/sci/ - Science & Math

How Can The Sum Of Natural Numbers Be Real If The Riemann-Zeta Function Isn't Real?

>>8562379
Is this some kind of complex joke?

>>8562378
Numberphile said it was legit. Proof enough for me.

>>8562381
no, it's a real joke

>>8562379
Jaden please

In math? Why not. In physical world? No.

>>8562398
It's used in QFT faggot.

>>8562378
Seems rational...

>>8562400
He said real world anon

If S is this sum, you can show that [math]0 \,=\, S \,-\, 2\,S \,+\, S \,=\, 1[/math]. Stop falling for Numbermemes.

>>8562410
Prove it.

>>8562385

Its true in the right context

>>8562414
Here's your spoonfeeding, Pajeet.

>>8562378
The result is a regularization.

Note [math]\sum\limits_{n = 1}^\infty {n{e^{ - \varepsilon n}}} = \frac{{{e^\varepsilon }}}{{{{\left( {{e^\varepsilon } - 1} \right)}^2}}} = \frac{1}{{{\varepsilon ^2}}} - \frac{1}{{12}} + O\left( {{\varepsilon ^2}} \right)[/math].

>>8562728

Just to flesh this out a bit:

https://www.youtube.com/watch?v=jcKRGpMiVTw

This video makes it much more clear what people mean when they say "the sum of the natural numbers is -1/12". It's a distorted statement and very clickbaity; I refuse to watch any channels that make this claim without qualifying the statement with some amount of exploration of the Riemann Zeta function.

>ITT: confusing a function for the analytic continuation of that function

>>8562960
yeah i saw that video too

>>8562676
This is a twisted conception of math.

If we agree that S is the sum of number till infinite, we should say it's infinite, and infinite minus 2 infinites plus infinite leads to and undefined result.
As long as the sum of numbers reaches a certain number, the theory this video shows goes to shit because:

1+2+3+4+....+ a + a+1 + ....+ k
0 -2 -4 -6 -....- 2(a-1)- 2*a -....- 2*(k -1) - 2*k
0+0+1+2+...+ a-2 + a-1 +... + k -2 + k-1 + k
= 1+0+0+0+...+ 0 + 0 +........+ 0 - k -1 + k = 1 -1 -k +k = 0

See? It's stupid to think the result is 1.
You don't handle infinites that way.
Much less is it's in a case like this where the sums are the same so therefore they "stop" when n = k, and if they're infinite a higher k is reached every time and the whole thing cancels itself with the "last" step.

Besides I hate asian faces so I had to prove him wrong.

>>8562676
This is a twisted conception of math.

If we agree that S is equal to S (damn I can't belive I have to explain this) as long as the sum of numbers reaches a certain number, the theory this video shows goes to shit because:

1+2+3+4+....+ a + a+1 + ....+ k
0 -2 -4 -6 -....- 2(a-1)- 2*a -....- 2*(k -1) - 2*k
0+0+1+2+...+ a-2 + a-1 +... + k -2 + k-1 + k
= 1+0+0+0+...+ 0 + 0 +........+ 0 - k -1 + k = 1 -1 -k +k = 0

See? It's stupid to think the result is 1.
You don't handle infinites that way.
Much less is it's in a case like this where the sums are the same so therefore they "stop" when n = k, and if they're infinite a higher k is reached every time and the whole thing cancels itself with the "last" step.

Besides I hate asian faces so I get all the cringes when they try to explain things.

>>8562676
This is a twisted conception of math.

If we agree that S is equal to S (damn I can't believe I have to explain this) as long as the sum of numbers reaches a certain number k, the theory this video shows goes to shit because:

1+2+3+4+....+ a + a+1 + ....+ k
0 -2 -4 -6 -....- 2(a-1)- 2*a -....- 2*(k -1) - 2*k
0+0+1+2+...+ a-2 + a-1 +... + k -2 + k-1 + k
= 1+0+0+0+...+ 0 + 0 +........+ 0 - k -1 + k = 1 -1 -k +k = 0

See? It's stupid to think the result is 1.
You don't handle infinites that way.
Much less in a case like this where the sums are the same so therefore they "stop" when n = k, and if they're infinite a higher k is reached every time and the whole thing cancels itself with the "last" step.

