Uniform bounds on locations of zeros of partial theta function

Abstract

We consider the partial theta function $\theta (q,z):={\sum }_{j=0}^{\infty }{q}^{j(j+1)/2}{z}^j$, where $(q,z)\in {\mathbb{C}}^2$, $|q|<1$. We show that for any $0<{\delta }_0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with ${\delta }_0\le |q|\le \delta$ and for any $n\ge n_0$ the function $\theta$ has exactly $n$ zeros with modulus $<|q{|}^{-n-1/2}$ counted with multiplicity.

References in corpus (3)

• Hardy-Petrovitch-Hutchinson's problem and partial theta function
• The leading root of the partial theta function
• Stabilization of the asymptotic expansions of the zeros of a partial theta function