P-adic numbers and the Ramanujan summation

[Spathi]

TL;DR Summary

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

In mathematics, there is the Ramanujan summation:

$$1+2+3+4+...=-\frac{1}{12}$$

https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯

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

https://en.wikipedia.org/wiki/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:

https://en.wikipedia.org/wiki/P-adic_number



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

 

[fresh_42]

Here is what you need to know to even understand the Ramanujan summation:
https://www.physicsforums.com/insights/the-extended-riemann-hypothesis-and-ramanujans-sum/
and here is explained what p-adic numbers are:
https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/

Ramanujan's summation is based on classical analysis, i.e. it depends on the Euclidean metric that induces an Archimedean evaluation. The only connection to p-adic numbers that I am aware of would be Hasse's principle that primarily deals with polynomials, i.e. Diophantic expressions, not series. Since
$$
1+2+3+4+\ldots \neq -\dfrac{1}{12}
$$
it is not even clear how Hasse's principle could be applied to the continuation of Riemann's zeta-function, let alone being "easier obtained". p-adic numbers have at first glance absolutely nothing to do with Riemann's zeta-function, so the connection between them alone would be quite a challenge.

 

[Infrared]

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

I'm a little confused. What exactly do you think is the relationship between the p-adics and Ramanujan summation?

In any event, $1+2+3+\ldots$ does not converge $p$-adically for any prime $p.$

 

[fresh_42]

A quick internet search gave me reasonable results about p-adic Dirichlet functions (L-functions), so the question is not completely out of thin air. However, I haven't searched for results in English. The idea is to generalize Euler's formula
$$
\zeta(1-n)\;=\;-\dfrac{B_n}{n}
$$
and consider $f(n) =(1-p^{n-1})\zeta(1-n)$ (under some technical conditions on the domain) as a unique continuous function $f\, : \,\mathbb{Z}_p\longrightarrow \mathbb{Q}_p.$

https://www.mathi.uni-heidelberg.de/~mfuetterer/texts/da_fuetterer.pdf

 

[Spathi]

I'm a little confused. What exactly do you think is the relationship between the p-adics and Ramanujan summation?

In any event, 1+2+3+… does not converge p-adically for any prime

If I am not mistaken, in the video I provided above, there it is shown, how with p-acics the following equation can be obtained:

$$1+x+x^2+x^3+x^4...=\dfrac{1}{1-x}$$

I don't remember how this equation is standardly obtained for x<1, somehow please remind. For x>1, if I am not mistaken, p-adics provide a proper way to get this sum. Maybe the Ramanujan summation and $$1+2+4+8+16...-1$$ are something very similar?
And one more question - maybe we should remove some axioms from "standard" mathematics to get alternative "hyberbolic" mathematics where these sums are fully correct?