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We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$.
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N. Revol and F. Rouillier, A library for arbitrary precision interval arithmetic, 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (2002)
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Computing degree 1 L L -functions rigorously
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T. S. Trudgian · 2011
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A zero density result for the Riemann zeta function
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Explicit estimates for the summatory function of Λ ( n ) / n \Lambda(n)/n from the one for Λ ( n ) \Lambda(n)
O. Ramaré · 2013
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Every odd number greater than 1 is the sum of at most five primes
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Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function
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Estimating π ( x ) \pi(x) and related functions under partial RH assumptions
J. Büthe · 2016
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On the first sign change of θ ( x ) − x \theta(x)-x
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Isolating some non-trivial zeros of zeta
D. J. Platt · 2017
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Explicit bounds for primes in arithmetic progressions
M.A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer · 2018
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Explicit estimates of some functions over primes
P. Dusart · 2018
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Corrigendum to “New bounds for ψ ( x ) \psi(x) ”
L. Faber and H. Kadiri · 2018
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Explicit zero density for the Riemann zeta function
H. Kadiri, A. Lumley, and N. Ng · 2018
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Effective approximation of heat flow evolution of the Riemann ξ \xi function, and a new upper bound for the de Bruijn–Newman constant
D. H. J. Polymath · 2019
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C. R. Smith · 2016
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Improvements to Turing’s Method II
T. S. Trudgian · 2016
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An analytic method for bounding ψ ( x ) \psi(x)
J. Büthe · 2017
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A practical analytic method for calculating π ( x ) \pi(x)
J. Franke, T. Kleinjung, J. Büthe, and A. Jost · 2017
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Arb: efficient arbitrary-precision midpoint-radius interval arithmetic
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X. Gourdon
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Explicit zero density estimate for the Riemann zeta-function near the critical line
A. Simonič · 2019
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Sharper bounds for the Chebyshev function θ ( x ) \theta(x)
S. Broadbent, H. Kadiri, A. Lumley, N. Ng, and K. Wilk · 2020
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NCI · 2020
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https://en.wikipedia.org/wiki/Riemann_hypothesis
Wikipedia. The Riemann hypothesis · 2020
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