Effects of the surrounding primordial black holes on the merger rate of
  primordial black hole binaries
  We develop an analytic formalism for computing the merger rate of primordial
black hole binaries with a general mass function by taking into account the
torques by the surrounding primordial black holes and linear density
perturbations. We find that $\alpha=-(m_{i}+m_{j})^2\partial^{2} \ln {\mathcal
R}(m_{i},m_{j})/\partial m_{i} \partial m_{j}=36/37$ is independent of the mass
function. Moreover, the ratio of the merger rate density of primordial black
hole binaries by taking into account the torques by the surrounding primordial
black holes to by the nearest primordial black hole is independent of the
masses of binaries.
