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On the order of the error function of the (k,r)-integers
Authors:MV Subbarao  D Suryanarayana
Institution:1. Department of Mathematics, University of Alberta, Edmonton 7, Alberta, Canada;2. Department of Mathematics, Andhra University, Waltair, India
Abstract:Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = akb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Qk,r(x) denotes the number of (k, r)-integers ≤ x, then Qk,r(x) = xζ(k)ζ(r) + Δk,r(x), where Δk,r(x) = O(x1rexp ?Blog35x (log log x)?15]), B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δk,r(x) and prove that
1x1αδk,r(t)dt=0(x1kω(x))or0(x3/(4r+1)ω(x))
, according as k ≤ (4r + 1)3 or k > (4r + 1)3, where ω(x) = exp B log x(log log x)?1].
Keywords:
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