Section 6.2 Chebyshev’s Estimates
Chebyshev gave the first concrete results toward proving the Prime Number Theorem (which at that time was a conjecture of Legendre and Gauss).
Remark 6.2.2.
Recall that \(f \asymp g\) means \(f = O(g)\) and \(g = O(f)\text{.}\) That is, there are constants \(C,D\) such that \(f(x) \leq C g(x)\) and \(g(x) \leq D f(x)\) for all sufficiently large \(x\text{.}\) And \(f \asymp g\) uniformly for \(x \geq x_0\) means this holds for all \(x \geq x_0\text{.}\)Corollary 6.2.3.
\(\theta(x) \asymp x\) and \(\pi(x) \asymp \frac{x}{\log x}\text{.}\) Explicitly, \(\theta(x) \leq (\log 4)x\) for \(x \geq 1\) and \(\theta(x) \gt \frac{x}{12}\) for all \(x \geq 2\text{;}\) \(\pi(x) \gt \frac{3 \log 2}{8} \frac{x}{\log x}\) for all \(x \geq 2\text{,}\) and \(\pi(x) \lt \frac{(6 \log 2) x}{\log x}\) for all \(x \geq 2\text{.}\)Remark 6.2.4.
\(\frac{3 \log 2}{8} = 0.2599301927\dotsc\text{,}\) \(6 \log 2 = 4.1588830834\dotsc\text{.}\)
The proofs are a bit long and we won’t go over them in class. (An exception is the proof that \(\theta(x) \leq (\log 4)x\text{,}\) which seems to be short and elementary, and very similar to the proof given in class of the Bertrand Postulate).
