Environmental Effect on the Associations of Background Quasars with   Foreground Objects: I. Analytic Investigation
  The associations of the angular positions of background quasars with foreground galaxies, clusters of galaxies and quasars are often attributed to the statistical lensing by gravitational potentials of the matter along the lines of sight, although it has been known that none of the individual objects (galaxies, clusters or quasars) are able to fully explain the reported amplitudes of the quasar number enhancements. This probably arises from the fact that the gravitational lensing effect by the environmental matter surrounding these objects has been ignored. In this paper we conduct an extensive study of the influence of the environmental matter on the prediction of quasar enhancement factor by employing the spatial two-point correlation function. Assuming a singular isothermal sphere for mass density profile in galaxy and cluster of galaxies, we estimate the average surface mass density $\overline{\Sigma}$ around galaxies, clusters and quasars from the galaxy-galaxy, cluster-cluster, cluster-galaxy and quasar-galaxy correlations. Our results show that the $\overline{\Sigma}$ induced quasar number enhancement in the scenario of gravitational magnification depends critically on the mass density parameters of galaxies ($\Omega_g$) and clusters of galaxies ($\Omega_c$) in the universe. For a flat cosmological model of $\Omega_0=1$ the environmental matter can indeed play an important role in the lensing origin of the quasar-quasar and quasar-galaxy associations if $\Omega_g\sim\Omega_c\sim\Omega_0$, while it is unlikely that $\overline{\Sigma}$ is sufficient to account for the reported quasar overdensity behind quasars/galaxies if galaxies and clusters of galaxies contribute no more than $25\%$ to the matter of the universe. Nonetheless, the 
Xiang-Ping Wu, Li-Zhi Fang, Zonghong Zhu and Bo Qin