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 $t\leq x$ in expected run time $O(x^{1/2+\varepsilon})$ for every $\varepsilon>0$. The method has been implemented and used to calculate the bound $\psi(t)  t \leq 0.94 \sqrt{t}$ for $11< t\leq 10^{19}$. In particular, this bound implies that $\operatorname{li}(t)  \pi(t) > 0$ for $t\in [2,10^{19}]$, which gives an improved lower bound for the Skewes number.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.02032
 Bibcode:
 2015arXiv151102032B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Numerical Analysis;
 11N05;
 11M26
 EPrint:
 17 pages, final version, to appear in Math. Comp