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arXiv · 1810.05684

Non vanishing of theta functions and sets of small multiplicative energy

Marc Munsch
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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 $\theta(x,\chi) \neq 0$ for any $x>0$. We give similar results for odd Dirichlet characters mod $p$.

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Marc Munsch. 2018-10-12. Non vanishing of theta functions and sets of small multiplicative energy. https://arxiv.org/abs/1810.05684

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