On cusp forms associated with binary theta series

Abstract.

Let $f \in \mathbb{Z}[x, y]$ be a primitive positive binary quadratic form with fundamental discriminant and let

$S(f, z) := \sum\limits_{n=1}^{\infty} a(f, n)e(nz)$

be the associated cusp form, i.e., the projection of the theta series of f onto the subspace of cusp forms. For any real $\beta > 0$ , the exact order of magnitude of the counting function $\sum\limits_{n \leq x} |a(f,n)|^{2 \beta}$ is given. For integral $\beta > 0$ , a meromorphic continuation of $\sum |a(f,n)|^{2 \beta}n^{-s}$ to the half plane $\Re s > 0$ is obtained. The number of sign changes of a(f, n) for $n \leq x$ is estimated.