We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3·1012. That is, all zeroes β+iγ of the Riemann zeta‐function with 0<γ⩽3·1012 have β=1/2. Moreover, all of these zeroes are simple.
We describe a rigorous implementation of the Lagarias and Odlyzko Analytic Method to evaluate the prime counting function and its use to compute unconditionally the number of primes less than 10 24 .
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