The convolution sum ∑ _m<n∕8σ(m)σ(n−8m)

Kenneth S. Williams

Vol. 228 (2006), No. 2, 387–396

DOI: 10.2140/pjm.2006.228.387

Abstract

The convolution sum ∑ _m<n∕8σ(m)σ(n−8m) is evaluated for all n ∈ ℕ. This evaluation is used to determine the number of representations of n by the quadratic form x₁² + x₂² + x₃² + x₄² + 2x₅² + 2x₆² + 2x₇² + 2x₈².

Keywords

divisor functions, convolution sums, Eisenstein series

Mathematical Subject Classification 2000

Primary: 11A25, 11E20, 11E25

Milestones

Received: 5 2005

Revised: 13 July 2005

Accepted: 28 July 2005

Published: 1 December 2006