In the first part below, the prime is used to denote a different function, so it's not a derivative. I tried using a tilde at first but it was invisible in the generated image, so it was impossible to distinguish \zeta ($\zeta$) from \tilde\zeta ($\tilde\zeta$).
sganesh88 said:
I'm surprised. Don't you think the series 1+1+1+.. should end up in infinity? How can it have a value -1/12?
Yes, it does diverge (i.e. "end up in infinity"). It doesn't have the value -1/12. As I explained above, there exists at least one function which can be defined through a summation like
\zeta(n) = \sum_{k = 1}^\infty \frac{1}{k^n}
on
some domain and then
extended to a larger domain which - technically - makes it a different function \zeta'(n). This does
not mean that on the larger domain, it can still be expanded in such a formal sum. So "a priori", the equality \zeta(n) = \zeta'(n) only holds on the domain of the original function, but in general takes different values outside that (for example, \zeta' may have finite values when the series expansion defining \zeta does not). Statements like "1 + 1 + ... = -1/12" arise from sloppy interpretation of such functions, where people (implicitly) do not distinguish between \zeta and \zeta'.
sganesh88 said:
May i know where such a result is used? In any particular field of physics?
In theoretical physics this is used in (closely related) techniques called "regularization" and "renormalization". In modern-day quantum field theories, infinities often arise where physical (i.e. finite) quantities are expected. As a simplified example, consider obtaining for some mass or energy scale the expression
\exp\left[ \hbar \sum_{k = 1}^\infty (2k) \right].
By replacing the infinite sum by the finite value 2 \zeta(-1) (which is, technically, the analytic continuation of the zeta function defined by the sum, in the sense of my earlier story, because the latter is not defined for x = -1) theoretical physicists
can make sense of such a value. The argumentation is that the "bare" value may be infinite, because of infinite quantum corrections; however when we actually
measure the quantity there will be quantum screening effects which will give us a finite outcome of the experiment (cf. bare and screened electron charge, for example).
I must admit, again, that although I understand the ideas of regularization and renormalization and see how they are useful, I still fail to see why choosing specifically this function - in the above case, the Riemann zeta function - is "correct" (as in: yielding the correct measurable value).