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The prime number theorem through one-sided Tauberian theorems

Gregory Debruyne

Abstract

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.

The prime number theorem through one-sided Tauberian theorems

Abstract

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.
Paper Structure (3 sections, 2 theorems, 16 equations)

This paper contains 3 sections, 2 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The one-sided Ingham-Karamata Tauberian theorem
  3. The prime number theorem

Key Result

Lemma 2.1

Let $\varepsilon > 0$ be arbitrary. There exists a real-valued function $\phi \in L^{1} \cap L^2$ satisfying the following properties: $\int^{\infty}_{-\infty} \phi(x) \mathrm{d}x = 1$, $\phi(x) \geq 0$ for positive $x$, while $\phi(x) \leq 0$ for negative $x$, $\int^{\infty}_{-\infty} x \phi(x) \ma

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 3.1