Sharper Bounds for the Chebyshev function $\theta (x)$

Abstract

In this article, we provide explicit bounds for the prime counting function $\theta (x)$ in all ranges of $x$. The bounds for the error term for $\theta (x)-x$ are of the shape $\epsilon x$ and $\frac{c_{k}x}{(\log x{)}^k}$, for $k=1,\dots ,5$. Tables of values for $\epsilon$ and $c_k$ are provided.

There is 41 page accompanying file to the main article which includes longer versions of Tables 8 to 15. This has the additional authors: Andrew Fiori and Josh Swidinsky