Consider the periodic Dirichlet series that has this iterative definition:
$\text{a1}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2}{\sqrt{6}}+...$
$\text{a2}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2+a1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2+a1}{\sqrt{6}}+...$
$\text{a3}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2+a1+a2}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2+a1+a2}{\sqrt{6}}+...$
$\text{a4}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2+a1+a2+a3}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2+a1+a2+a3}{\sqrt{6}}+...$
$...$
continuing this iteration, will it converge to zero?