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September 2019 Evaluation of the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48),(3,16),(1,54),(2,27)$
Şaban Alaca, Yavuz Kesicioglu
Funct. Approx. Comment. Math. 61(1): 27-45 (September 2019). DOI: 10.7169/facm/1742

Abstract

We determine the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48), (3, 16), (1,54), (2,27)$ for all positive integers $n$. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of $n$ by the octonary quadratic forms $k(x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2) + l(x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2)$ for $(k,l)=(1,16), (1,18), (2,9)$. A modular form approach is used.

Citation

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Şaban Alaca. Yavuz Kesicioglu. "Evaluation of the convolution sum $\sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,48),(3,16),(1,54),(2,27)$." Funct. Approx. Comment. Math. 61 (1) 27 - 45, September 2019. https://doi.org/10.7169/facm/1742

Information

Published: September 2019
First available in Project Euclid: 29 November 2018

zbMATH: 07126908
MathSciNet: MR4012360
Digital Object Identifier: 10.7169/facm/1742

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Rights: Copyright © 2019 Adam Mickiewicz University

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Vol.61 • No. 1 • September 2019
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