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?
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When you use complex numbers, things get very very nice. Specifically, there are very few functions with derivatives and so no two such functions can overlap except at isolated places. This means that if you have such a function defined on one part of the domain then there is basically exactly one way to extend it to larger parts of the complex plane. That is, you can extrapolate data perfectly – as long as you have the right tools. This is what Analytic Continuation *is*.
But that’s not really your question. Your question is about *computing* these values for – specifically – the Riemann Zeta Function. In general, there are two ways to do this, The first is pretty straightforward as there are formulas that hold for EVERY complex value (except 1, where it always diverges), such as [this integral formula for it](https://en.wikipedia.org/wiki/Riemann_zeta_function#Integral). This isn’t in terms of infinite sums, but in terms of integrals which should be understood as easy to calculate because computers can compute pretty much any integral numerically.
The other way is a bit more useful theoretically and we’ll assume that we can compute the infinite series in the places where is DOES converge for this. There are two important regions that would be considered in the “divergent” range for it. The first is everything left of the imaginary line (that is Re(z)<0) and the critical strip between (0<=Re(z)<=1) and these have slightly different ways to compute them. In the first region, we use the [Functional Equation](https://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann’s_functional_equation) to compute things. If you know Zeta(s), then you can use it to directly compute Zeta(1-s) which is evaluating at the point when you reflect it about the vertical line Re(s)=1/2. For the second region, the critical strip, we can actually use the [Dirichlet Eta Function](https://en.wikipedia.org/wiki/Dirichlet_eta_function), which is an “alternating” version of the Riemann Zeta Function. In the places where the zeta function converges, we have the formula Eta(s)=(1-2^(1-s))Zeta(s) but the eta function converges in the critical strip and so this can be used to extrapolate what Zeta(s) should be in the critical strip.
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As for the Riemann Hypothesis (and even less ELI5), these functions actually can initially be a distraction from what it is actually saying. In undergrad curriculum we have a heavy focus on calculus and analysis – including Complex Analysis – but very little emphasis in number theory and so the number theory is often avoided when introducing it. It’s easy to say “All the zeros of the Riemann Zeta Function are on the critical line” to people who have taken complex analysis but it’s harder to make it clear why this matters if you haven’t taken number theory. It’s all at the same “level”, it’s just that undergrad number theory isn’t really number theory.
But it isn’t all that hard. The real place to start with the Riemann Hypothesis is NOT zeta functions or anything, but the [Prime Number Theorem](https://en.wikipedia.org/wiki/Prime_number_theorem) which will introduce the Riemann Zeta Function as a natural thing linked to prime numbers. Roughly, the Prime Number Theorem says that the Nth prime is approximately located near the number Nln(N). Intuitively, you can understand the Riemann Hypothesis as follows: The Prime Number Theorem says that the Nth prime is about N*ln(N) with moderately sized error bars and the Riemann Hypothesis says that the Nth prime is about N*ln(N) with slightly more constrained sized error bars.
The Riemann Zeta Function helps with the Prime Number Theorem because it is [built from prime numbers](https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler’s_product_formula) and so has information about how they are distributed since how they are distributed controls how it converges. You can use it to get lots of info about prime numbers. For instance, that is diverges at s=1 means that there are infinitely many primes and *how fast* it diverges is connected to how many prime numbers there are (it gives a much better approximation than Euclid’s original proof of infinitely many prime numbers does).
So, effectively, the more we know about the Riemann Zeta Function, the more we know about primes. Specifically, because zeros of analytic functions in a way determine the function, the zeros of the Riemann Zeta Function are explicitly tied to the distribution of prime numbers. Like, there *is* a formula for the prime numbers and it uses the zeros of the Riemann Zeta Function in it. Basically, the zeros control periodic frequencies in the distribution of the primes and the more spread out the zeros are the more wildly these frequencies behave and so the more uncontrolled the primes are allowed to be. The Prime Number Theorem is proved by putting literally the most minimal constraints on where the zeros can be, which means that they are not allowed to be as wild as possible – there is some level of tameness to them. The Riemann Hypothesis is merely the hypothesis that the zeros are as constrained as they can possibly be, meaning that the primes have minimal wildness in how they are distributed.
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