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The Convolution Sum Σm<n/16σ(m)σ(n – 16m)

Published online by Cambridge University Press:  20 November 2018

Ayşe Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: [email protected]@[email protected]
Şaban Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: [email protected]@[email protected]
Kenneth S. Williams
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: [email protected]@[email protected]
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Abstract

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The convolution sum $\sum{_{m<n/16}\,\sigma (m)\sigma (n\,}-16m)$ is evaluated for all $n\,\in \,\mathbb{N}$. This evaluation is used to determine the number of representations of $n$ by the quadratic form $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2008

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