# How is $\zeta(0)=-1/2$? [duplicate]

Possible Duplicate:
Why does $1+2+3+\dots = {-1\over 12}$?

Fermat's Dream by Kato et al. gives the following:

1. $\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}$ (the standard Zeta function) provided the sum converges.

2. $\zeta(0)=-1/2$

Thus, $1+1+1+...=-1/2$ ? How can this possibly be true? I guess I'm under the impression that $\sum 1$ diverges.

• As you say, $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$ provided the sum converges. This says nothing directly about the value of $\zeta(s)$ when this sum diverges, for example when $s=0$. – Chris Eagle Sep 21 '11 at 17:04
• 1. is true for $\hbox{Re} \;s > 1$ only... 2. you will have to learn about "analytic continuation" to answer this. – GEdgar Sep 21 '11 at 17:04
• See this and this related question. – J. M. isn't a mathematician Sep 21 '11 at 17:05
• @Chris, go and try to explain that to well renowned physicists as Lubos Motl that still assert that the sum itself is what evaluates to minus one twelfth – lurscher Sep 21 '11 at 17:18
• However, i would be happy with that assertion if i would be shown evidence that any analytic continuation of that sum needs to be equal to the Riemann Zeta wherever it is well defined – lurscher Sep 21 '11 at 17:22

## 1 Answer

As GEdgar noted, the zeta function is extended to values for which the series diverges via an analytic continuation.

• Ok this helps, I guess I need to study analytic continuation. – Jason Smith Sep 21 '11 at 17:21