Vol. 15, No. 1, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18
Issue 7, 1221–1401
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author index
To appear
Other MSP journals
This article is available for purchase or by subscription. See below.
The Laplace transform of the second moment in the Gauss circle problem

Thomas A. Hulse, Chan Ieong Kuan, David Lowry-Duda and Alexander Walker

Vol. 15 (2021), No. 1, 1–27

The Gauss circle problem concerns the difference P2(n) between the area of a circle of radius n and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients P2(n)2, and prove that this series has meromorphic continuation to . Using this series, we prove that the Laplace transform of P2(n)2 satisfies 0P2(t)2etXdt = CX32 X + O(X12+𝜖), which gives a power-savings improvement to a previous result of Ivić (1996).

Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations r2(n + h)r2(n), where h is fixed and r2(n) denotes the number of representations of n as a sum of two squares. We use this Dirichlet series to prove asymptotics for n1r2(n + h)r2(n)enX, and to provide an additional evaluation of the leading coefficient in the asymptotic for nXr2(n + h)r2(n).

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Gauss circle problem, modular forms, automorphic forms, multiple Dirichlet series
Mathematical Subject Classification 2010
Primary: 11F30
Secondary: 11E45, 11F27, 11F37
Received: 30 June 2017
Revised: 17 June 2020
Accepted: 21 July 2020
Published: 1 March 2021
Thomas A. Hulse
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Chan Ieong Kuan
School of Mathematics
Sun Yat-Sen University
David Lowry-Duda
ICERM and Brown University
Providence, RI
United States
Alexander Walker
Department of Mathematics
Rutgers University
Piscataway, NJ
United States