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 $-7.51$).