Some explicit estimates for the error term in the prime number theorem

By combining and improving recent techniques results, we provide explicit estimates for the error terms |π(x)−li(x)|, |θ(x)−x| |ψ(x)−x| appearing in prime number theorem. For example, show all x≥2 that |ψ(x)−x|≤9.39x(log⁡x)1.515exp⁡(−0.8274log⁡x). Our rely heavily on zero-free regions zero-density Riemann zeta-function, improve existing bounds prime-counting functions large values of x.