On the first sign change of $\theta(x) - x$

TL;DR

This paper investigates the sign changes of the difference between the Chebyshev function θ(x) and x, establishing bounds for when θ(x) is less than or greater than x.

Contribution

It provides explicit bounds for the first sign change of θ(x) - x, extending previous results with new computational and theoretical insights.

Findings

θ(x) < x for 2 < x < 1.39×10^{17}

There exists x < exp(727.9513...) with θ(x) > x

Explicit bounds on the first sign change of θ(x) - x

Abstract

Let $\theta(x) = \sum_{p\leq x} \log p$. We show that $\theta(x)<x$ for $2<x< 1.39\cdot 10^{17}$. We also show that there is an $x<\exp(727.951332668)$ for which $\theta(x) >x.$