Chebyshev’s Theorem (weak version)

The goal

Pafnutij L’vovič Chebyshev, the mathematician who invented among the other things the “plantigrade machine” depicted alongside, is famous also for one his theorem which constitutes a first step towards the Prime Number Theorem, and which we’ll call simply “Chebyshev’s Theorem”. There are two versions of this Theorem, called “weak” and “strong” respectively; in this path we’ll focus on the weak one. Chebyshev’s Theorem and the Prime Number Theorem both say something about the relationship between the function \pi(x), which returns the number of prime numbers less than or equal to x, and the function \frac{x}{\log x}. By the weak version of Chebyshev’s Theorem, these two functions have the same order of magnitude, in the sense of Definition A.3. In order to understand this concept better, we can consider the case of polynomials: two polynomials have the same order of magnitude when they have the same degree, so, according to the weak version of Chebyshev’s Theorem, we can say that the functions \pi(x) and \frac{x}{\log x} resemble each other as much as two polynomials of the same degree, like x^2 and 3x^2 + x. With formulas, this concept is expressed as follows:

\pi(x) \asymp \frac{x}{\log x}

The Prime Number Theorem says something more accurate about the similarity between these two functions; on the other hand, the weak version of Chebyshev’s Theorem is much simpler to prove.

The path

The proof of the weak version of Chebyshev’s theorem consitutes an excellent application of some fundamental notions of number theory. On the basis of some properties of binomial coefficients, we’ll study some of the most important functions of number theory:

  • the function \theta^{\star}(x), which returns the product of the prime numbers less than or equal to x, and its logarithm \theta(x);
  • the function \psi^{\star}(x), which returns the least common multiple of the positive integers less than or equal to x, and its logarithm \psi(x);
  • finally, we’ll study the already cited function \pi(x), which returns the number of prime numbers less than or equal to x.

Map of paths for number theory
General introduction to paths