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 have a function on part of the complex plane, there are at most **one** reasonable way to extend it to the entire plane. “Extend” means we make a new function that have the same value as the old function, except that for certain input where the old function say “undefined” the new function actually give an output. And “reasonable” means the function is analytic, which, in simple but somewhat inaccurate terms, mean you can describe it using one nice formula that works everywhere. The process of making that new function is called analytic continuation.
The fact that there is at most only 1 way to extend has crucial important consequence: the values of a function at **any** tiny piece on the plane contains **all** information about the function everywhere else. This is what make analytic continuation very special. You can study the newly function potentially anywhere and obtain useful information about the old function.
In particular, for Riemann zeta function, the infinite sum is a partial function: it is only defined for input where the real part is >1. We extend the function by analytic continuation. The new function is NOT defined by this same infinite sum. However, all information we could know about this infinite sum can be obtained from anywhere on this new function.
So the question is which part of the new function is useful to study? By reflection formula, everything that happen when the real part of the input is <0 is pretty boring and completely predictable: there is a direct formula that relate the real part >1 side and the real part<0 side, point-by-point, so it’s not that useful. But the useful part is the critical strip, when the real part is between 0 and 1 (inclusive). We have a formula that relate behavior of each zero of the function on this region to aggregate behavior of the infinite sum across different input. Which basically let us transform a problem in one context into a problem in completely different context. The aggregate behavior of the sum is related to the distribution of primes, and instead of studying this aggregated behavior directly, we study the zero of the zeta function.
In fact, we had had small successes with this. The prime number theorem was proved by showing that there are no zeros occur on the boundary of the critical strip.
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