Vol. 1, No. 1, 2008

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
An asymptotic for the representation of integers as sums of triangular numbers

Atanas Atanasov, Rebecca Bellovin, Ivan Loughman-Pawelko, Laura Peskin and Eric Potash

Vol. 1 (2008), No. 1, 111–121
Abstract

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer n as the sum of k triangular numbers is asymptotically equivalent to the modified divisor function σ2k1(2n + k).

Keywords
modular form, triangular number, asymptotics
Mathematical Subject Classification 2000
Primary: 11F11
Milestones
Received: 30 October 2007
Accepted: 19 January 2008
Published: 28 February 2008

Communicated by Ken Ono
Authors
Atanas Atanasov
7653 Lerner Hall
Columbia University
New York, NY 10027
United States
Rebecca Bellovin
1603 Lerner Hall
Columbia University
New York, NY 10027
United States
Ivan Loughman-Pawelko
4067 Lerner Hall
Columbia University
New York, NY 10027
United States
Laura Peskin
5991 Lerner Hall
Columbia University
New York, NY 10027
United States
Eric Potash
4926 Lerner Hall
Columbia University
New York, NY 10027
United States