The ATESP radio survey III. Source counts

This paper is part of a series reporting the results of the ATESP radio sur vey obtained at 1.4 GHz with the Australia Telescope Compact Array. The sur vey consists of 16 radio mosaics with similar to8 " X 14 " resolution and u niform sensitivity (1 sigma noise level similar to 79 mu Jy) over the whole area of the ESO Slice Project redshift survey (similar to 26 so. degr. at delta similar to -40 degrees). The ATESP survey has produced a catalogue of 2960 radio sources down to a flux limit (6 sigma) of similar to0.5 mJy. In this Paper we present the 1.4 GHz log N-log S relation derived from the AT ESP radio source catalogue. The possible causes of incompleteness at the fa int end of the source counts are extensively discussed and their effects ar e quantified and corrected for. The ATESP counts are consistent with others reported in the literature, even though some significant discrepancies are present at low fluxes (below a few mJy). We investigate whether such discr epancies may be explained in terms of held-to-held anisotropies, considerin g the fact that ail the existing sub-mJy surveys cover small areas of sky ( from a fraction of square degree to a few square degrees). We stress that t he ATESP survey, covering 26 sq. degr., provides the best determination of source counts available today in the flux range 0.7 less than or similar to S-1.4 GHz less than or similar to 2 mJy.