Averages of singular series, or: when Poisson is everywhere

Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text…

I have recently posted on my web page a preprint concerning some averages of “singular series” (another example of pretty bad mathematical terminology…) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by Goldston-Pintz-Yildirim that there are infinitely many primes p for which the gap q-p between p and the next prime q is smaller than ε times the average gap, for arbitrary small ε>0.

This result refers to the behavior, on average over h=(h_1,…,h_k), of the constant S(h) which is supposed to be the leading coefficient in the conjecture

|{n<X | n+h_i is prime for i=1,…,k}|~S(h) X(log X)^-k

Gallagher showed that the average value of S(h) is equal to 1, and I’ve extended this in two ways…