Non vanishing of theta functions and sets of small multiplicative energy

Abstract

Let $\chi$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $\theta (x,\chi )$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author implies that there exists at least $\gg p/\log p$ characters such that the associated theta function does not vanish at a fixed point. Constructing a suitable mollifier, we improve this result and show that there exists at least $\gg p/\sqrt{\log p}$ characters such that $\theta (x,\chi )\ne 0$ for any $x>0$. We give similar results for odd Dirichlet characters mod $p$.