Is the Riemann Zeta Function of 0 Infinity or -1/2?

[dimension10]

The riemann zeta function of 0 is supposed to be -1/2. But isn't it infinity?

1/1^0 +1/2^0+1/3^0+1/4^0+...
1/1+1/1+1/1+1/1+...
1+1+1+1+...
infinity

But many sources claim that it is -1/2

 

[micromass]

The riemann zeta function of 0 is supposed to be -1/2. But isn't it infinity?

1/1^0 +1/2^0+1/3^0+1/4^0+...
1/1+1/1+1/1+1/1+...
1+1+1+1+...
infinity

But many sources claim that it is -1/2

Yes, you are correct, but the definition af the Riemann-zeta function as

$$\zeta (s)=\sum_{n=1}^{+\infty}{\frac{1}{n^s}}$$

is only defined as s>1 (or in the complex case, as Re(s)>1). The Riemann-zeta function for $s\leq 1$ is defined as the "analytic continuation" of the above function. That is: the unique function that looks most like $\zeta(s),s>1$.

 

[disregardthat]

is only defined as s>1 (or in the complex case, as Re(s)>1). The Riemann-zeta function for $s\leq 1$ is defined as the "analytic continuation" of the above function. That is: the unique function that looks most like $\zeta(s),s>1$.

The sum has infinitely many continuous extensions to the real line, but only one analytical extension to the complex plane.

 

[micromass]

The sum has infinitely many continuous extensions to the real line, but only one analytical extension to the complex plane.

Well, a meromorphic extension really, since there is a pole in 1.

 

[disregardthat]

Well, a meromorphic extension really, since there is a pole in 1.

You are right about that

 

[dimension10]

Yes, you are correct, but the definition af the Riemann-zeta function as

$$\zeta (s)=\sum_{n=1}^{+\infty}{\frac{1}{n^s}}$$

is only defined as s>1 (or in the complex case, as Re(s)>1). The Riemann-zeta function for $s\leq 1$ is defined as the "analytic continuation" of the above function. That is: the unique function that looks most like $\zeta(s),s>1$.

Thanks, but is there a formula for s<1?

 

[micromass]

There are several. Check out http://en.wikipedia.org/wiki/Riemann_zeta under the section "functional equation" and "representations"...

 

[dimension10]

There are several. Check out http://en.wikipedia.org/wiki/Riemann_zeta under the section "functional equation" and "representations"...

Thanks a lot.