Value of Ramanujan Summation In Quantum Mechanics

Question

In mathematics, sum of all natural number is infinity.

but Ramanujan suggests whole new definition of summation.

"The sum of $n$ is $-1/12$" what so called Ramanujan Summation.

First he find the sum, only Hardy recognized the value of the summation.

And also in quantum mechanics(I know), Ramanujan summation is very important.

Question. What is the value of Ramanujan summation in quantum mechanics?

This post imported from StackExchange Physics at 2014-03-17 04:00 (UCT), posted by SE-user 4545454545SI

Comments

What's the value of the golden ratio in Newtonian mechanics? What's the value of 1+1/2+1/4+1/8+... in general relativity? You're mixing up math with unrelated physics.

This post imported from StackExchange Physics at 2014-03-17 04:00 (UCT), posted by SE-user felix

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vixra.org/abs/1003.0235 here my paper on how can the zeta regularization and Ramanujan resummation be used to get finite values in quantum mechanics

This post imported from StackExchange Physics at 2014-03-17 04:00 (UCT), posted by SE-user Jose Javier Garcia

Answer 1

Lumo gives a very nice step by step calculation of this sum and a good discussion of the importance and application of such summation techniques in QFTs here:

http://motls.blogspot.com/2011/07/why-is-sum-of-integers-equal-to-112.html

Such mathematical calculations are NOT unrelated to physics; on the contrary they are important...