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B Ramanujan Summation

  1. Aug 11, 2017 #1
    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
    Last edited by a moderator: Aug 11, 2017
  2. jcsd
  3. Aug 11, 2017 #2


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    Have you read it and what do you know about the zeta function and analytic continuations?

    Also of interest:
  4. Aug 11, 2017 #3
    @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.
    Last edited by a moderator: Aug 11, 2017
  5. Aug 11, 2017 #4
    @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.
    Last edited by a moderator: Aug 11, 2017
  6. Aug 11, 2017 #5
    @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.
    Last edited by a moderator: Aug 11, 2017
  7. Aug 11, 2017 #6


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    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 [tex]\frac{1}{1-x}=1+x+x^2+x^3+....[/tex] for |x|<1. however the function is defined for all x, except x=1
  8. Aug 11, 2017 #7
    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?
    Last edited by a moderator: Aug 11, 2017
  9. Aug 12, 2017 #8


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    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: [tex]\frac{1}{1-x}[/tex] series evaluated at x=2 leads to 1+2+4+8+......=-1.
  10. Aug 13, 2017 #9
    Can you please explain how is it -1?
  11. Aug 13, 2017 #10
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