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Proceedings of the American Mathematical Society

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

 

 

Exponential sums related to binomial coefficient parity


Author: Alan H. Stein
Journal: Proc. Amer. Math. Soc. 80 (1980), 526-530
MSC: Primary 10A21; Secondary 05A10
DOI: https://doi.org/10.1090/S0002-9939-1980-0581019-4
MathSciNet review: 581019
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Abstract: Let $ \alpha (n)$ be the number of 1's in the binary expansion of n, $ z > 0$ and $ {\phi _z}(x) = {\Sigma _{n < x}}{z^{\alpha (n)}}$. Let $ {\theta _z} = (\log (1 + z))/\log 2,a(z) = \lim \inf {x^{ - {\theta _z}}}{\phi _z}(x),b(z) = \lim \sup {x^{ - {\theta _z}}}{\phi _z}(x)$ . Then $ 0 < a(z) \leqslant 1 \leqslant b(z) < 2$. Furthermore, if $ z \ne 1$, then $ a(z) < b(z)$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0581019-4
Article copyright: © Copyright 1980 American Mathematical Society