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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a certain sum in number theory. II


Author: Břetislav Novák
Journal: Trans. Amer. Math. Soc. 195 (1974), 357-364
MSC: Primary 10J25
DOI: https://doi.org/10.1090/S0002-9947-1974-0435002-1
MathSciNet review: 0435002
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Abstract: We derive ``exact order'' of the function

$\displaystyle \sum\limits_{k \leq \sqrt x } {{k^\rho }{{\min }^\beta }\left( {\frac{{\sqrt x }}{k},\frac{1}{{{P_k}}}} \right)}.$

Here $ \rho $ and $ \beta $ are nonnegative real numbers and, for given real $ {\delta _1},{\delta _2}, \cdots ,{\delta _r},{P_k} = {\max _j}\langle k{\delta _j}\rangle $ where $ \langle t\rangle $, for real t, denotes distance of t from the nearest integer. Using our results, we obtain the solution of the basic problem in the theory of lattice points with weight in rational many-dimensional ellipsoids.

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DOI: https://doi.org/10.1090/S0002-9947-1974-0435002-1
Keywords: Lattice points with weight in ellipsoids, simultaneous approximation
Article copyright: © Copyright 1974 American Mathematical Society

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