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

Jan Büthe

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arXiv · 1511.02032

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

Jan Büthe

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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.

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Jan Büthe. 2015-11-06. An analytic method for bounding $\psi(x)$. https://arxiv.org/abs/1511.02032

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