108 Vladimir Kostov

Vladimir Kostov (Université Côte d'Azur, France) // Analytic properties of the partial theta function // Abstract: We consider the partial theta function $\theta(q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x$ is a variable and $q$ a parameter ($|q|<1$). We deal with the two possible situations, when $q$ is real or complex. In the talk we focus on the analytic properties of $\theta$, such as asymptotic expansions for its zeros, its spectrum (i. e. the set of values of the parameter $q$ for which $\theta (q,.)$ has multiple zeros), behaviour of its zeros, especially of its complex conjugate pairs, when the parameter $q$ varies, separation in modulus of the zeros etc.