The error term in the prime number theorem

TL;DR

This paper explicitly refines the error term in the prime number theorem, reducing it to roughly the square root of previous bounds, and applies this to a classical inequality problem linked to Ramanujan.

Contribution

It makes explicit a theorem improving the error term in the prime number theorem and applies it to a longstanding problem related to Ramanujan.

Findings

Error term in prime number theorem improved to roughly square-root of previous bounds

Explicit version of Pintz's theorem provided

Application to Ramanujan's inequality problem

Abstract

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.