I would like to know why I can start supposing that $\sigma_{0}=0$ in the proof of the following theorem about Dirichlet series:
"Let $a_{n}\ge0$ for every $n\ge1$. Then the point $s=\sigma_{0}$ is a singularity of $f$."
Where $f$ is the Dirichlet series $\displaystyle\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$ and $\sigma_{0}$ is the abscissa of convergence.
If needed for more context I can indicate the full proof of the theorem.
Thank you in advance!
Say $f(s)=\sum\frac{a_n}{n^s}$. Define $g(s)=f(s+\sigma_0)$. Then $g(s)=\sum\frac{b_n}{n^s}$ where $b_n=a_n/n^{\sigma_0}\ge0$.