On a theorem of Apostol concerning Möbius functions of order k

D. Suryanarayana

Vol. 68 (1977), No. 1, 277–281

DOI: 10.2140/pjm.1977.68.277

Abstract

In 1970, Tom M. Apostol introduced a class of arithmetical functions μ_k(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,M_k(x) = Σ_nx −μ_k(n) = A_kx + O(x^1∕k log x), where A_k is a positive constant. In this paper we improve the above O-estimate to O(x^4k∕(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