# Generalized ZETA function

1. Oct 30, 2009

### zetafunction

given the function $$Z(s)= \prod _{k=0}^{\infty}\zeta (s+k)$$ with $$\zeta (s)$$ being the Riemann Zeta function

the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ??

what would be the Functional equation relating Z(1-s) and Z(s) ¿¿ from the definition of Riemann functional equation

2. Oct 30, 2009

### Petek

For which values of s does Z(s) exist? A necessary condition for an infinite product to converge is that the individual terms converge to 1. Thus, it must be true that

$$\zeta(s + k) \rightarrow 1$$ as $$k \rightarrow \infty$$

It's not clear to me for which values of s this holds.

Petek

3. Oct 31, 2009

### g_edgar

True for Re s > 1, right? And therefore true for all s ...

4. Nov 1, 2009

### Petek

Yes, that's right. Thanks!

Petek