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A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results

Alex Burgin

TL;DR

The paper develops an elementary, self-contained proof of Landau's Prime Ideal Theorem for number fields by extending Richter's orthogonality-based approach to the ideal setting. Central to the method is an explicit orthogonality identity that expresses the squared deviation of the divisor-count from its mean in terms of a GCD-based kernel $\Phi$, enabling equidistribution conclusions for $\Omega(\mathfrak m)$ modulo integers and, in particular, modulo $q$ and under irrational shifts. The authors establish Chebyshev-type bounds for prime ideals in $K$, supply combinatorial lemmas to construct structured families of primes and prime products, and connect these to a general framework that yields equidistribution results and Landau-type asymptotics. Appendix sections provide the analytical tools (Abel summation, Weyl's criterion) and an implication chain showing $L(x)=o(x)$ implies $M(x)=o(x)$, consolidating an elementary route to the prime-ideal distribution problem with potential extensions to Beurling-type systems.

Abstract

We give a new elementary proof of Landau's Prime Ideal Theorem. The proof is an extension of Richter's proof of the Prime Number Theorem. The main result contains other results related to the equidistribution of the prime ideal counting function.

A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results

TL;DR

The paper develops an elementary, self-contained proof of Landau's Prime Ideal Theorem for number fields by extending Richter's orthogonality-based approach to the ideal setting. Central to the method is an explicit orthogonality identity that expresses the squared deviation of the divisor-count from its mean in terms of a GCD-based kernel , enabling equidistribution conclusions for modulo integers and, in particular, modulo and under irrational shifts. The authors establish Chebyshev-type bounds for prime ideals in , supply combinatorial lemmas to construct structured families of primes and prime products, and connect these to a general framework that yields equidistribution results and Landau-type asymptotics. Appendix sections provide the analytical tools (Abel summation, Weyl's criterion) and an implication chain showing implies , consolidating an elementary route to the prime-ideal distribution problem with potential extensions to Beurling-type systems.

Abstract

We give a new elementary proof of Landau's Prime Ideal Theorem. The proof is an extension of Richter's proof of the Prime Number Theorem. The main result contains other results related to the equidistribution of the prime ideal counting function.
Paper Structure (11 sections, 14 theorems, 86 equations)

This paper contains 11 sections, 14 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation, Conventions, and Standard Definitions
  3. Background
  4. Density Constants
  5. Building Towards Theorem 1
  6. Proof of Proposition \ref{['prop2.2analogue']}
  7. Chebychev-Type Bounds for $\pi_K(x)$
  8. Combinatorial Ingredients
  9. Appendix A
  10. Appendix B
  11. Appendix C

Key Result

Theorem 1

Let $K$ be a number field. Then there exists a positive integer $N$ such that, for any $g:\mathbb{N}_0\rightarrow \mathbb{C}$ satisfying $|g (n)|\leq 1$ for all $n$, one has for each $k_1,k_2\geq N$.

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 1
  • Proposition 2
  • proof
  • Corollary 2: Equivalent Assertion to Proposition \ref{['orthogonalityProp']}
  • proof
  • Proposition 3
  • proof : Proof of Theorem \ref{['mainTheorem']} assuming Proposition \ref{['prop2.2analogue']}
  • Lemma 3
  • proof
  • ...and 16 more