A variance for $k$-free numbers in arithmetic progressions of given modulus

Abstract

LetS = {n ∈ N| there is no prime p with p k |n}, the set of k-free numbers. For some suitable main term η(q, a) to be defined soon enough we will study in this paper the object q a=1

On the variance of squarefree integers in short intervals and arithmetic progressions

“…Once extended to composite q our Theorem 2 improves on results by Warlimont [War80] and Vaughan [Vau05] who obtain an asymptotic formula with an additional averaging over q ≤ Q in the range x 2/3 ≤ Q = o(x). Moreover Theorem 2 improves on a succession of results by Blomer [Blo08], Nunes [Nun15] (see also [Par19]) and Le Boudec [LB18] who considered individual averages over (a, q) = 1 as we do in Theorem 2. In particular Nunes showed that (7) holds in the range x 31/41+ε ≤ q = o(x) and Le Boudec showed that the left-hand side of ( 7) is O ε ((x/q) 1/2+ε ) for all ε > 0 in the range x 1/2 ≤ q ≤ x.…”