Seonglae ChoWhile the natural number series 1+2+3+... diverges when summed directly, the Riemann zeta function is
defined as above, and through analytic continuation, values can be assigned even outside the "convergence region" .
Here, the "natural number series" is treated as the form where is substituted, not the actual sum of natural numbers. The reason why the sum of natural numbers is not -1/12 is because it's impossible to compare inequalities between divergent series and because the variables that determine divergence are different. The ordinary sum diverges by simply adding "without variables", while zeta regularization introduces a variable "s" and obtains the value through analytic continuation.
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere.
https://en.wikipedia.org/wiki/Riemann_zeta_function
