On the double zeros of a partial theta function

Abstract

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1), x∈R, and defines a partial theta function. For any fixed q∈(0,1) it has infinitely many negative zeros. For q taking one of the spectral values q˜1, q˜2, … (where 0.3092493386…=q˜1<q˜2<⋯<1, limj→∞⁡q˜j=1) the function θ(q,.) has a double zero yj which is the rightmost of its real zeros (the rest of them being simple). For q≠q˜j the partial theta function has no multiple real zeros. We prove that q˜j=1−π/2j+(log⁡j)/8j2+O(1/j2) and yj=−eπe−(log⁡j)/4j+O(1/j).