An analytic method for bounding \psi(x)

In this paper we present an analytic algorithm which calculates almost sharp bounds for the normalized remainder term (t-\psi (t))/\sqrt t for t\leq x in expected run time O(x^{1/2+\varepsilon }) for every \varepsilon >0. The method has been implemented and used to calculate such bounds for t\leq 10^{19}. In particular, these imply that li(x)-\pi (x) is positive for 2\leq x\leq 10^{19}.