Ramanujan Summation

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Leo Authersh

Main Question or Discussion Point

What does the equation ζ(−1) = −1/12 represent precisely?
It's impossible for that to be the sum of all natural numbers. And it is also mentioned in all the maths articles that the 'equal to' in the equation should not be understood in a traditional way.

If so, then why wikipedia article states that,

1+2+3+.... = - 1/12

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fresh_42
Mentor
It's impossible for that to be the sum of all natural numbers.
Yes.
And it is also mentioned in all the maths articles that the 'equal to' in the equation should not be understood in a traditional way.
Yes.
If so, then why are even wikipedia article states that,

1+2+3+.... = - 1/12
Have you read it and what do you know about the zeta function and analytic continuations?

God, I hate that video. The video is very misleading. I hoped they would be somewhat clear in it.

First of all, the series $1+2+3+4+...$ diverges. You will find no mathematician that disagrees with this. The most natural sum is $1+2+3+4+... = +\infty$.
Now, what is the $-1/12$ thing all about? Well, some mathematicians have found a way to associate a number to divergent series. I would not call that number the "sum" of the series, it is just a number associated to it. In this case, the number associated to $1+2+3+4+...$ is $-1/12$. Now, we often write $1+2+3+4+5+... = -1/12$, but that's where you should be careful, since that $=$ sign does not mean the classical one, in fact it means that we evaluate the series in a nonstandard way (like Ramanujan summation). Now in many circumstances, replacing $1+2+3+4+...$ with $-1/12$ is wrong and a very bad idea, but in some it might work out. It should then be shown why exactly we can replace the sum by $-1/12$.
Also of interest:
https://mathoverflow.net/questions/64898/values-of-the-riemann-zeta-function-and-the-ramanujan-summation-how-strong-is
https://en.wikipedia.org/wiki/Ramanujan's_sum

Leo Authersh
@fresh_42 As far I have understood from the topics I have studied earlier, this zeta regularization is used to define the type of the series.

ζ(−1) = −1/12 represents the value of the series in the complex plane at 1.

It is just the value of the series at a particular point (here it is 1) while in complex plane. It just defines the nature of the series in the complex plane. And those topics are strong with the point that the zeta function (extension of the series in complex plane) is continuous upto infinity.

So, as per my understanding, the wikipedia notion of writing the series as

1+2+3+4+5..... = -1/12 is wrong.

One thing that can be said is that Ramanujan based this discovery upon the already proven series

1+1-1+1-1+1... = 1/2

If you think about this series you can perceive that the value 1/2 is not the summation because the summation value alters infinitely between 1 and 0. But one can understand the nature of the series that the sum should be between 1 and 0 and hence the average value calculated as 1/2.

It's similar to the quantum physics, where they say that the chance of an electron to be present simultaneously in two different locations is not zero%. Some instance it can be 50℅ which can be interpreted numerically as the series above.

Again the common misinterpretation is that the 50% chance means the electron will be present in two different locations at the same time. But that's not true. It actually is that the possibility of an electron being in any one of the location at the same time is 50% (the probability of the electron present in a location is mutually dependent on its presence or absence in another location). And this is suggested by Schrödinger's paradox.

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Leo Authersh
@fresh_42 Note that the above answer is completely based on my understandings. And my understanding is incomplete and hence I asked this question in the thread for more comprehension on the subject.
I haven't yet read the links you posted. Will read it and let you know. Hopefully they will provide me better acquisition on the definition.

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Leo Authersh
@fresh_42 Nevertheless, I want to assert that the Numberphile video is nothing but a hypocrisy. Completely misleading people for the sake of making money through YouTube views. Especially the reactions given by both of them in the thumbnail of the video explicates the deception they execute.

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mathman
What does the equation ζ(−1) = −1/12 represent precisely?
It's impossible for that to be the sum of all natural numbers. And it is also mentioned in all the maths articles that the 'equal to' in the equation should not be understood in a traditional way.

If so, then why wikipedia article states that,

1+2+3+.... = - 1/12
The basic idea is analytic extension. The series is equal to some function where it converges. The function itself may be well defined outside the series convergence range. A very simple example $$\frac{1}{1-x}=1+x+x^2+x^3+....$$ for |x|<1. however the function is defined for all x, except x=1

Leo Authersh
The basic idea is analytic extension. The series is equal to some function where it converges. The function itself may be well defined outside the series convergence range. A very simple example $$\frac{1}{1-x}=1+x+x^2+x^3+....$$ for |x|<1. however the function is defined for all x, except x=1
Thank you for the explanation. If so, what is the range of convergence in the Ramanujan sum? And how can we have different ranges when the series is of natural numbers and not a variable?

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mathman
Thank you for the explanation. If so, what is the range of convergence in the Ramanujan sum? And how can we have different ranges when the series is of natural numbers and not a variable?
I am not familiar with the Ramanujan sum. The series of numbers results from evaluating the series at a particular value of the argument. For example: $$\frac{1}{1-x}$$ series evaluated at x=2 leads to 1+2+4+8+......=-1.

Leo Authersh
[tex said:
\frac{1}{1-x}[/tex] series evaluated at x=2 leads to 1+2+4+8+......=-1.
Can you please explain how is it -1?

$\frac{1}{1-2}=-1$