## Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 – \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by proving that $$\sum_{k = 2}^\infty \frac{\log ^2 k}{k^2} = \zeta”(2) \approx 1.98928$$ and then check with an aid of computer that for … Read more