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

TL;DR

This paper introduces an analytic algorithm that efficiently computes near-sharp bounds for the error term in the prime counting function, providing improved bounds for the Skewes number up to 10^19.

Contribution

The paper presents a novel analytic method with nearly optimal runtime to bound the error term of the prime counting function, extending the known bounds to very large values.

Findings

Bound |ψ(t) - t| ≤ 0.94√t for 11 < t ≤ 10^19

Li(t) - π(t) > 0 for t in [2, 10^19]

Improved lower bound for the Skewes number

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.