Functiones et Approximatio Commentarii Mathematici

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 and Yavuz Kesicioglu

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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.

Article information

Source
Funct. Approx. Comment. Math., Advance publication (2018), 19 pages.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1543460438

Digital Object Identifier
doi:10.7169/facm/1742

Subjects
Primary: 11A25: Arithmetic functions; related numbers; inversion formulas 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E25: Sums of squares and representations by other particular quadratic forms 11F11: Holomorphic modular forms of integral weight 11F20: Dedekind eta function, Dedekind sums 11F27: Theta series; Weil representation; theta correspondences

Keywords
convolution sums sum of divisors function Eisenstein series Eisenstein forms modular forms cusp forms Dedekind eta function octonary quadratic forms representations

Citation

Alaca, Şaban; Kesicioglu, Yavuz. 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., advance publication, 29 November 2018. doi:10.7169/facm/1742. https://projecteuclid.org/euclid.facm/1543460438


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