An average theorem for tuples of k-free numbers in arithmetic progressions

Abstract

The Barban-Davenport-Halberstam (BDH) Theorem and its refinement due to Hooley and Montgomery (HM) are important theorems in analytic number theory since they suggest what one believes to be the correct order of magnitude for the error term in the Prime Number Theorem for Arithmetic Progressions. The question of additive patterns in prime numbers is also a central problem, but a theorem of BDH type is out of reach -indeed it is not even known, at the time of writing, that there are infinitely many primes p such that p + 2 is also prime, never mind the Hardy-Littlewood Conjecture.Let k ≥ 2. If for a given n there is no prime p for which p k |n then n is said to be k-free. An asymptotic formula of similar shape to that in the BDH-HM Theorem is known for the k-free numbers, the current state of knowledge attained and summarised by Vaughan in [10]. Crucially, the corresponding question on additive patterns in the k-frees is accessible. For given non-negative integers 0 ≤ h 1 < • • • < h r we call n a k-free r-tuple associated to h := (h 1 , ..., h r ) if the n + h i are all k-free, and write R = R(h) for the set of all such n. The asymptotic count n≤x n∈R * d1•••dr>Y N d;h (t) = * d1•••dr≤Y