An analytic method for bounding $ψ(x)$
For number theorists, this provides a computational verification that li(t) - π(t) > 0 up to 10^19, improving the known lower bound for the Skewes number.
The paper presents an analytic algorithm that computes nearly sharp bounds for the normalized error term in the prime counting function ψ(x) with expected run time O(x^{1/2+ε}), achieving a bound |ψ(t)-t| ≤ 0.94√t for t up to 10^19, which improves the lower bound for the Skewes number.
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.