Applications of partial knowledge of the Riemann Hypothesis

We briefly describe how partial knowledge of the Riemann Hypothesis
can be obtained numerically, and present some applications
of such results. This includes a graded version of the well-known
conditional Schoenfeld bound for the prime counting function pi(x),
 which only assumes the correctness of the RH up to a certain height.
As an algorithmic application we provide an efficient method for
calculating limited range approximations to pi(x) and related functions.