Why are Ramanujan sums the same as the complex Zeta values?

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Discussion Overview

The discussion revolves around the relationship between Ramanujan sums and complex Zeta values, exploring whether there is a link between the two concepts. Participants consider examples of series that yield surprising results through both Zeta regularization and Ramanujan summation, questioning the nature of this relationship and its implications.

Discussion Character

  • Exploratory, Debate/contested

Main Points Raised

  • One participant notes that both Zeta regularization and Ramanujan summation yield results such as 1 + 2 + 3 + 4 + · · · = −1/12 and 1 + 1 + 1 + 1 + · · · = -1/2, suggesting a potential connection.
  • Another participant expresses curiosity about whether the observed relationship is always true, indicating an assumption that answering the first question might clarify this.
  • A participant provides a link to an external resource that may contain relevant information regarding the relationship between the Riemann Zeta function and Ramanujan summation.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the nature of the relationship between Ramanujan sums and complex Zeta values, and multiple competing views remain regarding the implications of their findings.

Contextual Notes

The discussion does not clarify the assumptions underlying the relationship between the two methods or the conditions under which the examples hold true.

ddd123
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Possibly a difficult question, but I've never found a discussion on the topic.

Thanks
 
Not much else to say, two examples:

1 + 2 + 3 + 4 + · · · = −1/12

1 + 1 + 1 + 1 + · · · = -1/2

This results both from Zeta regularization and Ramanujan summation. The two methods seem to me to be completely unrelated. It's an amazing coincidence so there must be a link between the two.

Another question would be: is it always true? But I'm assuming it is, or that the answer to the first question would answer this one too.
 
Last attempt!

Thanks
 

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