Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about

`\[c_k\frac{N^2}{\log^kN}\]`

for some constant $c_k$ dependent on k. The paper of Green and Tao gives a similar lower bound for all k (with a much worse constant, but still), and recent work by Green, Tao and Ziegler have established the correct asymptotic for k=3 and k=4.

I am looking for a reference which establishes an upper bound for all k - I'm sure I've heard of one, but I can't find mention of the relevant paper anywhere. Of course, if there is a simple proof, that would appreciated as well.

That is, I am looking for a reference and/or proof which establishes that the number of k-term arithmetic progressions of primes in [1,N] is at most
`\[c_k'\frac{N^2}{\log^kN}\]`

for some constant $c_k'$.