Thread: Chebyshev's Estimates View Single Post
 2009-01-22, 00:06 #1 brownkenny     Jan 2009 2×3 Posts Chebyshev's Estimates I've been working through Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" and I'm stuck on the proof of an upper bound for $\pi(x)$. For reference, it's Theorem 3 on page 11. The desired upper bound is $\pi(x) \leq \{ \log 4 + \frac{8 \log \log n}{\log n} \} \frac{n}{\log n}$ Using the bound $\prod_{p \leq n} p \leq 4^n$ it's easy to show that for $1 < t \leq n$ we have $\pi(n) \leq \frac{n \log 4}{\log t} + \pi(t)$ Tenenbaum then gives the bound $\pi(n) \leq \frac{n \log 4}{\log t} + t$ So far, so good. At this point in the proof, Tenenbaum says "The stated result follows by choosing $t = n / (\log n)^2$" and leaves the details to the reader. As much as I've looked at it, I still can't figure out how he arrives at the desired result. Any tips/suggestions? Thanks in advance.