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Optimal bounds for sums of non-negative arithmetic functions

Andrés Chirre, Harald Andrés Helfgott

TL;DR

The paper develops a sharp, general framework to bound sums of non-negative arithmetic functions using finite pole information of the Dirichlet series A(s)=∑ a_n n^{-s}, without relying on zero-free regions. It merges a Fourier-analytic Beurling–Selberg approach with contour shifting, yielding explicit, compactly supported formulas that express partial sums as sums over poles, including a main term from the pole at s=1 and a secondary spectral contribution from zeros. The method is then specialized to the von Mangoldt case via A(s)=−ζ′(s)/ζ(s), producing explicit bounds for ψ(x) and for sums involving Λ(n), with quantified error terms under RH up to height T. Across the paper, optimal weight functions and sharp error control are achieved through extremal L^1-approximation problems, providing a robust, computable framework for explicit analytic number theory results with direct implications for PNT-type bounds and zero-distribution analysis.

Abstract

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is natural to ask how such finite spectral information may be best used to estimate partial sums $\sum_{n\leq x} a_n$. Here, we prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-σ}$ for $a_n$ non-negative, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$, with no need for zero-free regions. We give not just bounds, but an explicit formula with compact support. Our bounds on $ψ(x)-x$ are, unsurprisingly, better and often simpler than a long list of existing explicit versions of the Prime Number Theorem. We treat the case of $M(x)$ and similar functions in a companion paper. Our solution mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type found in (Graham--Vaaler, 1981).

Optimal bounds for sums of non-negative arithmetic functions

TL;DR

The paper develops a sharp, general framework to bound sums of non-negative arithmetic functions using finite pole information of the Dirichlet series A(s)=∑ a_n n^{-s}, without relying on zero-free regions. It merges a Fourier-analytic Beurling–Selberg approach with contour shifting, yielding explicit, compactly supported formulas that express partial sums as sums over poles, including a main term from the pole at s=1 and a secondary spectral contribution from zeros. The method is then specialized to the von Mangoldt case via A(s)=−ζ′(s)/ζ(s), producing explicit bounds for ψ(x) and for sums involving Λ(n), with quantified error terms under RH up to height T. Across the paper, optimal weight functions and sharp error control are achieved through extremal L^1-approximation problems, providing a robust, computable framework for explicit analytic number theory results with direct implications for PNT-type bounds and zero-distribution analysis.

Abstract

Let be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of with , and their residues, for some large constant . It is natural to ask how such finite spectral information may be best used to estimate partial sums . Here, we prove a sharp, general result on sums for non-negative, giving an optimal way to use information on the poles of with , with no need for zero-free regions. We give not just bounds, but an explicit formula with compact support. Our bounds on are, unsurprisingly, better and often simpler than a long list of existing explicit versions of the Prime Number Theorem. We treat the case of and similar functions in a companion paper. Our solution mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type found in (Graham--Vaaler, 1981).

Paper Structure

This paper contains 45 sections, 56 theorems, 273 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic problem
  3. Results
  4. Context and methods
  5. Existing results
  6. Strategy
  7. Structure of the paper
  8. Notation
  9. Acknowledgements
  10. From a complex integral to an $L^1(\mathbb{R})$ approximation problem
  11. A smoothed Perron formula based on the Fourier transform
  12. Bounding unsmoothed sums from above and below
  13. An aside on optimality
  14. Extremal approximants to the truncated exponential
  15. Graham--Vaaler's one-sided approximants and their transforms
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $\{a_n\}_{n=1}^\infty$, $a_n\geq 0$ for all $n$. Assume that $A(s)=\sum_{n}a_n n^{-s}$ converges absolutely for $\Re{s}>1$ and extends meromorphically to $\mathbb{C}$, with a simple pole at $s=1$ and no other poles with $\Re s = 1$, $|\Im s|\leq T$ for some $T>0$. Assume that $A(s) T^s$ is bound where $\mathcal{Z}_{A}^+(T)$ is the set of poles $\rho$ of $A(s)$ with $0<\Im \rho< T$, $\mathcal{Z

Figures (5)

  • Figure 1: Beurling--Selberg majorant and minorant of $\mathds{1}_{(-\infty,0]}(u)$
  • Figure 2: Graham--Vaaler majorant and minorant of $\mathds{1}_{(-\infty,0]}(u)\cdot e^{\lambda u}$ for $\lambda = 1/4$
  • Figure 3: Lemma \ref{['lem:kolobrz2']}: contour-shifting and result. Only $G^\star(s)$ is integrated on $\mathcal{C}$.
  • Figure :
  • Figure :

Theorems & Definitions (114)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 104 more