  Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational
primes $p \leq P$ that split in the imaginary quadratic field
$\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various
unconditional and conditional (under existence of a Siegel zero) lower bounds
for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed
$\varepsilon>0$. This improves upon what is implied by work of Pollack and
Benli-Pollack.
  The second part of this paper is dedicated to proving an estimate for a
bilinear form involving Weyl sums for modular square roots (equivalently
Sali\'{e} sums). Our estimate has a power saving in the so-called
P{\'o}lya-Vinogradov range, and our methods involve studying an additive energy
coming from quadratic residues in $\mathbb{F}_q$.
  This bilinear form is inspired by the recent automorphic motivation: the
second moment for twisted $L$-functions attached to Kohnen newforms has
recently been computed by the first and fourth authors. So the third part of
this paper links the above two directions together and outlines the arithmetic
applications of this bilinear form. These include the equidistribution of
quadratic roots of primes, products of primes, and relaxations of a conjecture
of Erdos-Odlyzko-Sarkozy.
