A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

Abstract

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 well-known observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζ_𝒫(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.