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 similaritieswith 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. Itfinds applicationsin 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 theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $\mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric 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 thequestion how close to the origin the zeros of the function $\theta$ can be.In the generalcase when $q$ is complex it is truethat 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 containsthe 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 lefthalf-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 alsothat for $q\in (0,1)$, the function $\theta (q,.)$ has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).