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Criteria equivalent to the Riemann Hypothesis

J. Cisło, M. Wolf, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore VoronovAIP Conference ProceedingsPublished 2008doi 10.1063/1.3043867 Original︎ ↗Archived copy

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In this spirit in 1901, Koch proved [29] that the Riemann Hypothesis is equivalent to the following error term for the expression for the prime counting function: π(x) = Li(x) + O(√x ln(x)). Another similar criterion is RH ⇔ π(x) = Li(x) + O(x^{1/2+ε}) for each ε > 0.

The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.Verified

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Kind: conference paper
Authors: J. Cisło, M. Wolf, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov
Publisher: AIP Conference Proceedings
Published: 2008
DOI: 10.1063/1.3043867
Link: https://arxiv.org/abs/0808.0640
Archived copy: https://web.archive.org/web/20260204124415/https://arxiv.org/abs/0808.0640
Stored text: 4,158 characters, retrieved Nov 11, 2026 · 48 sections, 60 passages
Claims extracted: Not recorded
Versions: One known
Copies: None recorded
Corrections: None recorded
Facts checked: Nov 11, 2026

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