Gauss circle problem over smooth integers

Authors

Ankush Goswami, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

2-15-2024

Abstract

For a positive integer 𝑛, let 𝑟2(𝑛) be the number of representations of 𝑛 as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that 𝑅(𝑥) ∶= ∑ 𝑛≤𝑥 𝑟2(𝑛) ∼ 𝜋𝑥. Let 𝑃(𝑛) denote the largest prime factor of 𝑛 and let 𝑆(𝑥, 𝑦) ∶= {𝑛 ≤ 𝑥 ∶ 𝑃(𝑛) ≤ 𝑦}. In this paper, we study the asymptotic behavior of 𝑅(𝑥, 𝑦) ∶= ∑ 𝑛∈𝑆(𝑥,𝑦) 𝑟2(𝑛) for various ranges of 2 ≤ 𝑦 ≤ 𝑥. For 𝑦 in a certain large range, we show that 𝑅(𝑥, 𝑦) ∼ 𝜌(𝛼) ⋅ 𝜋𝑥 where 𝜌(𝛼) is the Dickman function and 𝛼 = log 𝑥∕ log 𝑦. We also obtain the asymptotic behavior of the partial sum of a generalized representation function following a method of Selberg.

Comments

Post-doc publication. Copyright the Author. Creative Commons Attribution 4.0 International License (CC BY 4.0)

Recommended Citation

Goswami, Ankush. "Gauss circle problem over smooth integers." New York J. Math 30 (2024): 270-294.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

New York Journal of Mathematics