The second and fourth moments of theta functions at their central point

Let χ range over the (p - 1)/2 even Dirichlet characters mod p ≥ 3, a prime. Let θ(x, χ) be the associated theta series. It is known that the square mean value of θ(1, χ) is asymptotic to p3/2/42 as p goes to infinity. We prove that the fourth mean value of θ(1, χ) is asymptotic to 3/16πp²logp as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.