6 research outputs found
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Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem
In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that
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The average order of the Möbius function for Beurling primes
In this paper, we study the counting functions ψP(x), NP(x) and MP(x) of a generalized prime system N. Here, MP(x) is the partial sum of the Möbius function over N not exceeding x. In particular, we study these when they are asymptotically well-behaved, in the sense that ψP(x)=x+O(xα+ϵ), NP(x)=ρx+O(xβ+ϵ) and MP(x)=O(xγ+ϵ), for some ρ>0 and α,β,γ<1. We show that the two largest of α,β,γ must be equal and at least 12