I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a way which can be written down as a single formula. (In principle, an analytic continuation can be written down as a formula, but no one ever seems to do it this way, and always end up appealing to drawings in the end.) Any formula will do, even a piecewise one, as long as there is a relatively elementary proof that the expression is equal to $\sum_{n\in\Bbb N_0}n^{-s}$ when $\Re[s]>1$ (and to constrain piecewise solutions which just use the standard definition in this region, you should also need to be able to show that it is a reasonable extension in some way, i.e. continuity or analyticity).
My current best bet is the Laruent expansion about the pole at $s=1$ (in terms of Stieltjes constants):
$$\zeta(s)=\frac1{s-1}+\sum_{n=0}^\infty\frac{(-1)^n(s-1)^n}{n!}\!\!\lim_{m\to\infty}\left[\sum_{k=1}^m\frac{(\ln k)^n}k-\frac{(\ln m)^{n+1}}{n+1}\right]$$
However, I was unable to find a proof online of the equality of this expression to the standard one, and I expect the proof to be rather involved. Does anyone have any suggestions for more workable formulas for $\zeta(s)$?
Considering the accepted answer for this related question and following Gerry Myerson's answer elsewhere, I found Sondow's paper.