Let'sw start from the 'standard' definition of the Riemann zeta function...

(1)

It is well known that (1) converges if . In particular has a singularity in and we suppose preliminarly that in all remaining complex plane it is analytic. In that case we can express the function 'somewhere around' a point in Taylor series...

(2)

... where...

(3)

... and the derivatives in (3) are given by ...

(4)

Now we start the analysis setting [why not?...] . Because we have supposed that the only singularity is in , the Taylor series around computed with (2), (3) and (4) converges in a circle centered in with radious , the 'red circle' in the figure...

The Taylor series converges in any interior point of the 'red circle' so that nobody forbids to expand in Taylor series in any point internal to it, using (2) to compute the derivatives of the function in the new 'central point' that call . Let suppose to determine by a ' counterclockwise rotation' of taking as 'pivot' [see figure...], we obtain a new Taylor expansion in a circle centered in with radious , i.e. the 'black circle' of figure. In that way we have in some way 'extended' the region in which is analytic. Proceeding we can hop to a new 'central pivot point' with a ' counterclockwise rotation' of ' and the same for and [the 'blue circle' in the figure...]. Of course is and that means that we have 'extended' the domain of on the real axis till to . At this point with four more ' counterclockwise rotation' we return to . At the end of the work we have considerably extended the domain of , in particular in part of the region where is . A spontaneous question: how to obtain a larger domain?... an obvious possibility is tho choose a larger value of like 3,4, ... or, why not, ... 'theoretically' we can extend the domain of to the whole complex plane, with the only exception of ...

Kind regards