On the Complex Conjugate Zeros of the Partial Theta Function

Abstract

We prove that (1) for any q ∈ (0, 1), all complex conjugate pairs of zeros of the partial theta function \(\theta (q,x): = \sum\nolimits_{j = 0}^\infty {{q^{j(j + 1)/2}}{x^j}}\) belong to the set {Re x ∈ (−5792.7,0), |Im x| < 132} ∪ {|x| < 18} and (2) for any q ∈ (−1,0), they belong to the rectangle {|Re x| < 364.2, |Im x| < 132}.