Effective de la Valle Poussin style bounds on the first Chebyshev function

TL;DR

The paper addresses making the historical de la Vallée Poussin bound on the first Chebyshev function fully explicit by leveraging fifty years of effective results. It introduces a simple lemma to convert bounds of the form $|\\vartheta(x)-x| < a x (\\ln x)^b \\exp(-c \\sqrt{\\ln x})$ into a de la Vallée Poussin style bound $|\\vartheta(x)-x| < \\tilde a x \\exp(-\\tilde c \\sqrt{\\ln x})$, enabling explicit constants and ranges of validity. The main contributions are two explicit bounds $|\\vartheta(x)-x| < x \\exp(-\\frac{1}{4}\\sqrt{\\ln x})$ for $x\\ge 2$ and $|\\vartheta(x)-x| < x \\exp(-\\frac{1}{3}\\sqrt{\\ln x})$ for $x\\ge 3$, plus variants that trade prefactor against larger domains and an asymptotically stringent, fully explicit bound. This work provides practical, thoroughly proven estimates for researchers needing explicit control over the Chebyshev function and clarifies how decades of related results yield simple, explicit bounds ready for application.

Abstract

In 1898 Charles Jean de la Valle Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form: \[ |θ(x)-x| = \mathcal{O}\left(x \exp(-K \sqrt{\ln x})\right). \] This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant. To the best of my knowledge this bound has never been made effective -- I have never yet seen this bound made fully explicit, with precise values being given for $x_0$ and $K$. Herein, using a number of effective results established over the past 50 years, I shall develop two very simple explicit fully effective bounds of this type: \[ |θ(x)-x| < \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 2). \] \[ |θ(x)-x| < \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 3). \] Many other fully explicit bounds along these lines can easily be developed. For instance one can trade off stringency against range of validity: \[ |θ(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 29), \] \[ |θ(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 41). \] With hindsight, some of these effective bounds could have been established almost 50 years ago.