A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

Abstract

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.

Locations

• Bulletin of the Australian Mathematical Society - View - PDF
• arXiv (Cornell University) - PDF
• Bulletin of the Australian Mathematical Society - View - PDF
• arXiv (Cornell University) - PDF