Improving bounds on prime counting functions by partial verification of the Riemann hypothesis

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