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

TL;DR

This paper derives explicit, effective bounds for the deviation of the first Chebyshev function from x, improving upon historical ineffective bounds and providing simple formulas with specific constants for large x.

Contribution

The paper develops the first fully explicit bounds of de la Valle Poussin's type, using recent effective results, with simple formulas and specific constants for large x.

Findings

Derived explicit bounds: | heta(x)-x| < x exp(-1/4 sqrt(ln x)) for x ≥ 2.

Derived explicit bounds: | heta(x)-x| < x exp(-1/3 sqrt(ln x)) for x ≥ 3.

Showed many other bounds can be similarly developed.

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: \[ |\theta(x)-x| = \mathcal{O}\left(x \exp(-K \sqrt{\ln x})\right). \] 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: \[ |\theta(x)-x| < \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 2). \] \[ |\theta(x)-x| < \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 3). \] Many other fully explicit bounds along these lines can easily be developed. For instance one can trade off stringency against range of validity: \[ |\theta(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 29), \] \[ |\theta(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 41). \] With hindsight, some of these effective bounds could have been established almost 50 years ago.