There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1\ (\text{mod}\ c)} e(\frac{ma+nd}{c})$ like $\sum_{m\in\mathbb F_c} S(m,n,c)^k$, $\sum\sum_{m_1,m_2\in\mathbb F_c} S(m_1,n,c)S(m_2,n,c)$, or other such averages over characters $e(nx)$. The recent work on sieve methods after Goldston, Pintz, Yildirim, and Zhang in particular uses cancellation beyond that in individual Kloosterman sums (proved by Weil).

On the other hand, sums of Kloosterman sums with varying denominator $c$, $\sum_{c\le x} S(m,n,c)/c$ and the related $\sum_{c\ge 1} S(m,n,c)/c^{2s}$ were studied by Linnik, Selberg and many others and are closely related to the Ramanujan-Petersson-Selberg and Linnik conjectures for congruence subgroups of SL(2,Z). The Weil bound implies a weak form of the Selberg conjecture, for instance. I would like to know whether some link is known between those 2 general problems. For instance does the Selberg conjecture imply better estimates than those used by Polymath, based on Weil's bound or Deligne's theory of weights for l-adic sheaves, toward the Elliott-Halberstam equidistribution conjecture?

Thank you.

EDIT: At 1st I used the term "Ramanujan sums" while I of course meant the more general "Kloosterman sums", sorry.