About Riemann's zeta function

Question

Is the riemann zeta function analytic? If so can it be expressed as a power series? Does it have a ratio of convergence ? Could it be said to have a center point of its ratio of convergence at +infinity where part of its circumference is the line RE(z)=1 ?

Answer 1

The Riemann zeta function is meromorphic, so it is analytic at every point except for the simple pole at $s=1$. Yes it can be expressed as a globally convergent Laurent series; look up the Stieltjes constants. The original p-series $\sum n^{-s}$ only converges in the abscissa ${\rm Re}(s)>1$, which may be thought of as a generalized circle around infinity with infinite radius and boundary ${\rm Re}(s)=1$.