Linear Equations in Primes, Green Tao

Question

Ben Green and Terence Tao, Linear Equations in Prime, Page 1764 and 1765. Are the products in Examples 5, 6, 7, 8 taken over the set of natural numbers or the set of prime numbers?

Answer 1

The products in Examples $5$, $6$, $7$ and $8$ are all taken over the set of primes, not of all of the natural numbers.

Note each example talks about primes, e.g., Example $5$ starts with "The number of $4$-tuples of primes $p_1 \lt p_2 \lt p_3 \lt p_4 \le N \ldots$". Also, the use of the symbol $p$, although it obviously doesn't involve primes in all situations, is nonetheless another indication here they are only dealing with primes. In addition, in Example $7$, the first product has $p \mid N^3 - N$ and the second product has $p \not\mid N^3 - N$, with this once again indicating the products are over the primes, with this example splitting between those which divide $N^3 - N$ and those which don't. Finally, and most conclusively, the results they provide are only true if the products are over the primes.