An analytic method for bounding $\psi (x)$

Abstract

In this paper we present an analytic altorithm which calculates almost sharp bounds for the normalized error term $(t-\psi (t))/\sqrt{t}$ for $t\le x$ in expected run time $O(x^{1/2+\epsilon })$ for every $\epsilon >0$. The method has been implemented and used to calculate the bound $|\psi (t)-t|\le 0.94\sqrt{t}$ for $11<t\le 10^{19}$. In particular, this bound implies that $\mathrm{li}(t)-\pi (t)>0$ for $t\in [2,10^{19}]$, which gives an improved lower bound for the Skewes number.

arXiv comment17 pages, final version, to appear in Math. Comp