Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The optimal Malliavin-type remainder for Beurling generalized integers

Frederik Broucke, Gregory Debruyne, Jasson Vindas

Abstract

We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $α\in (0,1]$ and $c>0$ (with $c\leq 1$ if $α=1$), a generalized number system is constructed with Riemann prime counting function $ Π(x)= \operatorname*{Li}(x)+ O(x\exp (-c \log^α x ) +\log_{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=ρx + Ω_{\pm}(x\exp(- c'(\log x\log_{2} x)^{\fracα{α+1}})$ for any $c'>(c(α+1))^{\frac{1}{α+1}}$, where $ρ>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].

The optimal Malliavin-type remainder for Beurling generalized integers

Abstract

We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given and (with if ), a generalized number system is constructed with Riemann prime counting function and whose integer counting function satisfies the extremal oscillation estimate for any , where is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].

Paper Structure

This paper contains 13 sections, 12 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Construction of the continuous example
  3. Analysis of the saddle points
  4. The saddle points
  5. The steepest path through $s_{0}$
  6. The contribution from $s_{0}$
  7. The steepest paths through $s_{m}$, $m\neq0$.
  8. The contributions from $s_{m}$, $m\neq0$
  9. The remainder in the contour integral
  10. Connecting the steepest paths
  11. Returning to the line $[\kappa-\mathrm{i} T, \kappa+\mathrm{i} T]$
  12. Conclusion of the analysis of the continuous example
  13. The discrete example

Key Result

Theorem 1.2

We have $\beta^{\ast}(\alpha)=\alpha/(1+\alpha)$ for any $\alpha\in (0,1]$.

Theorems & Definitions (20)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 10 more