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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sums of quotients of additive functions


Author: Jean-Marie De Koninck
Journal: Proc. Amer. Math. Soc. 44 (1974), 35-38
MSC: Primary 10H25
DOI: https://doi.org/10.1090/S0002-9939-1974-0332683-3
MathSciNet review: 0332683
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Abstract: Denote by $ \omega (n)$ and $ \Omega (n)$ the number of distinct prime factors of $ n$ and the total number of prime factors of $ n$, respectively. Given any positive integer $ \alpha $, we prove that

$\displaystyle \sum\limits_{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\... ... = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} $

where $ {a_1} = \sum\nolimits_p {1/p(p - 1)} $ and all the other $ {a_i}$'s are computable constants. This improves a previous result of R. L. Duncan.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0332683-3
Keywords: Additive functions, factorization of integers
Article copyright: © Copyright 1974 American Mathematical Society