Density results for the zeros of zeta applied to the error term in the prime number theorem

Abstract

We improve the unconditional explicit bounds for the error term in the prime counting function ψ(x). In particular, we prove that, for all x > 2, we haveand that, for all log x ≥ 3 000, |ψ(x) − x| < 4.47 • 10 −15 x. This compares to results of Platt & Trudgian (2021) who obtained 4.51 • 10 −13 x. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning π(x) will appear in a follow up work.

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

“…Compared with the results of this paper, [FKS22] contains better bounds on |ψ(x) − x| for lower values of X (X ≤ 10000) and at present, does not feature any corresponding results for |π(x) − li(x)|. However, as [FKS22] is currently unpublished, and our results on |ψ(x) − x| for low X are used to establish subsequent results, we have refrained from using the work in [FKS22] here. Importantly though, H. Kadiri and G. Hiary brought to our attention an unreliable result in [Hia16] which was previously used in this work.…”

“…Upon announcing the first version of this article, H. Kadiri informed us that she was working on similar work with A. Fiori and J. Swidinsky. Subsequently, they announced their work [FKS22]. Compared with the results of this paper, [FKS22] contains better bounds on |ψ(x) − x| for lower values of X (X ≤ 10000) and at present, does not feature any corresponding results for |π(x) − li(x)|.…”

“…Subsequently, they announced their work [FKS22]. Compared with the results of this paper, [FKS22] contains better bounds on |ψ(x) − x| for lower values of X (X ≤ 10000) and at present, does not feature any corresponding results for |π(x) − li(x)|. However, as [FKS22] is currently unpublished, and our results on |ψ(x) − x| for low X are used to establish subsequent results, we have refrained from using the work in [FKS22] here.…”