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

Holley-Reid, John; Rouse, Jeremy

doi:10.2140/involve.2023.16.727

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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

DOI: 10.2140/involve.2023.16.727

Abstract

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

r_{n} (n) \sim \frac{B \cdot A^{n}}{\sqrt{n}},

with $B \approx 0.2821$ and $A \approx 4.133$ .

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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

Vol. 16, No. 5, 2023