Correction to: “Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem” [J. Number Theory 130 (2010) 707–715]
Year: 2024
Type: erratum
Abstract: We discuss the main result of [1] which is concerned with the study of generalised prime systems for which the integer counting function N P ( x ) is asymptotically very well-behaved, in the sense that N P ( x ) = ρ x + O ( x β ) , where ρ is a positive constant and β < 1 2 . For such systems, the associated zeta function ζ P ( s ) is holomorphic for σ = ℜ s > β . It was claimed that for β < σ < 1 2 , ∫ 0 T | ζ P ( σ + i t ) | 2 d t = Ω ( T 2 − 2 σ − ε ) for (i) any ε > 0 , and (ii) for ε = 0 for all such σ except possibly one value. The proof of these statements contains a flaw however, and in this Corrigendum we indicate where the mistake occurred but show that the proof can be rectified to still obtain (i) and get a slightly weaker result for (ii). The resulting Corollary 2 of [1] concerning the Dirichlet divisor problem for generalised integers remains essentially correct.
Source: Journal of Number Theory
Author: Titus W. Hilberdink
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,210
OpenAlex ID: W4404949213