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For a positive integer 𝑛, let π‘Ÿ2(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(π‘₯) ∢= βˆ‘ 𝑛≀π‘₯ π‘Ÿ2(𝑛) ∼ πœ‹π‘₯. Let 𝑃(𝑛) denote the largest prime factor of 𝑛 and let 𝑆(π‘₯, 𝑦) ∢= {𝑛 ≀ π‘₯ ∢ 𝑃(𝑛) ≀ 𝑦}. In this paper, we study the asymptotic behavior of 𝑅(π‘₯, 𝑦) ∢= βˆ‘ π‘›βˆˆπ‘†(π‘₯,𝑦) π‘Ÿ2(𝑛) for various ranges of 2 ≀ 𝑦 ≀ π‘₯. For 𝑦 in a certain large range, we show that 𝑅(π‘₯, 𝑦) ∼ 𝜌(𝛼) β‹… πœ‹π‘₯ where 𝜌(𝛼) is the Dickman function and 𝛼 = log π‘₯βˆ• log 𝑦. We also obtain the asymptotic behavior of the partial sum of a generalized representation function following a method of Selberg.


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

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

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New York Journal of Mathematics

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Mathematics Commons



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