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A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

2010
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Bulletin of the Australian Mathematical Society
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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

doi:10.1017/s0004972709000884
fatcat:65svirvi6jf5tbqp2esb62sihm