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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> extending the domain of an infinite series using analytic continuation
The phrase “using analytic continuation” is pretty misleading. There is not in fact a specific “method” of analytic continuation. Analyticity is simply a property that implies a function satisfying this property has **at most one** extension to an analytic function on a larger region. It does *not* assure you that such an extension is possible, and it might not be.
A function defined on a subset of the complex plane is called analytic when it can be expressed around every point in that subset as a power series. It turns out that two analytic functions on a subset of C that are equal on some tiny disc in that subset must be equal everywhere in the subset. (For the experts, I am glossing over the issue of connectedness of the subset for simplicity.) This may not be true for functions that only satisfy some weaker conditions than being analytic, such as continuity: if two continuous functions on a subset of C are equal on a tiny disc in that subset, it does *not* mean they have to be equal on the whole subset. Mathematicians say the property of being analytic is much more “rigid” than the property of being continuous.
Now for “analytic continuation”… if I have a subset S of the complex plane containing a disc and a larger subset T, and I can extend an analytic function on S to an analytic function on T by two methods, then the two functions I get on T *must be the same* because they are equal everywhere on the original set S (which contains a tiny disc) and two analytic functions on T that are equal on a tiny disc must be equal everywhere on T. Have we shown analytic functions on S can always be extended to analytic functions on T? Not at all. We only showed each analytic function on S can have *at most one* extension to an analytic function on T. We did not show there must be an analytic extension of the function on S to T at all, and maybe there isn’t. The term “analytic continuation” just refers to this “at most one” property of the extended function on T. It does not tell you at all how to find the extended function.
The demonstration that a specific analytic function on S can be extended to an analytic function on T requires serious work. There’s no guarantee it can be done “by analytic continuation”. All that we can say ahead of time is that if we can extend an analytic function on S to an analytic function on T, then there is just one result: two approaches to doing this must lead to the same function on T. Actually building an analytic function on T that extends some analytic function on S may be easy or it may be hard. In Riemann’s paper on the zeta function, he gave two methods of extending the zeta function from the half plane Re(s) > 1 to C – {1}. His methods look quite different, but the two functions they lead to on C – {1} have to be the same function (so his methods lead to the same function on C – {1} given by two rather different-looking formulas) because of the “at most one extension” property of analytic functions being extended to larger subsets.
The extension of the zeta function beyond the half plane Re(s) > 1 does not use the infinite series expression for the zeta function, since that series doesn’t converge elsewhere. Instead we use other (more complicated-looking) formulas for the zeta function, typically in terms of integrals instead of infinite series, and those other formulas have the advantage that by massaging them in the right way we can see they make sense on C – {1}. We don’t have to worry about different formulas for the zeta function on Re(s) > 1 leading to different extensions of the function on C – {1}, even if the resulting extension formulas look different, because if we can check the extended functions are analytic then we are guaranteed that those different-looking formulas on C – {1} must be the same function by what I wrote about in previous paragraphs.
Verifying that a specific analytic function on a specific subset of C can be extended to an analytic function on some larger subset of C is often hard work (and in many cases is still an unsolved problem). We can’t wave some magic wand and say “we did it by analytic continuation”. The property of analyticity assures us that an extension of the function is unique if it exists, but it doesn’t tell us an extension of the function exists (while maintaining the property of analyticity).
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