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Note on fractional sum of the divisor function
Abstract
Let d(n) be the divisor function and denote by [t] the integral part of the real number t. We prove that ∑n≤ x1/cd([x/nc])=dcx1/c+Oε,c(xθc+ε), where dc is a suitable constant, θc<2/(3c+2) for 0<c<2/11. This result constitutes an improvement upon that of Y. Feng (2024).
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BibTeX
@misc{yang2026note,
title = {{Note on fractional sum of the divisor function}},
author = {Zhiyuan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04579},
eprint = {2610.04579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04579},
}
RIS
TY - UNPB
TI - Note on fractional sum of the divisor function
AU - Zhiyuan Yang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.04579
UR - https://arxiv.org/abs/2610.04579
AB - Let d(n) be the divisor function and denote by [t] the integral part of the real number t. We prove that ∑_n≤ x^1/cd([x/n^c])=d_cx^1/c+O_ε,c(x^θ_c+ε), where d_c is a suitable constant, θ_c<2/(3c+2) for 0<c<2/11. This result constitutes an improvement upon that of Y. Feng (2024).
ER -