  We give a version of the Artin-Tate formula for surfaces over finite fields
not assuming Tate's conjecture. It gives an equality between terms related to
the Brauer group on the one hand and terms related to the Neron-Severi group on
the other hand. We give estimates on the terms appearing in the formula and use
this to gives sharp estimates on the size of the Brauer group of abelian
surfaces depending on the p-rank.
