Let k denote a fixed positive
integer. We define an arithmetical function μ_{k}, the Möbius function of order k, as
follows:
μ_{k}(1)  = 1,  
 μ_{k}(n)  = 0 if p^{k+1}n for some prime p,  
 μ_{k}(n)  = (−1)^{r} if n = p_{
1}^{k}⋯p_{
r}^{k} ∏
_{i>r}p_{i}^{ai
},0 ≦ a_{i} < k,  
 μ_{k}(n)  = 1 otherwise.   
In other words, μ_{k}(n) vanishes if n is divisible by the (k + 1)st power of some prime;
otherwise, μ_{k}(n) is 1 unless the prime factorization of n contains the kth powers of
exactly r distinct primes, in which case μ_{k}(n) = (−1)^{r}. When k = 1,μ_{k}(n) is the
usual Möbius function, μ_{1}(n) = μ(n).
This paper discusses some of the relations that hold among the functions μ_{k} 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 M_{1}(x) = ∑
_{n≦x}μ(n).
