AGN and QSOs in the eROSITA All-Sky Survey - I. Statistical properties

{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 active galactic nuclei (AGN) and quasars/quasi-stellar objects (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 about 1 {\×} 10⁻¹⁴ergs⁻¹ cm⁻² will be achieved in the 0.5-2.0 keV band. With this sensitivity, eROSITA is expected to detect about 3 million AGN on the extragalactic sky (|b| {\>} 10{\}). The median redshift of the eRASS AGN will be z {\} 1 with approximately 40\% of the objects in the z = 1-2 redshift range. About 10⁴-10⁵ AGN are predicted beyond redshift z = 3 and about 2000-30 000 AGN beyond redshift z = 4, the exact numbers depend on the poorly known behavior of the AGN XLF in the high-redshift and high-luminosity regimes. Of the detected AGN, the brightest 10\% will be detected with more than {\}38 counts per point-spread-function (half-energy width), while the faintest 10\% will have fewer than {\}9 counts. The optical counterparts of approximately 95\% of the AGN will be brighter than IAB = 22.5 mag. 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 0.5-2.0 keV sensitivity of {\}2 {\×} 10⁻¹⁴ to {\}2 {\×} 10⁻¹³ergs⁻¹ cm⁻², respectively. }