If one's goal were to calculate one value of the zeta function, say $\zeta(-1)$ is at all feasible to do this explicitly by calculating Taylor series and doing an analytic continuation "by hand" - or will the work involved therein be tantamount to developing e.g. the functional equation given below?
Suppose you start with $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ which is defined on the real line for $s > 1$ and your goal is to calculate some other value of the analytic continuation, say at $s = -1$. I've read many great expositions that ultimately lead to the functional equation $$\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s)\zeta(1-s).$$ The derivations aren't too bad, and as a partial solution, it is quite direct (albeit somewhat clever) to extend $\zeta(s)$ with the $\eta$ function, which would work if you wanted to evaluate somewhere with $\mathrm{Re}(s)\geq 0$.
Anyways, to my question: In theory, without much ingenuity, one could do the following:
- Note that the given definition for $\zeta(s)$ defines an analytic function on the half plane Re$(s) \geq 1$. We've extended $\zeta(s)$ a bit.
- We calculate the Taylor series for $\zeta(s)$ at the point $s=6+12i$. One would discover a radius of convergence of $13$. Doing so requires either calculating derivatives or could be done by integration.
- One has now extended $\zeta(s)$ to include e.g. the point $s = -4+12i$. One could then calculate the Taylor series around that point and would find a radius of convergence of $13$ once again.
- This last Taylor series would be sufficient for calculating $\zeta(-1)$.
Are these calculations feasible? I suppose by that I mean, would we be able to write down enough detail in steps 2 and 3 so that in step 4 we could see that $\zeta(-1) = -1/12$? I'm wondering if perhaps the answer to this question is: Yes, but in doing so, you would essentially develop everything in the functional equation above. It's no "easier" to do it this way. Thanks for any thoughts.