Number Theory

    

On the Gauss Circle Problem

Authors: Theophilus Agama

Using the compression method, we prove an inequality related to the Gauss circle problem. Let $mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2bigg(1+frac{1}{4}sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r}) leq mathcal{N}_r leq 8r^{2}bigg(1+sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r})$$ for all $r>1$. This implies that the error function $E(r)$ of the counting function $mathcal{N}_rll r^{1-epsilon}$ for any $epsilon>0.$

Comments: 13 Pages. A few corrections have been implemented

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Submission history

[v1] 2022-12-22 03:24:28
[v2] 2024-03-31 07:27:16
[v3] 2025-01-03 22:58:15

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