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CONVOLUTION SUMS INVOLVING THE DIVISOR FUNCTION

Published online by Cambridge University Press:  09 November 2004

Nathalie Cheng  and
Kenneth S. Williams
Nathalie Cheng
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada ([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, Canada ([email protected])
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Abstract

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The series

\begin{alignat*}{2} L_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ M_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_3(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ N_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_5(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \end{alignat*}

are evaluated and used to prove convolution formulae such as

$$ \sum_{m\le n}\sigma(4m-3)\sigma(4n-(4m-3))=4\sigma_3(n)-4\sigma_3(\tfrac12n). $$

AMS 2000 Mathematics subject classification: Primary 11A25

Keywords

divisor functionconvolution sumarithmetic identity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 3 , October 2004 , pp. 561 - 572
Copyright
Copyright © Edinburgh Mathematical Society 2004