An analytic method for bounding 𝜓(𝑥)

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