Vol. 32, No. 1, 1970

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ISSN: 0030-8730
Möbius functions of order k

Tom M. (Mike) Apostol

Vol. 32 (1970), No. 1, 21–27
Abstract

Let k denote a fixed positive integer. We define an arithmetical function μk, the Möbius function of order k, as follows:

μk(1) = 1,
μk(n) = 0 if pk+1|n for some prime p,
μk(n) = (1)r if n = p 1kp rk i>rpiai ,0 ai < k,
μk(n) = 1 otherwise.
In other words, μk(n) vanishes if n is divisible by the (k + 1)-st power of some prime; otherwise, μk(n) is 1 unless the prime factorization of n contains the k-th powers of exactly r distinct primes, in which case μk(n) = (1)r. When k = 1k(n) is the usual Möbius function, μ1(n) = μ(n).

This paper discusses some of the relations that hold among the functions μk for various values of k. We use these to derive an asymptotic formula for the summatory function

        ∑
Mk(x) =    μk(n)
n≦x

for each k 2. Unfortunately, the analysis sheds no light on the behavior of the function M1(x) = nxμ(n).

Mathematical Subject Classification
Primary: 10.43
Milestones
Received: 11 April 1969
Published: 1 January 1970
Authors
Tom M. (Mike) Apostol