Chebyshev Functions and Inclusion-Exclusion

Abstract

The Chebyshev functions θ and ψ are defined by
θ(x)=∑log p, ψ(x)=∑log p
where the sums are taken over primes and prime powers respectively. Clearly θ(x)≤ψ(x) and it is not difficult, using Chebyshev's theorem in a weak form, to prove that
0≤ψ(x)-θ(x)≤θ(x^1/2)log x/log 2
  ≤π(x^1/2)(log x)²/2 log 2
  ≤4x^1/2log x/log 2.