Is the Green-Tao theorem a consequence of the Euler's theorem?

Terence Tao has a paper on his web site

http://www.math.ucla.edu/~tao/preprints/Expository/gy-corr.dvi

explaining that the only information about prime numbers needed for the Green-Tao proof is that $\zeta(s)$ has a simple pole at $s=1$.

That behavior of zeta is equivalent to the "prime zeta function" $\sum \frac{1}{p^s}$ having a logarithmic singularity at the same point, so that $\sum \frac{1}{p^s} - \log (\frac{1}{s-1})$ is analytic near $s=1$. This is slightly more than divergence of $\sum \frac {1}{p}$, but it can be proved by the same method as Euler's proof of the divergence.

A loose interpretation of this is that in many places in the Green-Tao paper, natural density assumptions can (probably) be replaced by weaker statements about Dirichlet density.