A Remark on Partial Sums Involving the Mobius Function

Let 〈P〉 ⊂ N be a multiplicative subsemigroup of the natural numbers N= {1, 2, 3, . . .} generated by an arbitrary set P of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈P〉:n≤x (μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈P (1− (1/p)) (the case where P is all the primes is a wellknown observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζP (s) := ∏ p∈P (1− (1/p s))−1 on the line {Re(s)= 1}. As equivalent forms of the first inequality, we have | ∑ n≤x :(n,P)=1(μ(n))/n| ≤ 1, | ∑ n|N :n≤x (μ(n))/n| ≤ 1, and | ∑ n≤x (μ(mn))/n| ≤ 1 for all m, x, N , P ≥ 1. 2000 Mathematics subject classification: primary 11A25.