AxiomOfChoice
AxiomOfChoice
Joined: Sep 12, 2012
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March 2nd, 2014 at 11:55:57 AM permalink
Quote: s2dbaker

Then I'm just missing the second definition of the Zeta function. It might help me to understand better if I knew the definition of the Zeta function that WolframAlpha uses when n is less than 1. Anyone know what that is?



There is only one Reimann zeta function (it's not "the definition that Wolfram Alpha uses", it's just "the definition"). It's defined and explained here: http://en.wikipedia.org/wiki/Reimann_Zeta_function.

The math is non-trivial. You should probably also read and understand this: http://en.wikipedia.org/wiki/Analytic_continuation and read and REALLY understand this: http://en.wikipedia.org/wiki/Divergent_series

Note that there is difference between saying "A method assigns the value x to the sum" and "The value of the sum is x", because different methods can assign different values to the same sum.
24Bingo
24Bingo
Joined: Jul 4, 2012
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March 2nd, 2014 at 11:58:06 AM permalink
We've told you. The definition is the unique meromorphic function that's equal to that sum wherever it converges. That doesn't mean that the sum converges wherever the function is defined.

One way to calculate this for numbers less than zero is the identity ζ(x)*Γ(x/2)/πx/2 = ζ(1-x)*Γ((1-x)/2)/π(1-x)/2, the derivation of which is spelled out here. (For numbers between zero and one, it turns out that (1-21-x)-1*Σ((-1)n-1/nx) is equal to that sum where it converges, and that gives you those values.)
The trick to poker is learning not to beat yourself up for your mistakes too much, and certainly not too little, but just the right amount.
andysif
andysif
Joined: Aug 8, 2011
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October 3rd, 2015 at 4:03:27 AM permalink
I understand that infinite sums are only defined when they converge. I would have shouted "bull" to this -1/12 thing as well.

But this guy seems to be a math professor and he seems to know what he is talking about.

https://www.youtube.com/watch?v=0Oazb7IWzbA

The way I get it, he is not saying 1+2+3+... EQUALS -1/12, but what he is trying to say is, when questions arise in physics that involve infinite sums like the sum of all paths in quantum mechanics, one can often substitute it with -1/12 and get the "correct" answer. Sort of like a back door trick, but the important point is that IT WORKS.

Anyone know if this is real? Any other examples?
TheGrimReaper13
TheGrimReaper13
Joined: Sep 25, 2015
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October 3rd, 2015 at 3:11:43 PM permalink
Gee, with a bit of work to consider also the negative integers, maybe we can mold the -1/12 into say +/- 1/6.
So much bullshit; so little time!

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