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The number of representations of $n$ as a growing number of squares

John Holley-Reid and Jeremy Rouse

Vol. 16 (2023), No. 5, 727–735
Abstract

Let rk(n) denote the number of representations of the integer n as a sum of k squares. We give an asymptotic for rk(n) when n grows linearly with k. As a special case, we find that

rn(n) B An n ,

with B 0.2821 and A 4.133.

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Keywords
sums of squares, circle method, saddle-point method
Mathematical Subject Classification 2010
Primary: 11E25
Secondary: 41A60
Milestones
Received: 8 October 2019
Revised: 11 May 2022
Accepted: 9 November 2022
Published: 9 December 2023

Communicated by Filip Saidak
Authors
John Holley-Reid
Department of Mathematics
Wake Forest University
Winston-Salem, NC
United States
Jeremy Rouse
Department of Mathematics
Wake Forest University
Winston-Salem, NC
United States