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Uniform bounds on locations of zeros of partial theta function (1607.05453v1)

Published 19 Jul 2016 in math.CV

Abstract: We consider the partial theta function $\theta (q,z):=\sum _{j=0}{\infty}q{j(j+1)/2}zj$, 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 $\delta _0\leq |q|\leq \delta$ and for any $n\geq n_0$ the function $\theta$ has exactly $n$ zeros with modulus $<|q|{-n-1/2}$ counted with multiplicity.

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