I’m basically a layman trying to understand the Riemann Hypothesis, and in my journey so far I’ve made it past the zeta function, somewhat understand the part about using complex numbers as exponents, and up to the part about ‘extending the domain of an infinite series using analytic continuation’.
How exactly are mathematicians finding values for diverging series using it? Are they basically winging it by using geometry or is there a specific way?
In: 1
Are you familiar with [Taylor series](https://en.wikipedia.org/wiki/Taylor_series)? An infinite polynomial, made from all derivatives of a function at one point, which converges to said function around that point?
Analytic continuation basically uses the Taylor series to “bunny-hop” around “bad points” (like `zeta(1)`, where it blows up) to define the function everywhere, except at those “bad” points.
You take a point where your function and derivatives exist and construct a Taylor series there. The series gives you values around that point (including complex), but eventually diverges. You pick one point, where it still converges, and construct a second Taylor series, which defines more points, you pick one and construct third Taylor series… and so on, until you reach any point you want.
I recommend [this video](https://youtu.be/CjSKmcWRFzE?t=523) about analytic continuation. Actually, I recommend the whole channel: they have good videos exactly on the Riemann Hypothesis. They even [have a video](https://youtu.be/oVaSA_b938U), that explains the hypothesis without zeta function.
Edit: Zeta function can be extended a bit even without analytic continuation. You just need to isolate “bad point” at `zeta(1)`. See [eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function), it is defined for x>0, and has a formula, that gives zeta.
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