Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis

In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that | ψ ( x ) − x | and | ϑ ( x ) − x | are bounded from above by x log ⁡ x ( log ⁡ x − log ⁡ log ⁡ x ) 8 π for all x ≥ 101 and x ≥ 2 657 respectively, where ψ ( x ) and ϑ ( x ) are the Chebyshev ψ and ϑ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.