1. ## Big Oh question

I am trying to prove an idientity I have come accross in a book:

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

I have managed to prove,

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

using various facts such as

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

But I can't prove the above more difficult identity. Can anyone help me with this?

2. ## Re: Big Oh question

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

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

I have managed to prove,

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

using various facts such as

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

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 $\displaystyle t$ then $\displaystyle \log(t)\gg \log(\log(t)) +O\left(\frac{\log\log t}{\log t}\right)$ so:
$\displaystyle \log\left(\log (t)-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)\right)=(\log \log (t)) + \log\left(1+\frac{-\log\log (t)+O\left(\frac{\log\log (t)}{\log (t)}\right)}{\log(t)}\right)$
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 $\displaystyle O(\frac{\log \log t}{(\log t)^2})$ term absorbs most of them.