On Mon, 4 Jul 2005, Jeffrey N. Cook wrote:

> I am a novice with the Zeta Function. Could someone help me understand

> something...

>

> If I set S = 1, and calculate up to 1/7 and 1/7 only,

>

> Z(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s

This expression for the zeta function only converges when the real

part of s is greater than one. In Calculus this was called the

"p-test" (at least in the books I use).

> Is this what is meant "that no zeros could lie on the line Re(z) =

> 1" ? Does S need to be greater than 1 here?

Nope. That refers to the analytic continuation of the sum above--that

means there is a function that converges on most of the complex plane

that agrees with the sum above (where it converges). This continuation is

called the Riemann zeta function and its the one that all the talk is

about

CC