Does 1+2+3+... Really Equal -1/12?
Here’s what’s really going on:<p>You can define a function f from a subset of the complex numbers to the complex numbers where f(z) = \sum_{n=1}^\infty 1&#x2F;n^z.
Be careful with the domain of this function: the series does not converge for all z. You can plug in -1 and see that symbolically, f(-1) = 1 + 2 + 3 + .... But the series does not converge for z = -1, and it is simply not true that \sum_{n=1}^\infty n = -1&#x2F;12; the series does not converge; equating it to something is a nonsensical thing to do.<p>What is going then? Even though f is not defined for all complex numbers, there exist functions from the complex numbers to the complex numbers that -- restricted to the domain of f -- are equal to f. They &quot;continue&quot; f to all of the complex numbers. And if one imposes a restriction on these continuations (namely that they are analytic), then it turns out that there is a unique analytic continuation of f: the Riemann zeta function. And zeta(-1) = -1&#x2F;12.<p>Don’t confuse the definitions here. The Riemann zeta function can be defined as the analytic continuation of the series, the series is not defined in terms of the Riemann zeta function!