How to show $\sum 1/n^s$ doesn't converge for $0\leq Re(s) \leq 1$?

Question

There are many sources which show fairly easily that for $\sum 1/n^s$ with complex $s$:

• the infinite sum converges for $Re(s)>1$
• the infinite sum diverges for $Re(s)<0$

However I can't find accessible proofs that show:

• the infinite sum diverges for $0 \leq Re(s)<1$
• the infinite sum doesn't converge (oscillates?) for $Re(s)=1$ and $s\neq 1$

To be clear, I am not asking about the analytic continuation of the Riemann Zeta function, only the power series $\zeta(s)=\sum_{n=1}^{\infty}1/n^s$.

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Notes:

I have asked a related question (link) which is not as precise as this one. That one wasn't answered fully.

I have searched stack exchange and math overflow (and academic papers/notes) and have not found accessible proofs. Where this question is discussed the commentary glosses over the working.