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Additive Sums of Generalized Shifted Ternary Divisor Functions
Abstract
In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum Dℐ,𝒥(X,1)=∑n ≤ X τℐ(n) τ𝒥(n + 1), where ℐ={α1,α2,α3} and 𝒥 = {β1, β2, β3}, with |αi|, |βi|≤ δ for i=1,2,3, and 0<δ<1/44. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function τ𝒥 in arithmetic progressions up to the level 2/3.
Related
- Shifted convolutions of divisor functions
- Statistics of divisor functions over function fields
- Triple divisor type functions in arithmetic progressions to prime power moduli
- A Joint Local Limit Theorem for Prime Divisor Counting Functions
- Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums
BibTeX
@misc{hamieh2026additive,
title = {{Additive Sums of Generalized Shifted Ternary Divisor Functions}},
author = {Alia Hamieh and Do Nhat Tan Vo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10966},
eprint = {2610.10966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10966},
}
RIS
TY - UNPB
TI - Additive Sums of Generalized Shifted Ternary Divisor Functions
AU - Alia Hamieh
AU - Do Nhat Tan Vo
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10966
UR - https://arxiv.org/abs/2610.10966
AB - In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum D_ℐ,𝒥(X,1)=∑_n ≤ X τ_ℐ(n) τ_𝒥(n + 1), where ℐ={α_1,α_2,α_3} and 𝒥 = {β_1, β_2, β_3}, with |α_i|, |β_i|≤ δ for i=1,2,3, and 0<δ<1/44. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function τ_𝒥 in arithmetic progressions up to the level 2/3.
ER -