How is $3 + 4\cos \theta + \cos 2\theta \geq 0$ related to $\zeta(s)^3|\zeta(s + it)^4\zeta(s + 2it)| \geq 1$? The inequality
$$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$
follows from
$$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0.$$
How is that done? What is the relationship between zeta and the trigonometry?

The inequality you state does not seem to make too much sense.
Anyway, you can find a related discussion in the book Riemann's Zeta Function, by H.M. Edwards on pages 79/80.
Google books link: http://books.google.com/books?id=ruVmGFPwNhQC&pg=PA79
Snapshot:

Hope that helps.
