 

Ankush Goswami
Gauss circle problem over smooth integers
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Published: 
February 15, 2024. 
Keywords: 
circle problem, divisor problem, smooth numbers, asymptotic estimate, largest prime factor. 
Subject [2020]: 
11M06, 11L20, 11L40, 11N37, 11P05. 


Abstract
For a positive integer n, let r_{2}(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), the sum of r_{2}(n) for n ≤ x, is comparable to 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), the sum of r_{2}(n) for n in S(x,y), for various ranges of 2 ≤ y ≤ x. For y in a certain large range, we show that R(x,y) is comparable to 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.


Acknowledgements
I thank the anonymous referee for their meticulous feedback, which improved the exposition and fixed several mathematical and typographical errors. In particular, I appreciate his pointing out the error in the proof of Theorem 2.3 in the earlier version, and suggesting a direct application of Perron's formula. Finally, I thank Krishna Alladi, George Andrews, Bruce Berndt and Atul Dixit for their feedback on the manuscript.


Author information
Ankush Goswami
School of Mathematical and Statistical Sciences
The University of Texas Rio Grande Valley
1201 W. University Dr.
Edinburg, TX 78539, USA
ankush.goswami@utrgv.edu

