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 as p goes to infinity. We prove that the fourth mean value of θ(1,χ) is asymptotic to as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.