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

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Hilberdink, T. W. (2010) Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem. Journal of Number Theory, 130 (3). pp. 707-715. ISSN 0022-314X

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To link to this article DOI: 10.1016/j.jnt.2009.09.008

Abstract/Summary

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 .

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical and Physical Sciences > Department of Mathematics and Statistics
ID Code:23360
Uncontrolled Keywords:Beurling's generalised primes; Dirichlet divisor problem
Publisher:Elsevier

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