Besides I hate asian faces so I get all the cringes when they try to explain things.

>>8562400
That is evidence that QFT is on the wrong track. Obviously any math that doesn't correspond to reality should be thrown out, because the point of math is to describe/predict/model reality. It's obvious that in reality if you have an uncountable positive number of something, that is not the same as having negative one twelfth of that thing.

>>8562378
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

>>8563139
QFT is not like String Theory. Things like this aren't just used because they give mathematically "nice" results, but because they result in predictions that align with experimental values.

>>8562728
That approaches infinity as epsilon tents to infinity, fool.

>>8563139
>That is evidence that QFT is on the wrong track

Actually it's evidence of the opposite. If the predictions made by QFT using this formula weren't vindicated, then then formula would have been thrown out. The formula in fact does describe/predict/model reality quite well in the appropriate context, which is why it continues to be used in the theory.

https://en.wikipedia.org/wiki/Casimir_effect

>>8563148
Yes it does. But in QFT you deal with stuff like that all the time by setting bounds like [math]\frac{1}{{{\varepsilon ^2}}} < \Lambda [/math], determined by experiment. This then allows you to follow through and make calculations.

>>8563097
>is given proof that you can't sum naturals in a linear and stable fashion
>rejects the proof by summing in a completely different way (the correct one)
Are you deeply retarded or have you been half-asleep each time you've been exposed to Numberphile memes?

>>8563139
The summation index is countable though.

>>8562378
It's not true.
Proof :
By definition, [math]\zeta(s)[/math] is only defined for [math]\mathrm{Re}(s) > 1[/math]. The sum [eqn]1 + 2 + 3 + ... = \frac{-1}{12}[/eqn]
is obtained by computing [math]\zeta(-1)[/math] . Since [math] ]\mathrm{Re}(-1) = -1 \ngtr 1[/math], using the zeta function to compute this sum is a mistake.
A funny mistake, but a mistake.

>>8562440
THAT WAS THE JOKE >>8562381 MADE YOU FUCKING FAGGOT.
REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

>>8563626
>[math]\ngtr[/math]

>>8563657
good reaction image

>>8563676
Thanks if not sarcastic.

Here is a great video where this sum is mentioned.
https://youtube.com/watch?v=sD0NjbwqlYw
You have to see it in the context of the Riemann Zeta function, otherwise it is incorrect.

>>8562410
More simply, you CANNOT change the order of terms in a divergeant sum.

>>8563142
yall bitches read this

smoke tarry tao ery day

>>8563626
>>8563657
*<
sorry :/

>>8563657
>>8563676
at least post the real judy.

>>8565464
Without changing the order. See >>8562676

>>8562378
By inducyion on the number of adds!
Base case (1 add): 0 + 1 = 1 > 0 is positive
Now suppose the sum up to K is positove. We add K+1 which is >= 2 and therefore positove. So the sum cannot be negative assuming omega consistency

t. CS first year

>>8565556
so what's anorexia gonna do when she hits 30?

>>8565556

>>8562381
>>8562379
this was amazing actually

https://m.youtube.com/watch?v=sD0NjbwqlYw
best math channel

>>8563626
you actually have no idea what you are talking about mate

>>8567009
>>8566996
Anzu is a pure princess. she is very healthy!

>>8562378
>Do you guys believe this,if not demonstrate why not

It's not really a sum...

It's the y intercept of a slope that represents the infinite series.

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

>>8563139
– the sum isn't uncountable
–math doesn't have a purpose fuck u
– although the sum doesn't equal \frac{-1}{12} it behaves like it algebraically so that's what physicists do to wave some hands and make some progress in their nonsensical (imo) field

>>8562728
>>8562938
ty for being the voice of reason, anon

>>8565556
>dat cute little ass
*heavy breathing*

>>8562378
The conclusion isn't irrational

>>8563848
I was sincere

>>8567439
>math doesn't have a purpose

Yes it does. There's a reason we construct math so that 1 doesn't equal 2 -- we want math to coorespond to reality. We could construct it any way we want, but we choose the method we do because it works best as a tool. There's nothing else to it.

>>8562938
Hey that's my math lecturer!