The Erdős-Turán conjecture states that
If $A\subset\mathbb{N}$ is such that $$ \sum_{n\in A} \frac{1}{n} = \infty,$$ then $A$ contains arithmetic progressions of any given length.
I'm interest when $A$ is the set of all prime number . In this case, we know that, via Euler's theorem, $$ \sum_{p \text{ prime}} \frac{1}{p} = \infty.$$
So we have the Green-Tao Theorem:
The sequence of prime numbers contains arbitrarily long arithmetic progressions.
Is the Green-Tao theorem a consequence of the Euler's theorem? In other words, in the proof of the Green-Tao theorem is used 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.