Vol. 235, No. 1, 2008
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Powers of theta functions

Heng Huat Chan & 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

Primary: 11E25

Secondary: 33E05, 11F11, 05A19
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