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It seems aptly named because Ramanujan also dealt with infinite series. One of his most famous equations is the ramanujan summation [1] where he derives that th

It seems aptly named because Ramanujan also dealt with infinite series. One of his most famous equations is the ramanujan summation [1] where he derives that the sum of infinite series 1 + 2 + 3 + 4 + ... = - 1 / 12 [1] https://en.wikipedia.org/wiki/Ramanujan_summation https://en.wikipedia.org/wiki/Ramanujan_summation

amelius 7y ago
I still don't understand why this derivation is such an achievement. If anything, it seems like dirty hacking, but this time done by a mathematician.
- nurettin 7y ago
  I read somewhere that the result is confirmed in applied physics where it somehow relates to the casimir force, but I'm not a physicist.
  - mathandpoop 7y ago
    It's a theorem of analytic number theory. It's just a special value of the ζ-function. People misunderstand how analytic continuation works and therefore (wrongly) interpret ζ(-1) as the divergent series 1+2+3+... (The correct interpretation is as the unique analytic continuation of the holomorphic function $\sum_{n\geq 1}\frac{1}{n^s}$ defined on the half-plane $\Re(s)>1$ to $\mathbb{C}\setminus\{1\}$.) It's actually a pretty simple consequence of the functional equation for ζ and a few special values of Γ. That in turn comes from a theta-function identity which can be proven using Poisson's summation formula. What I'm trying to say is that it's legit maths, that has been distorted due to the shock value of writing the equation "1+2+3+... = -1/12". If you're looking for a reference, go to Davenport's "Multiplicative Number Theory". It's short, self-contained, and extremely well-written. Serre's "A Course in Arithmetic" should also work.
eximius 7y ago
Except he didn't. He created a way of assigning surrogate values to forms that otherwise wouldn't have one. To quote from the Wikipedia article you linked: > assigning a value to divergent infinite series Every time someone claims that series is EQUAL to -1/12, someone loses faith in math.

amelius 7y ago

I still don't understand why this derivation is such an achievement. If anything, it seems like dirty hacking, but this time done by a mathematician.

nurettin 7y ago
I read somewhere that the result is confirmed in applied physics where it somehow relates to the casimir force, but I'm not a physicist.
- mathandpoop 7y ago
  It's a theorem of analytic number theory. It's just a special value of the ζ-function. People misunderstand how analytic continuation works and therefore (wrongly) interpret ζ(-1) as the divergent series 1+2+3+... (The correct interpretation is as the unique analytic continuation of the holomorphic function $\sum_{n\geq 1}\frac{1}{n^s}$ defined on the half-plane $\Re(s)>1$ to $\mathbb{C}\setminus\{1\}$.) It's actually a pretty simple consequence of the functional equation for ζ and a few special values of Γ. That in turn comes from a theta-function identity which can be proven using Poisson's summation formula. What I'm trying to say is that it's legit maths, that has been distorted due to the shock value of writing the equation "1+2+3+... = -1/12". If you're looking for a reference, go to Davenport's "Multiplicative Number Theory". It's short, self-contained, and extremely well-written. Serre's "A Course in Arithmetic" should also work.

eximius 7y ago

Except he didn't. He created a way of assigning surrogate values to forms that otherwise wouldn't have one. To quote from the Wikipedia article you linked: > assigning a value to divergent infinite series Every time someone claims that series is EQUAL to -1/12, someone loses faith in math.