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Upper bound of $ \sum 1/(a\sb i\log a\sb i)$ for primitive sequences


Authors: Paul Erdős and Zhen Xiang Zhang
Journal: Proc. Amer. Math. Soc. 117 (1993), 891-895
MSC: Primary 11B05
DOI: https://doi.org/10.1090/S0002-9939-1993-1116257-4
MathSciNet review: 1116257
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Abstract: A sequence $ A = \{ {a_i}\} $ of positive integers $ {a_1} < {a_2} < \cdots $ is said to be primitive if no term of $ A$ divides any other. The senior author conjectures that, for any primitive sequence $ A$,

$\displaystyle \sum\limits_{a \leqslant n,a \in A} {\frac{1} {{a\,\log \,a}}} \l... ...um\limits_{p \leqslant n} {\frac{1} {{p\,\log \,p}}} \quad {\text{for}}\;n > 1,$

where $ p$ is a variable prime. In our two previous papers we partially proved this conjecture. The main result of this paper is: for any primitive sequence $ A$,

$\displaystyle \sum\limits_{a \in A} {\frac{1} {{a\,\log \,a}} < 1.84.} $

We also give a necessary and sufficient condition for this conjecture, i.e.,

$\displaystyle \sum\limits_{b \in B} {\frac{1} {{b\,\log \,b}} \leqslant \sum {\frac{1} {{p\,\log \,p}}} } $

for any primitive sequence $ B$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1116257-4
Keywords: Primitive sequences
Article copyright: © Copyright 1993 American Mathematical Society