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The error term in the prime number theorem

Dave Platt, Tim Trudgian

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.

The error term in the prime number theorem

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.

Paper Structure

This paper contains 5 sections, 10 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preparatory lemmas
  3. Pintz's method
  4. Proof of Corollary \ref{['white']}
  5. Application to Ramanujan's inequality

Key Result

Theorem 1

Let $R = 5.573412$. For each row $\{X, A, B, C, \epsilon_0\}$ from Table tab3 we have and for all $\log x \geq X$.

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 3 more