1/2,0.56]$, no CCPs are born to the left of $\mathcal{L}_{12}(q)$. For $q=0.56$, there are no CCPs with Re $x\leq-4479.9\ldots$, and for $q\in[\tilde{q}_{1},0.56]$, there are no CCPs with Re $x\leq-5792.6\ldots$.

One has $\mathcal{L}_{14}(0.6)~{}:~{}$Re $x=-1647.4\ldots$ and $\mathcal{L}_{16}(0.6)~{}:~{}$Re $x=-4576.1\ldots$. For $q\in[\tilde{q}_{1},0.6]$, the line $\mathcal{L}_{14}(q)$ encounters no zeros of $\theta$ and it does not intersect $I$, so there are no CCPs on and to the left of it. One has $\mathcal{L}_{16}(0.63)~{}:~{}$Re $x=-2045.8\ldots$ and $\mathcal{L}_{18}(0.63)~{}:~{}$Re $x=-5154.6\ldots$. Thus there are no CCPs on and to the left of $\mathcal{L}_{16}(q)$ for $q\in[0.6,0.63]$. One has $\mathcal{L}_{18}(2/3)~{}:~{}$Re $x=-1810.0\ldots$, therefore there are no CCPs on and to the left of $\mathcal{L}_{18}(q)$ for $q\in[0.63,2/3]$, hence there are no CCPs in Re $x\leq-5792.6\ldots$ for $q\in K_{2}$.

Consider the numbers $b_{n}:=-(1-1/n)^{-4(n+1)-1/2}$, $\mathbb{N}\ni n\geq 2$. These are the values of $-q^{-\ell_{n}-1/2}$ for $q=1-1/n\in K_{n}$. The sequence $b_{n}$ is increasing, with $b_{2}=-5792.6\ldots$ and $b_{3}=-804.4\ldots$. Hence for $n\geq 3$, $b_{n}\in I$. For $q\in[2/3,1)$, there is at least one number $-q^{-\nu_{0}-1/2}$, $\nu_{0}\in\mathbb{N}$, in the interval $(-5792.6\ldots,-1226)$; this follows from $1226\cdot(3/2)<5792.6$. The corresponding line $\mathcal{L}_{\nu_{0}}(q)$ contains no zeros of $\theta$ for $q\in K_{n}$; this results from Lemma 2 and from $b_{n}\in I$ for $n\geq 3$. For each line $\mathcal{L}_{\nu}(q)$, $\nu\geq\nu_{0}$, it is true that for $q\in K_{n}$, there are no CCPs on and to the left of it. As $q$ increases in $K_{n}$, the line $\mathcal{L}_{\nu_{0}}(q)$ might begin to intersect the interval $I$, but it is always true that any line $\mathcal{L}_{\nu_{0}+k}(q)$, $k\in\mathbb{N}$, which does not intersect $I$ has no CCPs on it and to its left (and there is at least one such line for Re $x\geq-5792.6\ldots$). Hence there are no CCPs for $q\in[\tilde{q}_{1},1)$ and Re $x\leq-5792.6\ldots$. ∎

Proof of part (2).

We use again a result of [12]: For any $q\in(-1,0)$, all zeros of $\theta(q,.)$ belong to the strip $\{|{\rm Im}~{}x|<132\}$. We have to show that the real part of any complex zero of $\theta$ belongs to the interval $(-364.2,364.2)$. For $q\in(-1,0)$, we consider in the same way the intervals $J_{n}:=[-1+1/(n+1),\bar{q}_{1}]$, $n\geq 3$. We set $\ell_{n}:=4(n+1)$. For $q\in J_{n}$, we consider the vertical lines $\mathcal{L}_{\nu}(q)~{}:~{}$Re $x=-|q|^{-\nu-1/2}$ and $\mathcal{R}_{\nu}(q)~{}:~{}$Re $x=|q|^{-\nu-1/2}$, $\mathbb{N}\ni\nu\geq\ell_{n}$. As $|q|$ increases, the line $\mathcal{L}_{\nu}(q)$ moves to the right and $\mathcal{R}_{\nu}(q)$ moves to the left. We compare the factors $\mu:=1-q^{m}$, $\lambda:=1+xq^{m}$ and $\chi:=1+q^{m-1}/x$ in cases A) $q=q_{*}\in(0,1)$, Re $x=-(q_{*})^{-\nu-1/2}$ and B) $q=-q_{*}\in(-1,0)$, Re $x=\pm|q_{*}|^{-\nu-1/2}$; we denote them by $\mu_{A}$, $\mu_{B}$, $\lambda_{A}$, $\lambda_{B}$, $\chi_{A}$ or $\chi_{B}$ according to the case. Clearly, $|$Im $\lambda_{A}|=|$Im $\lambda_{B}|$, $|$Im $\chi_{A}|=|$Im $\chi_{B}|$ and either $\mu_{B}=1-q_{*}^{m}=\mu_{A}\in(0,1)$ or $\mu_{B}=1+q_{*}^{m}>1>\mu_{A}$, either $|$Re $\lambda_{B}|=|1-q_{*}^{-\nu-1/2+m}|=|$Re $\lambda_{A}|$ or $|$Re $\lambda_{B}|=|1+q_{*}^{-\nu-1/2+m}|>|$Re $\lambda_{A}|$ and either $|$Re $\chi_{B}|=|1-q_{*}^{-\nu-3/2+m}/|x|^{2}|=|$Re $\chi_{A}|$ or $|$Re $\chi_{B}|=|1+q_{*}^{-\nu-3/2+m}/|x|^{2}|>|$Re $\chi_{A}|$. Hence $|\Theta^{*}(-q_{*},\pm|q_{*}|^{-\nu-1/2})|>|\Theta^{*}(q_{*},-(q_{*})^{-\nu-1/2})|$.

The majoration (1) of $|G|$ remains valid. The quantity $1/(|x|-1)$ takes the same values for $x=\pm|q_{*}|^{-\nu-1/2}$, so $|\Theta^{*}|>|G|$ along $\mathcal{L}_{\nu}(q)$ and $\mathcal{R}_{\nu}(q)$. The double zero $y_{j}$ of $\theta(\bar{q}_{j},.)$, $j\in\mathbb{N}$, belongs to the interval $\tilde{I}:=[-237,237]$ (with $y_{1}=-2.9\ldots$ and $y_{2}=2.9\ldots$, see [11]); this follows from $|\bar{q}_{1}|>1-1/(12\alpha_{0})=0.69\ldots$ and $\gamma_{13}=89.9\ldots<237$, see Lemma 3 and its proof.

One has $b_{4}=-364.1\ldots$, $b_{5}=-236.7\ldots$, so for $n\geq 5$, $b_{n}\in\tilde{I}$. For $q\in J_{n}$, there exists a number $|q|^{-\nu-1/2}\in(237,364)$ with $\nu\geq\ell_{n}$ (because $237/364<|\bar{q}_{1}|$) and a line $\mathcal{L}_{\nu}(q)$ (resp. $\mathcal{R}_{\nu}(q)$) with no CCPs on and to the left (resp. right) of it. By analogy with the case $q\in(0,1)$ one shows that for $q\in(-1,\bar{q}_{1}]$, the moduli of the real parts of the complex zeros of $\theta$ are bounded by $|b_{4}|<364.2$. ∎