An analytic method for bounding $ψ(x)$

Abstract

In this paper we present an analytic altorithm which calculates almost sharp bounds for the normalized error term $(t-ψ(t))/\sqrt{t}$ 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 the bound $|ψ(t) - t| \leq 0.94 \sqrt{t}$ for $11< t\leq 10^{19}$. In particular, this bound implies that $\operatorname{li}(t) - π(t) > 0$ for $t\in [2,10^{19}]$, which gives an improved lower bound for the Skewes number.