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

Effects of the surrounding primordial black holes on the merger rate of primordial black hole binaries