Context. The main element of the observing program of the Spectrum-Roentgen-Gamma orbital observatory is a four-year all-sky survey, in the course of which the entire sky will be scanned eight times. Aims. We analyze the statistical properties of AGN and QSOs that are expected to be detected in the course of the eROSITA all-sky survey (eRASS). Methods. According to the currently planned survey strategy and based on the parameters of the Galactic and extragalactic X-ray background as well as on the results of the recent calculations of the eROSITA instrumental background, we computed a sensitivity map of the eRASS. Using the best available redshift-dependent AGN X-ray luminosity function (XLF), we computed various characteristics of the eRASS AGN sample, such as their luminosity- and redshift distributions, and the brightness distributions of their optical counterparts. Results. After four years of the survey, a sky-average sensitivity of ~1x10^(-14) erg/s/cm^2 will be achieved in the 0.5-2.0keV band. With this sensitivity, eROSITA is expected to detect ~3 million AGN on the extragalactic sky (|b|>10deg). The median redshift of the eRASS AGN will be z~1 with ~40% of the objects in the z=1-2 redshift range. About 10^4 - 10^5 AGN are predicted beyond redshift z=3 and about 2 000 - 30 000 AGN beyond redshift z=4, the exact numbers depend on the poorly known behavior of the AGN XLF in the high-redshift and luminosity regimes. Of the detected AGN, the brightest 10% will be detected with more than ~38 counts per PSF HEW, while the faintest 10% will have fewer than ~9 counts. The optical counterparts of ~95% of the AGN will be brighter than I_(AB)=22.5mag. The planned scanning strategy will allow one to search for transient events on a timescale of half a year and a few hours with a 0.5-2.0keV sensitivity of ~2x10^(-14) to ~2x10^(-13) erg/s/cm^2, respectively.
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