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

Author: David A. Clark
Journal: Proc. Amer. Math. Soc. 123 (1995), 363-365
MSC: Primary 11B83
MathSciNet review: 1243164
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Abstract: A sequence $ A = \{ {a_i}\} $ of positive integers is called primitive if no term of the sequence divides any other. Erdös conjectures that, for any primitive sequence A,

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

where the sum is over all primes less than or equal to n. We show that

$\displaystyle \sum\limits_{a \in A} {\frac{1}{{a\log a}} \leq {e^\gamma } < 1.7811,} $

where $ \gamma $ is Euler's constant.

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

  • [1] P. Erdös and Z. Zhang, Upper bound of $ \sum {1/({a_i}\log {a_i})} $ for primitive sequences, Proc. Amer. Math. Soc. 117 (1993), 891-895.
  • [2] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689

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