Did I make a mistake? Or is analytic continuation just a retarded meme?
Did I make a mistake? Or is analytic continuation just a retarded meme?
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1+2+3+... != -1/12 no matter how many times you watch the numberphile video
That's not even my question
Your mistake was at Σn = -1/12 and the guy who replied to you was on point.
Oh and at Σn^3=ζ(-3)
Τhat's not what ζ(-3) is.
what is it then?
is the riemann zeta not the summation of the reciprocal of all natural numbers raised to the power of s (in this case -3)?
I worded really poorly, but you know what i mean
It is not. The zeta function is defined as Σ1/n^s for s with Re(s)>1 and then, for the rest of the complex plane, you take the (unique) entire function which agrees with Σ1/n^s for s with Re(s)>1.
For Re(s)
>!= as "not equal"
Stay pleb Veeky Forums
is it not true though that
1 + 2 + 3 + 4 + ...
has an "assigned value" of -1/12 because that is where the analytic continuation of the riemann zeta function lands at -1/12 for s = -1, even it doesn't literally equal it?
And if so, does my problem not remain intact in the sense that multiple values are assigned to the same thing?
I don't know much about this form of math (I dropped out of high school and never studied maths at a higher level) so forgive me if I come across as a complete moron.
>is it not true though that
1 + 2 + 3 + 4 + ...
Yes, it is true. Since that continuation is unique(it's a bit hard to prove that, but it is true), you can say that Σn corresponds to -1/12. But it is not equal to it and you can't do operations with -1/12 and get truths about Σn. Your picture in the OP shows that.
>forgive me if I come across as a complete moron.
You definitely don't sound like a moron. A moron wouldn't ask such questions.
To be clear you can use -1/12 to get info about the behavor of Σn, but I haven't studied about that. I remember hearing that it was being used in string theory or something.
You can also take a look at math.stackexchange.com
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Let's do the following proof by contradiction:
>Assume that one can assign [math]\sum{n^s}[/math] the value of [math]\zeta(s)]/math] for Re(s) < 1
>(a) 1 + 2 + 3 ... = -1/12
>(b) 1 + 1 + 1 ... = -1/2
>Subtracting (b) from (a) yields 0 + 1 + 2 + 3 ... = -1/12 - (-1/2)
>0 + 1 + 2 + 3 ... is still 1 + 2 + 3 ...
>5/12 = -1/12
See something wrong there?
> != isn't 'not equal'
>Did I make a mistake?
Yes.
>Or is analytic continuation just a retarded meme?
Yes.
Terry Tao explains the values of zeta at negative values as the constant value appearing in the
(not necessarily convergent) asymptotic expansion using a sufficiently decaying cutoff function.
Shit, repost of
My bad
:(
We eat the same food
He should have XOR'd
OP is sloppy OR he believes the Law of excluded middle.
Cs fag go home