  We present a derivation of the numerical phenomenon that differences between
the Riemann zeta function's nontrivial zeros tend to avoid being equal to the
imaginary parts of the zeros themselves, a property called statistical
"repulsion" between the zeros and their differences. Our derivation relies on
the statistical properties of the prime zeta function, whose singularity
structure specifies the positions of the Riemann zeros. We show that the prime
zeta function on the critical line is asymptotically normally distributed with
a covariance function that is closely approximated by the logarithm of the
Riemann zeta function's magnitude on the 1-line. This creates notable negative
covariance at separations approximately equal to the imaginary parts of the
Riemann zeros. This covariance function and the singularity structure of the
prime zeta function combine to create a conditional statistical bias at the
locations of the Riemann zeros that predicts the zero-difference repulsion
effect. Our method readily generalizes to describe similar effects in the zeros
of related L-functions.
