On a generalisation sum involving the Euler function

2024/08/02 by Z. Ye, Zhefeng Xu, Ye, Zhaoxi +1
Mathematics · #11A25 #11L05 #11N37 #Advanced Mathematical Identities #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.01015

openalex publication_date 2024/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let j ≥1, k≥ 0 be real numbers and φ(n) be the Euler function. In this paper, we study the asymptotical behaviour of the summation function S_j,k(x):=∑_{n≤ x}\fracφ ( [ (x)/(n) ]^j ) [ (x)/(n) ]^k as x→ ∞ , where [ ⋅ ] is the integral part function. Our results combine and generalize the recent work of Zhai, Wu and Ma.

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