On multiple zeros of a partial theta function

Abstract

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.