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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 ?

anon
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user128932
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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 ${\rm Re}(s)=1$.

anon
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  • Can the Riemann zeta function have two distinct regions of divergence ,each region connected; in the right-half plane? – user128932 Apr 17 '14 at 01:59
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    @user128932, yes. This is implicit in sea's answer: the zeta function is analytic in the right half plane in $$;\left{z\in\Bbb C;;;0\le\text{Re},z<1\right}\cup\left(\left{z\in\Bbb C;;;\text{Re},z\ge 1\right}\setminus{1}\right)$$ – DonAntonio Apr 17 '14 at 02:19
  • I should have wrote, can the right half plane be divided into at least three regions, each one separately connected, one being {z an element of C; 0<=REz <1} and two distinct circular regions of DIVergence that are subsets of ({z an element of C;REz>=1}{1})?? – user128932 Apr 17 '14 at 02:30
  • @user128932 Divergence of what? Like I said, the original p-series converges in the half plane Re(s)>1. Are you looking for a different series expression for the Riemann zeta function? Also, asking for two distinct circular regions (so, like, circles or annuluses?) seems extremely strange and random; I expect it would be topologically trivial not to mention meaningless with the right series expression. – anon Apr 17 '14 at 20:35
  • Sorry I should have said the inverse of the Riemann zeta function. Does this have a connected region of points = W such that any point a+bi , where 'zeta'(a+bi) is an element of W and it approaches infinite, in other words ,divergent? – user128932 Apr 18 '14 at 02:09
  • Can the inverse of the Riemann zeta function for any z ( an element of C) ; Re >= 1 \ {1} ; have two distinct regions of convergence? – 201044 Sep 14 '15 at 03:59