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The method of Pintz for the Ingham question about the connection of distribution of ζ-zeros and order of the error in the PNT in the Beurling context

Révész, Szilárd (2024) The method of Pintz for the Ingham question about the connection of distribution of ζ-zeros and order of the error in the PNT in the Beurling context. MICHIGAN MATHEMATICAL JOURNAL. ISSN 0026-2285

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Abstract

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term Δ(x):=ψ(x)−x, given any zero-free region D(η):={s=σ+it∈C : σ:=Rs≥1−η(t)}. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes P, the generated arithmetical semigroup G and the corresponding integer counting function N(x), and the corresponding error term ΔG(x):=ψG(x)−x in the PNT of Beurling, where ψG(x) is the Beurling analog of ψ(x). First we prove that if the Beurling zeta function ζG does not vanish in D(η), then the extension of Pintz' result holds: |ΔG(x)|≤xexp(−(1−ε)ωη(x)) (x>x0(ε)), where ωη(y) is the naturally occurring conjugate function to η(t), introduced into the field by Ingham. In the second part we prove a converse: if ζG has an infinitude of zeroes in the given domain, then analogously to the classical case, |ΔG(x)|≥xexp(−(1+ε)ωη(x)) holds "infinitely often". This also shows that both main results are sharp apart from the arbitrarily small ε>0. The classical results of Pintz used many facts about the Riemann zeta function. Recently we worked out a number of analogous results–including some construction of quasi-optimal integration paths, a Riemann-von Mangoldt type formula, a Carlson-type density theorem and a Turán type local density theorem–for the Beurling context, too. These, together with Turán’s power sum theory, all play some indispensable role in deriving the main results of the paper.

Item Type: Article
Uncontrolled Keywords: Beurling zeta function, analytic continuation, arithmetical semigroups, Knopfmacher’s Axiom A, Beurling prime number theorem, zero of the Beurling zeta function, oscillation of remainder term, Mellin transform
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 04 Apr 2024 07:19
Last Modified: 04 Apr 2024 07:19
URI: https://real.mtak.hu/id/eprint/191558

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