Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

Gao, Peng; Zhao, Liangyi

doi:10.1112/mtk.12219

Mathematika

2023

DOI: 10.1112/mtk.12219

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Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

Peng Gao,

Liangyi Zhao

Abstract: We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of are simple.

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Cited by 3 publications

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An annotated bibliography for comparative prime number theory

Martin,

Yang,

Bahrini

et al. 2025

Expositiones Mathematicae

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An annotated bibliography for comparative prime number theory

Martin,

Yang,

Bahrini

et al. 2025

Expositiones Mathematicae

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Negative discrete moments of the derivative of the Riemann zeta‐function

Bui,

Florea,

Milinovich

2024

Bulletin of London Math Soc

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We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.

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A discrete mean value of the Riemann zeta function

Benli,

Elma,

2024

Mathematika

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In this work, we estimate the sum over the nontrivial zeros of the Riemann zeta function where is a complex number with and and are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where is replaced by for all . The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann hypothesis, we obtain the lower bound which was previously known under the generalized Riemann hypothesis, only in the case .

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Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

Abstract: We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all under the Riemann hypothesis and the assumption that all zeros of are simple.

Cited by 3 publications

References 25 publications

An annotated bibliography for comparative prime number theory

An annotated bibliography for comparative prime number theory

Negative discrete moments of the derivative of the Riemann zeta‐function

A discrete mean value of the Riemann zeta function

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