Vol. 68, No. 1, 1977

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On a theorem of Apostol concerning Möbius functions of order k

D. Suryanarayana

Vol. 68 (1977), No. 1, 277–281
Abstract

In 1970, Tom M. Apostol introduced a class of arithmetical functions μk(n) for all positive integral k, as a generalization of the Möbius function μ(n) = μ1(n) and established the following theorem: For k 2,Mk(x) = Σnx μk(n) = Akx + O(x1∕k log x), where Ak is a positive constant. In this paper we improve the above O-estimate to O(x4k∕(4k2+1) ω(x)) on the assumption of the Riemann hypothesis, where ω(x) = exp{Alog x(log log x)1},A being a positive absolute constant.

Mathematical Subject Classification
Primary: 10H25, 10H25
Milestones
Received: 2 November 1976
Published: 1 January 1977
Authors
D. Suryanarayana