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 = p1k⋯prk∏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 = 1,μk(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
for each k ≧ 2. Unfortunately, the analysis sheds no light on the behavior of the
function M1(x) =∑n≦xμ(n).
