Sunday, 14 October 2018

sequences and series - Intuition behind zeta(1) = frac112

When I first watched numberphile's 1+2+3+... = 112 I thought the sum actually equalled 112 without really understanding it.



Recently I read some wolframalpha pages and watched some videos and now I understand (I think), that 112 is just an associative value to the sum of all natural numbers when you analytically continue the riemann-zeta function. 3Blue1Brown's video really helped. What I don't really understand is why it gives the value 112 specifically. The value 112 seems arbitrary to me and I don't see any connection to the sum of all natural numbers. Is there any intuition behind why you get 112 when analytically continue the zeta function at ζ(1)?



EDIT(just to make my question a little clearer):
I'll use an example here. Suppose you somehow didn't know about radians and never associated trig functions like sine to π but you knew about maclaurin expansion. By plugging in x=π to the series expansion of sine, you would get sine(π) = 0. You might have understood the process in which you get the value 0, the maclaurin expansion, but you wouldn't really know the intuition behind this connection between π and trig functions, namely the unit circle, which is essential in almost every branch of number theory.



Back to this question, I understand the analytic continuation of the zeta function and its continued form for s<0 ζ(s)=2sπs1sinπs2Γ(1s)ζ(1s) and how when you plug in s = -1, things simplify down to 112 but I don't see any connection between the fraction and the infinite sum. I'm sure there is a beautiful connection between them, like the one between trig functions and π, but couldn't find any useful resources on the internet. Hope this clarified things.

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