Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis
Ethan Simpson Lee, Paweł Nosal
TL;DR
The paper derives explicit, RH-conditional bounds for the errors in the prime number theorem, notably proving $|\psi(x) - x| \le \frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$ for all $x \ge 101$ and the analogous bound for $|\vartheta(x) - x|$ for $x \ge 2657$. It achieves this via a combination of an explicit Goldston-type method and a smoothed explicit formula for $\psi(x)$, together with rigorous bounds on zero-distribution terms and auxiliary lemmas, supplemented by numerical verification for smaller $x$ (via Büthe). These techniques yield sharpened, explicit descriptions of the errors in Mertens' theorems and stronger, computable ranges where lower-order terms can be omitted. The results have practical impact for sieve methods and analytic estimates that rely on precise error control in PNT under RH. Overall, the work provides tighter, fully explicit RH-based bounds that improve prior Schoenfeld-type results and deepen the quantitative understanding of prime-counting errors.
Abstract
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 $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8π}$$ for all $x\geq 101$ and $x \geq 2\,657$ respectively, where $ψ(x)$ and $\vartheta(x)$ are the Chebyshev $ψ$ and $\vartheta$ 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.