# Big Oh question

• Nov 22nd 2011, 08:52 PM
aukie
Big Oh question
I am trying to prove an idientity I have come accross in a book:

$\log\left(\log t-\log\log t+O\left(\frac{\log\log t}{\log t}\right)\right)=\log\log t-\frac{\log\log t}{\log t}-\frac{1}{2}\left(\frac{\log\log t}{\log t}\right)^{2}+O\left(\frac{\log\log t}{\left(\log t\right)^{2}}\right)$

I have managed to prove,

$\log\left(\log t+O\left(\log\log t\right)\right)=\log\log t+O\left(\frac{\log\log t}{\log t}\right)$

using various facts such as

• $O\left(O\left(f\left(z\right)\right)\right)=O\left (f\left(z\right)\right)$,
• $g\left(z\right)O\left(f\left(z\right)\right)=O(g(z )f(z))$
• and $\log z=O\left(z\right)$.

But I can't prove the above more difficult identity. Can anyone help me with this?
• Nov 23rd 2011, 12:10 AM
CaptainBlack
Re: Big Oh question
Quote:

Originally Posted by aukie
I am trying to prove an idientity I have come accross in a book:

$\log\left(\log t-\log\log t+O\left(\frac{\log\log t}{\log t}\right)\right)=\log\log t-\frac{\log\log t}{\log t}-\frac{1}{2}\left(\frac{\log\log t}{\log t}\right)^{2}+O\left(\frac{\log\log t}{\left(\log t\right)^{2}}\right)$

I have managed to prove,

$\log\left(\log t+O\left(\log\log t\right)\right)=\log\log t+O\left(\frac{\log\log t}{\log t}\right)$

using various facts such as

• $O\left(O\left(f\left(z\right)\right)\right)=O\left (f\left(z\right)\right)$,
• $g\left(z\right)O\left(f\left(z\right)\right)=O(g(z )f(z))$
• and $\log z=O\left(z\right)$.

But I can't prove the above more difficult identity. Can anyone help me with this?
I think you need to observe that for large enough $t$ then $\log(t)\gg \log(\log(t)) +O\left(\frac{\log\log t}{\log t}\right)$ so:
$\log\left(\log (t)-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)\right)=(\log \log (t)) + \log\left(1+\frac{-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)}{\log(t)}\right)$
This is precisely the part I was stuck on, was not sure how to deal with all terms that arise from the 3rd order expansion of the "big" log term on the right. But then I realized the $O(\frac{\log \log t}{(\log t)^2})$ term absorbs most of them.