What are the periodic Dirichlet series?

Question

My question is: What are the periodic Dirichlet series? Does the Riemann zeta function $ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ and the alternating zeta function $η(s)=\sum_{n=1}^\infty (-1)ⁿ⁻¹/n^{s}$ are periodic?

Answer 1

Look in that thesis by Tapas Chatterjee, where you find the answer is yes for both the Riemann zeta function and the alternating zeta function.