Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{ 12}$, we provide strong estimates on $\pi(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|\pi(x)-\text{ li}(x)|<\sqrt{ x}\log x/8\pi$ for $2657\leq x\leq 1.101\cdot 10^{ 26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.
Categories: math.NT