I'm really not a π day hater, but I do roll my eyes when I read articles exclaiming π "goes on forever!" as if that is a rare and notable property that makes it unique among numbers. I mean, 1/3 "goes on forever" too.
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Replying to @divbyzero
What's really needed is a better appreciation of Gotthold Eisenstein's penetrating analysis of functions of period 1. When everyone knows the wonders lurking in ... + 1/(x - 2) +1/(x - 1) + 1/x + 1/(x + 1) + 1/(x + 2) + ... then we will indeed all have cause to celebrate.
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Replying to @MathPrinceps @divbyzero
That's pi*cot(pi*x), you can prove it using complex analysis
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Some Googling says this was first discovered by Euler (presumably using methods other than complex analysis)
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Replying to @akivaw @divbyzero
Eisenstein's methods reach extremely far, and indeed inaugurated a profound sequence of developments in analytic number theory that continue to this day. Yet they relied essentially on "only high school algebra." It's not the tool that matters; it's the touch of a master's hand.
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Replying to @MathPrinceps @divbyzero
I see. I oughta look this stuff up. (Just noticed that if you plug in x=¼ you get the Leibniz series for pi)
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Replying to @akivaw @divbyzero
Indeed, this approach also provides the most natural elementary explanation of Euler's answer to the Basel problem -- that is, to why the sum of the reciprocal squares is [(pi)^2]/6.
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Replying to @MathPrinceps @divbyzero
This is the log-derivative of the product formula for the sine, right? Which is connected to both the Basel problem and the Wallis product. (Basel, Wallis, and Leibniz united under one formula, woot)
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Replying to @akivaw @divbyzero
Yeah. And the key here is that Eisenstein unites all these threads by stressing periodicity, which he enforces via what is now called "averaging over the group" (in this case, the additive group of the integers.) Hard to exaggerate how far these ideas reach.
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Replying to @MathPrinceps @divbyzero
So what does this look like for other groups?
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Exactly the right question! The next group Eisenstein studied was the translation group in two dimensions, and its discrete subgroup of "integral translations." That gave him his theory of elliptic functions. Then you look at the modular group, and get automorphic functions.
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Replying to @MathPrinceps @divbyzero
Ah, yeah. Weierstrass p and all that. I started reading a book about this stuff. It's way over my head, but as far as I can tell, it's just magic. The fact that holomorphic functions on compact Riemann surfaces are constant is so overpowered.
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Replying to @akivaw @divbyzero
In some sense, the Liouville theorem and its generalizations are at the heart of everything; it is on their account that all sorts of crucial vector spaces are provably finite-dimensional -- and from this a nearly limitless set of astounding (number-theoretical) things follow.
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