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A further discussion on quasar-galaxy associations from gravitational lensing

Quasar-galaxy associations, if they result from the effect of gravitational lensing by foreground galaxies, depend sensitively on the shape of the quasar number counts. Two kinds of quasar number-magnitude relations are predicted to produce quite different properties in quasar-galaxy associations: the counts of Boyle, Shanks and Peterson (1988; BSP) provide both positive and ``negative" associations between distant quasars and foreground galaxies, relating closely with the knee ($B\approx19.15$) in these counts. However, Hawkins and Véron (1993; HV) quasar data lead to only a positive magnitude-independent quasar-galaxy association. The current observational evidence on quasar-galaxy associations, either positive or null, is shown to be the natural result of gravitational lensing if quasars follow the BSP number-magnitude relation. On the other hand, the HV counts are unable to produce the reported associations by the mechanism of gravitational lensing. It is emphasized that special attention should be paid to the limiting magnitudes in the selected quasar samples when one works on quasar-galaxy associations.