# A domain free of the zeros of the partial theta function

### Abstract

The partial theta function is the sum of the series

\medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}

\medskip\noi where $q$

is a real or complex parameter ($|q|<1$). Its name is due to similarities

with the formula for the Jacobi theta function

$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$.

The function $\theta$ has been considered in Ramanujan's lost notebook. It

finds applications

in several domains, such as Ramanujan type

$q$-series, the theory

of (mock) modular forms, asymptotic analysis, statistical physics,

combinatorics and most recently in the study of section-hyperbolic polynomials,

i.~e. real polynomials with all coefficients positive,

with all roots real negative and all whose sections (i.~e. truncations)

are also real-rooted.

For each $q$ fixed,

$\theta$ is an entire function of order $0$ in the variable~$x$. When

$q$ is real and $q\in (0,0.3092\ldots )$, $\theta (q,.)$ is a function of the

Laguerre-P\'olya

class $\mathcal{L-P}I$. More generally,

for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a real

polynomial

without real zeros and a function of the class $\mathcal{L-P}I$. Thus it is

an entire function with

infinitely-many negative, with no positive and with finitely-many complex

conjugate zeros. The latter are known to belong

to an explicitly defined compact domain containing $0$ and

independent of $q$ while the negative zeros tend to infinity as a

geometric progression with ratio $1/q$. A similar result is true for

$q\in (-1,0)$ when there are also infinitely-many positive zeros.

We consider the

question how close to the origin the zeros of the function $\theta$ can be.

In the general

case when $q$ is complex it is true

that their moduli are always larger than $1/2|q|$.

We consider the case when $q$ is real and prove that for any $q\in (0,1)$,

the function $\theta (q,.)$ has no zeros on the set

$$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap \{x\in\mathbb{C}\colon {\rm Re} x\leq 0\}

\cap \{x\in\mathbb{C}\colon |{\rm Im} x|\leq 3/\sqrt{2}\}$$

which contains

the closure left unit half-disk and is more than $7$ times larger than it.

It is unlikely this result to hold true for the whole of the left

half-disk of radius~$3$.

Similar domains do not exist for $q\in (0,1)$, Re$x\geq 0$, for

$q\in (-1,0)$, Re$x\geq 0$ and for $q\in (-1,0)$, Re$x\leq 0$. We show also

that for $q\in (0,1)$, the function $\theta (q,.)$

has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).

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*Matematychni Studii*,

*58*(2), 142-158. https://doi.org/10.30970/ms.58.2.142-158

Copyright (c) 2022 V. Kostov

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