Vol. 235, No. 1, 2008

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Powers of theta functions

Heng Huat Chan and Shaun Cooper

Vol. 235 (2008), No. 1, 1–14
Abstract

The Ramanujan–Mordell Theorem for sums of an even number of squares is extended to other quadratic forms. A number of explicit examples is given. As an application, the value of the convolution sum

  ∑
σ(m)σ(n− 23m )
1≤m <n∕23

is determined, where σ(m) denotes the sum of the divisors of m.

Keywords
sum of squares, Ramanujan, convolution sum, modular form, Eisenstein series
Mathematical Subject Classification 2000
Primary: 11E25
Secondary: 33E05, 11F11, 05A19
Milestones
Received: 6 November 2007
Revised: 7 December 2007
Accepted: 10 December 2007
Published: 1 March 2008
Authors
Heng Huat Chan
Department of Mathematics
National University of Singapore
Kent Ridge 119260
Singapore
Shaun Cooper
Institute of Information and Mathematical Sciences
Massey University – Albany
Private Bag 102904, North Shore Mail Centre
Auckland
New Zealand