On two conjectures concerning the partial sums of the harmonic series

Author:
Stephen M. Zemyan

Journal:
Proc. Amer. Math. Soc. **95** (1985), 83-86

MSC:
Primary 40A05

DOI:
https://doi.org/10.1090/S0002-9939-1985-0796451-8

MathSciNet review:
796451

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Abstract: Let denote the th partial sum of the harmonic series. For a given positive integer , there exists a unique integer such that . It has been conjectured that is equal to the integer nearest , where is Euler's constant. We provide an estimate on which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which and differ from .

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0796451-8

Article copyright:
© Copyright 1985
American Mathematical Society