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Old 2009-01-22, 00:06   #1
brownkenny
 
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Jan 2009

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Default 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.
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