Is the Riemann Zeta function negative in the critical strip?

Just saw a question on "how to prove that the Riemann Zeta function is negative in the critical strip". What is meant by Zeta(s) < 0?

Does it mean that its real part is negative, or both real and imaginary parts are negative? I thought a complex function has no sign.

• $\zeta(s) < 0\;$ for real $0\le s < 1$ – gammatester Mar 1 '16 at 14:52
• Does 0 <= s < 1 mean 0 <= Re(s) < 1 ? I am not familiar with inequalities in complex numbers – Hass Saidane Mar 1 '16 at 15:12
• I mean real $s$ – gammatester Mar 1 '16 at 15:15
• @Hass : if you want to understand $\zeta(s)$ maybe you should start studying a complex analysis course. $\zeta(x)(x-1)$ is real for $x \in [0;\infty[$ and the functional equation tells us it has no pole (it is an entire function) hence it is real for every $x \in \mathbb{R}$. – reuns Mar 1 '16 at 15:19
• Thanks for the clarification. – Hass Saidane Mar 1 '16 at 15:39