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1.1. In the paper [Fu], we generalized the Hasse zeta functions ζA(s) of commutative finitely generated rings A over the ring Z of integers, to noncommutative rings (concerning the definition, see below), and computed the zeta functions of some rings. The examples suggested a strong relationship between the convergence of the Hasse zeta functions and the Gelfand-Kirillov dimensions of the rings (concerning the definition of the Gelfand-Kirillov dimension, see section 2). So in [Fu], we conjectured the relationship between them (see section 2). In the present paper, we consider this conjecture in the case of group rings. For a (not necessarily commutative) finitely generated ring A over Z, in [Fu] we defined the Hasse zeta function CA(S) of A by

On Hasse zeta functions of group algebras of almost nilpotent groups