Is this $\sum_{n=1}^{\infty}n^{\zeta(s)}=1$ ,for $s\to1$ ?

Question

let $s$ be a complex variable in the neighborhood of $1$, some calcualtion in wolfram alpha show that the series is convergent but this not convince me i want to know how do i show the bellow result if it is true ?

Question: Is this :$\sum_{n=1}^{\infty}n^{\zeta(s)}=1$ ,for $s\to1$ ?

Answer 1

The serie is not convergent. just try approaching from the right and you will get it totally unbound, so plug $1.0000001$ in wolfram to check it.

The phases of $\zeta(s)$ at the pole depend where you are coming from. From the left it tends to $-\infty$ and that is why it looked convergent.