I need to determine the asymptotic behavior of $$a_n=\sum_{k=2}^{n-2}\frac1{\ln k\ln(n-k)}$$ as $n\to\infty$. I know some elementary methods that might help. For example, split the index $\lvert k-n/2\rvert\le m$ for some appropriate $m$, but I want to study further analytic methods. My idea is the following:
Let $F(z)=\sum_{n=2}^\infty z^n/\ln n$, then $a_n$ is the coefficient of $z^n$ in the Taylor series of $F(z)^2$ at $z=0$. The radius of convergence is $1$. By Cauchy's integral formula, $$a_n=\int_\gamma\frac{F(z)^2dz}{z^{n+1}}$$ where $\gamma$ is the positive oriented unit circle distorted around $1$. Apparently, the major part of the integral is the contribution when $z$ is near $1$. However, I have no idea how to proceed since I have no prior experience on this type of asymptotic estimation. I need some guide on this.
Any idea? Thanks.
we can prove this limit $$\lim_{n\to\infty}\dfrac{\log^2{n}}{n}\sum_{k=2}^{n-2}\dfrac{1}{\log{k}\log{(n-k)}}=1$$
This is a International Competition in Mathematics for Universtiy Students in Plovdiv, Bulgaria 1994 last problem: can see this solution:http://www.imc-math.org.uk/imc1994/prob_sol.pdf