I am confused about an approximation that I see in a paper on the quadratic sieve.
I have the following result (given from prior calculations):
$\log u \approx \frac{1}{2}(\log ( \log X))$,
where $u$ and $X$ can be seen as variables. From this, the author derives directly
$u \approx \sqrt{\frac{2\log X}{\log (\log X)}}$.
I have been trying to understand this result but I can't seem to see what are the steps from going to $\log u$ to $ u$ (obviously, here is not just a matter of taking $e^{\log u}$). Is there any relation between the $\log$ and the square root ?
Pomerance exact reference:
"We are looking at the simplier expression \begin{equation*} X^{1/u}u^u \end{equation*}
We would like to choose $u$ so as to minimize this expression. Take logarithms: so we are to minimize \begin{equation*} \frac{1}{u}\log X + u \log u \end{equation*} The derivative is $0$ when $u^2(\log u + 1) = \log X$. Taking the $\log$ of this equation, we find that $\log u \approx \frac{1}{2}\log\log X$, so that \begin{equation*} u \approx (2\log X / \log\log X)^{1/2}" \end{equation*}
It's not true. Let's work backwards. $$ u \approx \sqrt{\frac{2\log X}{\log(\log X)}}$$ i.e. $$ u = (1 + o(1)) \sqrt{\frac{2 \log X}{\log(\log X)}}$$ would imply $$ \eqalign{\log u &= \log(1 + o(1)) + \frac{1}{2} \log \log X - \frac{1}{2} \log \log \log X + \frac{1}{2} \log 2\cr &= \frac{1}{2} \log \log X - \frac{1}{2} \log \log \log X + \frac{1}{2} \log 2 + o(1)}$$ which implies, but is not implied by, $$\log u \approx \frac{1}{2} \log \log X$$ I suspect you're misinterpreting Pomerance. What is the exact quote?
EDIT: OK, so there is more to it than just $\log u \approx \dfrac{1}{2} \log \log X$, there is an equation $$ u^2 (\log u + 1) = \log X \tag{1}$$ That does indeed imply $$ \log u \approx \frac{1}{2} \log \log X$$ and then you put this in for $\log u$ (or $\log u + 1$) in (1) to get $$\frac{u^2 \log \log X}{2}\approx \log X$$ multiply by $2/\log \log X$ and then take square roots.