Riemann Zeta Function and Pi in Infinite Series

[Kevin_Axion]

I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me.

Suppose we have an infinite series of the form:

[itex]\sum^_{n = 1}^{\infty} 1/n^\phi[/itex]

where [itex]\phi[/itex] is some even natural number, it appears that it is always convergent to a rational multiple of [itex]\pi[/itex].

Now if we take this series and change it slightly:

[itex]\sum^_{n = 1}^{\infty} 1/n^\alpha[/itex]

where [itex]\alpha[/itex] is some even natural number, it appears that it is always convergent to the Riemann Zeta Function evaluated at [itex]\alpha[/itex] i.e. [itex]\zeta(\alpha)[/itex].

Can someone explain the relationship expressed in these infinite series?

EDIT: What's wrong with my LaTeX for the infinite series?

 

[Mute]

I don't see what the difference between the two series you posted is, except changing phi to alpha. The LaTeX is perhaps not working because you wrote an extra ^ after the \sum, which shouldn't be there.

The reason that the sum "converges" to the Riemann Zeta function is that for integer exponents that series is the definition of the Riemann Zeta function. It is then extended to arbitrary complex exponents by analytic continuation. If you compute [itex]\zeta(2n)[/itex] for n a postive integer, you would find it is a multiple of [itex]\pi[/itex] to some power.

 

[micromass]

Here's the post with fixed LaTeX...
Maybe there should be something different between the first and two series?

I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me.

Suppose we have an infinite series of the form:

[itex]\sum_{n = 1}^{\infty} \frac{1}{n^\phi}[/itex]

where [itex]\phi[/itex] is some even natural number, it appears that it is always convergent to a rational multiple of [itex]\pi[/itex].

Now if we take this series and change it slightly:

[itex]\sum_{n = 1}^{\infty} \frac{1}{n^\alpha}[/itex]

where [itex]\alpha[/itex] is some even natural number, it appears that it is always convergent to the Riemann Zeta Function evaluated at [itex]\alpha[/itex] i.e. [itex]\zeta(\alpha)[/itex].

Can someone explain the relationship expressed in these infinite series?

EDIT: What's wrong with my LaTeX for the infinite series?

 

[Kevin_Axion]

Very cool, the only difference I made was that the exponent is even in one and odd in the other.

 

[micromass]

Very cool, the only difference I made was that the exponent is even in one and odd in the other.

Ah, then you probably made a typo

Anyway, the first series will also converge to [itex]\zeta(\phi)[/itex], and this will have a nice characterization as a rational function of pi.
If alpha is odd, then all we know is that the value is [itex]\zeta(\alpha)[/itex]. These values are very mysterious and not well understood. For example, [itex]\zeta(3)[/itex] is called Apery's constant and shows up in some physics problems.

But you'll see more of this in your future math major!

 

[Kevin_Axion]

Haha! Engineering !