On the order of the error function of the (k,r)-integers |
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Authors: | MV Subbarao D Suryanarayana |
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Institution: | 1. Department of Mathematics, University of Alberta, Edmonton 7, Alberta, Canada;2. Department of Mathematics, Andhra University, Waltair, India |
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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 , where , 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 , according as or , where ω(x) = exp B log x(log log x)?1]. |
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