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The Riemann hypothesis is true up to $3\cdot 10^{12}$

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arxiv 2004.09765 v1 pith:BIEVZBOS submitted 2020-04-21 math.NT

classification math.NT
keywords riemannbetacdotgammahypothesistruearithmeticcdot10
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abstract

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $\beta + i\gamma$ of the Riemann zeta-function with $0<\gamma\leq 3\cdot 10^{12}$ have $\beta = 1/2$.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An explicit uniform cubic wedge for consecutive Toeplitz minors of the Riemann xi coefficients

    math.NT 2026-07 accept novelty 7.0 of 10

    For every r ≥ 2 and k ≥ 10^18 r^3, the consecutive Toeplitz minor D_{r,k} of the Riemann xi coefficients is strictly positive, proved without using verified zeta zeros.

  2. Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis

    math.GM 2026-07 unverdicted novelty 6.0 of 10

    Under simple zeros, Φ(e^{-x})=Σ μ(n)e^{-nx} equals a zero sum plus trivial-zero series with a log term; O(x^{-1/2}) implies RH unconditionally.

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