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- Title
On some sums involving the integral part function.
- Authors
Liu, Kui; Wu, Jie; Yang, Zhishan
- Abstract
Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès.
- Subjects
INTEGRAL functions; REAL numbers; PRIME numbers; NATURAL numbers; PARTITION functions
- Publication
International Journal of Number Theory, 2024, Vol 20, Issue 3, p831
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S179304212450043X