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# p-adic numbers and the Ramanujan summation

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In mathematics, there is the Ramanujan summation:

1+2+3+4…=-1/12

This sum is used in physics for predicting the Casimir effect:

I have also heard that this sum was used in the string theory (more precisely, in the original bosonic string theory).

Then, in mathematics the p-adic numbers are used:

My question is: can the Ramanujan summation be relatively easily obtained using the p-acid numbers?

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On 7/9/2023 at 10:54 PM, Spathi said:

My question is: can the Ramanujan summation be relatively easily obtained using the p-adic numbers?

That depends on what you mean by "relatively easily"🤣

Edited by OceanBreeze
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P-adic functions exist in an exoteric area of mathematics seldom breached by engineers. So, you may take what I, a marine engineer, have to say about this with an atom of salt.

P-adic numbers once were the domain of pure numbers-theory but have found their way into more and more Physics, thanks to some brilliant mathematical physicists.

Reverting back to the question asked in the OP, I can only find one equation which ties together p-adic functions ( actually only closely related L-adic functions ) and the Ramanujan function .

I believe by solving this sigma summation in terms of an L-function of the Deligne motive:

L(s) = ∑nτ (n)n−s, will yield the Ramanujan function  τ (n).

Based on that, I have to say that obtaining the Ramanujan function by means of p-adic numbers cannot be said to be “relatively easy”.

But again I stress that the answer depends on your definition of “relatively easily” and that depends greatly on who is doing the calculating!

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