Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal bounds for sums of bounded arithmetic functions

Andrés Chirre, Harald Andrés Helfgott

TL;DR

The paper develops a sharp, general framework to bound sums ∑_{n≤x} a_n from finite pole data of a Dirichlet series A(s) with |Im s|≤T, without relying on information beyond height T. It blends a Fourier-based smoothed Perron method with Beurling–Selberg extremal approximants to produce optimal L^1 weights, allowing explicit residue formulas that capture the contributions of poles up to height T. Applied to bounded a_n and, in particular, the Möbius function μ(n), the authors derive explicit M(x) and m(x) bounds, including a concrete M(x) bound |M(x)| ≤ (3/(π·10^{10})) x + 11.39 √x (and improved constants via square-free refinements). The work also provides a detailed, computation-backed treatment of square-free numbers, refined constants, and generalizations to other Dirichlet series, continuous weights, and coprimality constraints, with practical implications for explicit bounds in analytic number theory.

Abstract

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data to give explicit estimates on sums $\sum_{n\leq x} a_n$? The problem of giving explicit bounds on the Mertens function $M(x) = \sum_{n\leq x} μ(n)$ illustrates how open this basic question was. Bounding $M(x)$ might seem equivalent to estimating $ψ(x) = \sum_{n\leq x} Λ(n)$ or the number of primes $\leq x$. However, we have long had fairly good explicit bounds on prime counts, while bounding $M(x)$ remained a notoriously stubborn problem. We prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-σ}$ for $a_n$ bounded, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$ and no data on the poles above. Our bounds on $M(x)$ are stronger than previous ones by many orders of magnitude. (Similar results for $ψ(x)$ are given in a companion paper.) Using rigorous residue computations by D. Platt, we obtain, for $x\geq 1$, $$|M(x)|\leq \frac{3}{π\cdot 10^{10}}\cdot x + 11.39 \sqrt{x}.$$ This is a corollary of our main result, essentially an explicit formula with the contribution of each pole clearly stated; we shall discuss how this finer structure can be useful. Our proof mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type (Carneiro--Littmann, 2013); for $σ=1$, the approximant in (Vaaler, 1985) reappears. While we proceed independently of existing explicit work on $M(x)$ and $ψ(x)$, our method has an important step in common with work on another problem by (Ramana--Ramaré, 2020).

Optimal bounds for sums of bounded arithmetic functions

TL;DR

The paper develops a sharp, general framework to bound sums ∑_{n≤x} a_n from finite pole data of a Dirichlet series A(s) with |Im s|≤T, without relying on information beyond height T. It blends a Fourier-based smoothed Perron method with Beurling–Selberg extremal approximants to produce optimal L^1 weights, allowing explicit residue formulas that capture the contributions of poles up to height T. Applied to bounded a_n and, in particular, the Möbius function μ(n), the authors derive explicit M(x) and m(x) bounds, including a concrete M(x) bound |M(x)| ≤ (3/(π·10^{10})) x + 11.39 √x (and improved constants via square-free refinements). The work also provides a detailed, computation-backed treatment of square-free numbers, refined constants, and generalizations to other Dirichlet series, continuous weights, and coprimality constraints, with practical implications for explicit bounds in analytic number theory.

Abstract

Let be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of with for some large constant . What is the best way to use such finite spectral data to give explicit estimates on sums ? The problem of giving explicit bounds on the Mertens function illustrates how open this basic question was. Bounding might seem equivalent to estimating or the number of primes . However, we have long had fairly good explicit bounds on prime counts, while bounding remained a notoriously stubborn problem. We prove a sharp, general result on sums for bounded, giving an optimal way to use information on the poles of with and no data on the poles above. Our bounds on are stronger than previous ones by many orders of magnitude. (Similar results for are given in a companion paper.) Using rigorous residue computations by D. Platt, we obtain, for , This is a corollary of our main result, essentially an explicit formula with the contribution of each pole clearly stated; we shall discuss how this finer structure can be useful. Our proof mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type (Carneiro--Littmann, 2013); for , the approximant in (Vaaler, 1985) reappears. While we proceed independently of existing explicit work on and , our method has an important step in common with work on another problem by (Ramana--Ramaré, 2020).

Paper Structure

This paper contains 52 sections, 48 theorems, 225 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic problem
  3. Dichotomy
  4. Results
  5. Estimates for $a_n$ bounded
  6. Context and methods
  7. Existing results on the Mertens function
  8. Strategy
  9. Structure of the paper
  10. Notation
  11. Acknowledgements
  12. From a complex integral to an $L^1(\mathbb{R})$ approximation problem
  13. A smoothed Perron formula based on the Fourier transform
  14. From sums to $L^1$ norms.
  15. General bound
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $A(s)=\sum_n a_n n^{-s}$ extend meromorphically to $\mathbb{C}$. Assume $a_\infty = \sup_n |a_n|<\infty$. Let $T\geq 4\pi$. Assume $A(s) T^s$ is bounded on some $S$ as in eq:sapli. Then, for any $\sigma\in \mathbb{R}$, $x>e^2 T$, for $\delta = \frac{\pi}{2 T}$. Here $\mathcal{Z}_{A}(T)$ is the set of poles $\rho$ of $A(s)$ with $|\Im \rho|\leq T$, and where $\mathrm{tanhc}(x)$ equals $\frac{

Figures (4)

  • Figure 1: A ladder $S$: climb it leftwards to shift a contour to $\Re s = -\infty$.
  • Figure 2: Vaaler approximant to $\mathds{1}_{(-\infty,0]}(u)$
  • Figure 3: Carneiro--Littmann approximant to $\mathds{1}_{(-\infty,0]}(u)\cdot e^{\lambda u}$ for $\lambda = 1/4$
  • Figure 4: Proof of Lemma \ref{['lem:kolobrz']}: contour-shifting and result

Theorems & Definitions (100)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 90 more