  In this work we show that based on a conjecture for the pair correlation of
integers representable as sums of two squares, which was first suggested by
Connors and Keating and reformulated here, the second moment of the
distribution of the number of representable integers in short intervals is
consistent with a Poissonian distribution, where "short" means of length
comparable to the mean spacing between sums of two squares. In addition we
present a method for producing such conjectures through calculations in prime
power residue rings and describe how these conjectures, as well as the above
stated result, may by generalized to other binary quadratic forms. While
producing these pair correlation conjectures we arrive at a surprising result
regarding Mertens' formula for primes in arithmetic progressions, and in order
to test the validity of the conjectures, we present numericalz computations
which support our approach.
