Sums of weighted averages of gcd-sum functions II

2018/01/11 by Isao Kiuchi, Kiuchi, Isao, Sumaia Saad Eddin +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1801.03653

openalex publication_date 2018/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely ∑_{k≤ x}(1)/(k^s) ∑_j=1^kslogΓ((j)/(k^s)) ∑_\substackd|k
d^s|jf*μ(d) \rm and ∑_{k≤ x}(1)/(k^s)∑_j=0^k^s-1 B_m∑_\substackd|k
d^s|j f*μ(d) with any fixed integer s> 1 and any arithmetical function f. We give asymptotic formulas for them with various multiplicative functions f. We also consider several formulas of the Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.

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