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Proceedings of the American Mathematical Society

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

 

 

An elementary method for estimating error terms in additive number theory


Author: Elmer K. Hayashi
Journal: Proc. Amer. Math. Soc. 52 (1975), 55-59
MSC: Primary 10J99
DOI: https://doi.org/10.1090/S0002-9939-1975-0376586-8
MathSciNet review: 0376586
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Abstract: Let $ {R_k}(n)$ denote the number of ways of representing the integers not exceeding $ n$ as the sum of $ k$ members of a given sequence of nonnegative integers. Using only elementary methods, we prove a general theorem from which we deduce that, for every $ \epsilon > 0$,

$\displaystyle {R_k}(n) - c{n^\beta } \ne o({n^{\beta (1 - \beta )(1 - 1/k)/(1 - \beta + \beta /k) - \epsilon }})$

where $ c$ is a positive constant and $ 0 < \beta < 1$.

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DOI: https://doi.org/10.1090/S0002-9939-1975-0376586-8
Article copyright: © Copyright 1975 American Mathematical Society