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A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

Published online by Cambridge University Press:  13 January 2010

TERENCE TAO*
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (email: [email protected])
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Abstract

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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〉:nx(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 ζ𝒫(s)≔∏ p∈𝒫(1−(1/ps))−1 on the line {Re(s)=1}. As equivalent forms of the first inequality, we have ∣∑ nx:(n,P)=1(μ(n))/n∣≤1, ∣∑ nN:nx(μ(n))/n∣≤1, and ∣∑ nx(μ(mn))/n∣≤1 for all m,x,N,P≥1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Diamond, H., ‘Elementary methods in the study of the distribution of prime numbers’, Bull. Amer. Math. Soc. 7 (1982), 553589.CrossRefGoogle Scholar
[2]Granville, A. and Soundararajan, K., ‘Negative values of truncations to L(1,χ)’, in: Analytic Number Theory: A Tribute to Gauss and Dirichlet, Clay Mathematics Proceedings, 7 (American Mathematical Society, Providence, RI, 2007).Google Scholar
[3]Hildebrand, A., ‘On Wirsing’s mean value theorem for multiplicative functions’, Bull. London Math. Soc. 18 (1986), 147152.Google Scholar
[4]Landau, E., ‘Über den Zusammenhang einiger neuer Sätze der analytischen Zahlentheorie’, Wiener Sitzungsberichte, Math. Klasse 115 (1906), 589632.Google Scholar
[5]Skałba, M., ‘On Euler–von Mangoldt’s equation’, Colloq. Math. 69 (1995), 143145.CrossRefGoogle Scholar