PreprintCC BY-ND 4.0HAL
ON THE GAUSS CIRCLE PROBLEM
Abstract
Using the \emph{compression} method, we prove an inequality related to the Gauss circle problem. Let 𝒩r denote the number of integral points in a circle of radius r>0. We have 2r2(1+1/4∑ 1≤ k≤ ⌊ (log r)/(log 2)⌋1/(22k−2))+O(r/(log r)) ≤ 𝒩r ≤ 8r2(1+∑ 1≤ k≤ ⌊ (log r)/(log 2)⌋1/(22k−2))+O(r/(log r)) for all r>1. This implies that the error function E(r) of the counting function 𝒩r≪ r1−ϵ for any ϵ>0.
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BibTeX
@misc{agama2026gauss,
title = {{ON THE GAUSS CIRCLE PROBLEM}},
author = {Theophilus Agama},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05777107v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - ON THE GAUSS CIRCLE PROBLEM
AU - Theophilus Agama
PY - 2026
DA - 2026/10/04
PB - HAL
UR - https://hal.science/hal-05777107v1
AB - Using the \emph{compression} method, we prove an inequality related to the Gauss circle problem. Let 𝒩_r denote the number of integral points in a circle of radius r>0. We have 2r^2(1+1/4∑ _1≤ k≤ ⌊ (log r)/(log 2)⌋1/(2^2k−2))+O(r/(log r)) ≤ 𝒩_r ≤ 8r^2(1+∑ _1≤ k≤ ⌊ (log r)/(log 2)⌋1/(2^2k−2))+O(r/(log r)) for all r>1. This implies that the error function E(r) of the counting function 𝒩_r≪ r^1−ϵ for any ϵ>0.
ER -