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The function is defined as follows

[tex] \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} [/tex]

where [tex] s \in \mathbb{C} [/tex]

we know that

[tex] n^s = exp(s\;ln\;n)[/tex]

so I can write

[tex] \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{exp(s\;ln\;n)} [/tex]

but since

[tex] \frac{1}{exp(s\;ln\;n)} = 1 + \frac{1!}{(s\;ln\;n)} + \frac{2!}{(s\;ln\;n)^2} + ... =

1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} [/tex]

how can I write this ''sum within a sum''? ζ(s) here, if I am correct, would be an infinite sum of terms which are infinite sums.

Thank you for taking the time to help!

edit :

Could I say

[tex]1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} = a_k[/tex]

then

[tex] \zeta (s) = \sum_{k=1}^{\infty} a_k[/tex]

Does that make any sense?