From this post @Olivier Oloa gives the closed form for this sum $(1)$
$$\sum_{n=1}^{\infty}\frac{H_n^2-(\gamma+\ln n)^2}{n}=\frac{5}{3}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2\tag1$$
I was trying to generalises $(2)$
$$\sum_{n=1}^{\infty}\frac{H_{kn}^2-[\gamma+\ln(kn)]^2}{n}=F(k)\tag2$$ where $k\ge1$
but I couldn't only got so far finding out the closed form for $k=2$
$$\sum_{n=1}^{\infty}\frac{H_{2n}^2-[\gamma+\ln(2n)]^2}{n}=\frac{11}{12}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2+\eta(1)[\eta(2)-\gamma^2-2\gamma_1]\tag3$$
where $\eta(k)$ is the Dirichlet eta function
Does anyone knows how to find the closed form for $(2)$?
