Note on a sum involving the divisor function

2025/05/03 by Liuying Wu, Wu, Liuying
Mathematics · #Advanced Mathematical Theories #Algebraic and Geometric Analysis #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.01645

openalex publication_date 2025/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d(n) be the divisor function and denote by [t] the integral part of the real number t. In this paper, we prove that ∑_{n≤ x^1/c}d([(x)/(n^c)])=d_cx^1/c+O_ε,c (x^max\(2c+2)/(2c²+5c+2),5/(5c+6)\+ε), where d_c=∑_k≥1d(k)(\frac1k^1/c-\frac1(k+1)^1/c) is a constant. This result constitutes an improvement upon that of Feng.

Citations

• The fractional sum of small arithmetic functions
• A variant of the prime number theorem
• On certain sums of number theory
• On a sum involving the Euler function
• Mean square of zeta function, circle problem and divisor problem revisited

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• On a sum involving general arithmetic functions and the integral part function
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• Further study on MacMahon-type sums of divisors
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