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- TL;DR Summary
- I wrote $$\zeta (x+yi)$$ as ##\zeta(x)\zeta(yi) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^x} *(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n})##. I want to simplify the second term in terms of the zeta function so I can solve for ##\zeta (x)##.

How do I re-write $$\displaystyle\sum_{n=1}^\infty \frac{1}{n^x} *(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n})$$ in terms of ##\zeta (x)## ? I want to solve for ##\zeta (x)## and simplifying the above expression in terms of ##\zeta (x)## would avoid repetition.

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