Twins of \(k\)-free numbers in arithmetic progressions (Q555507)

The author derives new results on the variance of twins of \(k\)-free numbers in arithmetic progressions. Let \(\mu_k(n)\) be the characteristic function of k-free integers, and \[ A_k(x; q,a) =\sum_{\substack{ n\leq x \\ n\equiv a\pmod{q}}} \mu_k(n)\mu_k(n + 1) . \] In [Mich. Math. J. 47, 173--190 (2000; Zbl 0987.11061)] by \textit{J. Brüdern, A. Perelli} and \textit{T. D. Wooley}, it is shown that \[ A_k(x; q,a) = q^{-1}g(q,a)x + O \left(x^{\frac2{k+1}+\varepsilon}\right) \] where \(g(q, a)\) is a certain convergent infinite series, and they obtain an upper bound for the variance \[ Y_k(x,Q) =\sum_{q\leq Q}\, \sum^q_{a=1}| A_k(x; q,a)-q^{-1}g(q, a)x| ^2. \] The aim of the present paper is to establish another upper bound for \(Y_k(x,Q)\) that gives an improvement when \(k > 2\) and \(Q \gg x^{3/4}.\) An estimate for \(\sum_{q\leq Q}\,\sum^q_{a=1} A^2_k(x; q, a)\) is found using the Hardy-Littlewood method employed by \textit{R. C. Vaughan} in [Proc. London Math. Soc. (3) 91, 573--597 (2005; Zbl 1117.11052)]. The bounds for the other terms in \(Y_k(x,Q)\) depend in the first instance on considering various congruences. The details of the proofs are very complicated and intricate.