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On certain properties of harmonic numbers.
- Source :
-
Journal of Number Theory . Jun2017, Vol. 175, p66-86. 21p. - Publication Year :
- 2017
-
Abstract
- Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log p ) ; (b) δ ( T p ) exists and 1 − ( 2 log p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0022314X
- Volume :
- 175
- Database :
- Academic Search Index
- Journal :
- Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 121276463
- Full Text :
- https://doi.org/10.1016/j.jnt.2016.11.027