2018/10/12 by Munsch, Marc
#05D05 #11B30 #11L40 #11N37. Secondary : 05C42 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11M99
paper · doi:10.48550/arxiv.1810.05684
Let χ range over the (p-1)/2 even Dirichlet characters modulo a prime p and denote by θ(x,χ) 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 ≫ 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 ≫ p/ √(log p) characters such that θ(x,χ) ≠ 0 for any x>0. We give similar results for odd Dirichlet characters mod p.