On the zeros of a partial theta function

The series converges for q∈[0,1), x∈R and defines a partial theta function. For any q∈(0,1) fixed it has infinitely many negative zeros. For countably many values of q said to form the spectrum of θ (where , ) the function θ(q,.) has a double zero which is the rightmost of its real zeros (the rest of them being simple). For it has no multiple real zeros. For the function θ(q,.) has exactly N complex conjugate pairs of zeros counted with multiplicity (we set ). If denote the zeros of ∂θl/∂xl(q,.) in the order of decreasing, then and .