The Riemann hypothesis is true up to 3*10^12

en · official · sha256:0b3ca80db019457e16dc18582212fa7586872415d82fab1df4919e7fbf8fa020

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

Referencing norms: HeightWithinRegion, IntervalWithinRegion, PlattTrudgianClaim, PlattTrudgianEvidence, VerifiedRegion, candidate_height, candidate_interval, height_within_region, interval_within_region, pt-asserts, pt-height, pt-method, pt-region-lower, pt-region-upper, pt-source-verified, region_lower_bound_exclusive, region_upper_bound_inclusive, verification_method

en · official · sha256:f101c1f2b3e6bb5deae5625b012b731cbfaf9b7d94f2fb999682b0262620b40f

The Riemann zeta-function zeta(s) has trivial zeroes at s = -2, -4, -6 ..., and non-trivial zeroes in the strip 0 < sigma < 1, where here, and hereafter s = sigma + it. The Riemann hypothesis asserts that all non-trivial zeroes rho = beta + i*gamma have beta = 1/2. In the absence of a proof, it is extremely important to obtain partial verifications of the Riemann hypothesis. To that end, define H as the largest number for which it is known that all zeroes rho = beta + i*gamma with 0 < gamma <= H have beta = 1/2.

en · official · sha256:73c4dc0fd2037d2d4c674d2ca6253717d3fb3c962834a91288797d872a852c54

The Riemann hypothesis is true up to 3 * 10^12 Dave Platt School of Mathematics, University of Bristol, Bristol, UK Tim Trudgian School of Science, The University of New South Wales Canberra, Australia April 22, 2020

Referencing norms: PlattTrudgianRegion, PlattTrudgianSource