This post continues the discussion we had in my previous post.
I discussed only $k=0$ case before dismissing the others as uninteresting. They aren't after all. After some further work, I think I've figured out how the first DGF given above generalized to the other cases as well.
Following the same notations given before we have,
$\displaystyle\sum_{n=1}^\infty\frac{\sigma_k(m\cdot n)}{n^s}=\zeta(s)\zeta(s-k) \prod_{p_j|m}\left(1+(p_j^{k}+p_j^{2k}+p_j^{3k}+\cdots+p_j^{m_jk})(1-p_j^{-s})\right)$
Quite simple after all!! But I don't think the second DGF simplifies as nicely. That should be one for another post. But there is something else.
The first formula given in the previous post works even when $n$ is replaced by $n^2$. We are gonna make that replacement because of the closed form DGF we have for $\sigma_k(n^2)$.
Again following the same notations, this time we have,
$\displaystyle\sum_{n=1}^\infty\frac{\sigma_k(m\cdot n^2)}{n^s}=\frac{\zeta(s)\zeta(s-2k)\zeta(s-k)}{\zeta(2s-2k)} \prod_{p_j|m}\left(1+(p_j^{k}+p_j^{2k}+\cdots+p_j^{m_jk})\left(\frac{1-p_j^{-s}}{1+p_j^{-(s-k)}}\right)\right)$
As before, these DGFs can be used in combination with Dirichlet's Hyperbola method to find their summatory functions. Hope you find these interesting and worth your while. See ya soon.
Till then
Yours Aye,
Me
I discussed only $k=0$ case before dismissing the others as uninteresting. They aren't after all. After some further work, I think I've figured out how the first DGF given above generalized to the other cases as well.
Following the same notations given before we have,
$\displaystyle\sum_{n=1}^\infty\frac{\sigma_k(m\cdot n)}{n^s}=\zeta(s)\zeta(s-k) \prod_{p_j|m}\left(1+(p_j^{k}+p_j^{2k}+p_j^{3k}+\cdots+p_j^{m_jk})(1-p_j^{-s})\right)$
Quite simple after all!! But I don't think the second DGF simplifies as nicely. That should be one for another post. But there is something else.
The first formula given in the previous post works even when $n$ is replaced by $n^2$. We are gonna make that replacement because of the closed form DGF we have for $\sigma_k(n^2)$.
Again following the same notations, this time we have,
$\displaystyle\sum_{n=1}^\infty\frac{\sigma_k(m\cdot n^2)}{n^s}=\frac{\zeta(s)\zeta(s-2k)\zeta(s-k)}{\zeta(2s-2k)} \prod_{p_j|m}\left(1+(p_j^{k}+p_j^{2k}+\cdots+p_j^{m_jk})\left(\frac{1-p_j^{-s}}{1+p_j^{-(s-k)}}\right)\right)$
As before, these DGFs can be used in combination with Dirichlet's Hyperbola method to find their summatory functions. Hope you find these interesting and worth your while. See ya soon.
Till then
Yours Aye,
Me
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