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An upper bound of $\sum 1/(a_ i\log a_ i)$ 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, \[ \sum \limits _{a \leq n,a \in A} {\frac {1}{{a\log a}} \leq \sum \limits _{p \leq n} {\frac {1}{{p\log p}},\quad {\text {for}}\;n > 1,} } \] where the sum is over all primes less than or equal to n. We show that \[ \sum \limits _{a \in A} {\frac {1}{{a\log a}} \leq {e^\gamma } < 1.7811,} \] where $\gamma$ is Euler’s constant.

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

    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.
  • J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689

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Article copyright: © Copyright 1995 American Mathematical Society