Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

Abstract: In this paper we study generalised prime systems for which the integer counting function is asymptotically well behaved, in the sense that , where ρ is a positive constant and . For such systems, the associated zeta function is holomorphic for . We prove that for , for any , and also for for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in , which is for some . Letting denote the infimum of such θ, we show that . [Copyright &y& Elsevier]