Study the character of the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$

Question

Discuss the character of the series

$$\sum_{n=1}^{\infty} \left|\frac{1}{n^{2z}}\right|^2$$

where $z\in \mathbb C$ and $|z|=\frac{1}{4}$.

Any suggestions please?

Thank you very much

Answer 1

Note that $$ \left|n^{-2z}\right|^2=\left|n^{-2\Re (z)-2 i \Im (z)}\right|^2=\left(n^{-2 \Re(z)}\right)^2=n^{-4\Re(z)} $$ for $n\ge 1$, so the sum is $$ \sum_{n=1}^{\infty}n^{-4\Re(z)}=\zeta\left(4\Re(z)\right), $$ with convergence only for $4\Re(z) > 1$. If we are given that $\left|z\right|=1/4$, clearly $\Re(z) \le 1/4$, and so the series diverges on the specified circle in the complex plane.