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A Wiener-Ikehara type theorem and its application to Chebyshev bounds for Beurling primes

Yarne Tranoy, Jasson Vindas

TL;DR

This work develops a new Wiener-Ikehara-type Tauberian theorem that yields two-sided exponential bounds $e^{x} \ll S(x) \ll e^{x}$ from mild boundary behavior of the Laplace transform. It then applies the theorem to Beurling generalized prime number systems, deriving a criterion for Chebyshev bounds under weaker boundary conditions on the Beurling zeta function near $s=1$. The method integrates distributional boundary values, local pseudofunctions, and an explicit decomposition of boundary values to obtain transparent Tauberian conclusions. Collectively, the results extend and simplify existing criteria (notably Diamond–Zhang) for Chebyshev bounds in Beurling prime systems and suggest broader applicability of the approach to complex Tauberian problems.

Abstract

We provide a new version of the Wiener-Ikehara theorem where one deduces bounds $$ 0< \liminf_{x\to\infty} \frac{S(x)}{e^{x}}\leq \limsup_{x\to\infty} \frac{S(x)}{e^{x}} <\infty $$ for (in particular) a non-decreasing function $S$ from a mild hypothesis on the boundary behavior of its Laplace transform on a vertical segment containing $s=1$. As an application, we establish new criteria for the validity of Chebyshev bounds for Beurling generalized prime number systems under weaker conditions than were known so far.

A Wiener-Ikehara type theorem and its application to Chebyshev bounds for Beurling primes

TL;DR

This work develops a new Wiener-Ikehara-type Tauberian theorem that yields two-sided exponential bounds from mild boundary behavior of the Laplace transform. It then applies the theorem to Beurling generalized prime number systems, deriving a criterion for Chebyshev bounds under weaker boundary conditions on the Beurling zeta function near . The method integrates distributional boundary values, local pseudofunctions, and an explicit decomposition of boundary values to obtain transparent Tauberian conclusions. Collectively, the results extend and simplify existing criteria (notably Diamond–Zhang) for Chebyshev bounds in Beurling prime systems and suggest broader applicability of the approach to complex Tauberian problems.

Abstract

We provide a new version of the Wiener-Ikehara theorem where one deduces bounds for (in particular) a non-decreasing function from a mild hypothesis on the boundary behavior of its Laplace transform on a vertical segment containing . As an application, we establish new criteria for the validity of Chebyshev bounds for Beurling generalized prime number systems under weaker conditions than were known so far.
Paper Structure (8 sections, 9 theorems, 53 equations)

This paper contains 8 sections, 9 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributions and Fourier transform
  4. Pseudofunctions, pseudomeasures, and boundary values
  5. Two basic lemmas from Wiener's theory
  6. Controlled decrease: Tauberian conditions
  7. The Tauberian theorem
  8. Application: Chebyshev bounds for Beurling primes

Key Result

Theorem 1.1

Let $S\in L^{1}_{loc}[0,\infty)$. Then, the asymptotic relation eq: asymp S W-I holds if and only if $S$ is log-linearly slowly decreasing, its Laplace transform is convergent for $\Re e\: s>1$, and the analytic function $G$ given by eq: Laplace extension W-I has local pseudofunction boundary behavi

Theorems & Definitions (15)

  • Theorem 1.1: Exact Wiener-Ikehara theorem D-V2016
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2: korevaar1965
  • Lemma 2.3: Wiener's division lemma korevaarbook
  • Definition 2.4
  • Lemma 2.5: D-V2016
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • ...and 5 more