A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

TERENCE TAO
2010 Bulletin of the Australian Mathematical Society  
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
more » ... ence 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. 2000 Mathematics subject classification: primary 11A25.
doi:10.1017/s0004972709000884 fatcat:65svirvi6jf5tbqp2esb62sihm