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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 ,
Şaban Alaca  and
Kenneth S. Williams
Ayşe Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: aalaca@math.carleton.casalaca@math.carleton.cawilliams@math.carleton.ca
Şaban Alaca
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6 e-mail: aalaca@math.carleton.casalaca@math.carleton.cawilliams@math.carleton.ca
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: aalaca@math.carleton.casalaca@math.carleton.cawilliams@math.carleton.ca
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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 $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$.

Keywords

11A2511E2011E25divisor functionsconvolution sumsEisenstein series
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 3 - 14
Copyright
Copyright © Canadian Mathematical Society 2008

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