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

Abstract

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