Evaluation of the convolution sum $$\displaystyle \varvec{\sum _{al+bm=n} \sigma (l)\sigma _ 3 (m) }$$ ∑ a l + b m = n σ ( l ) σ 3 ( m ) for $$\varvec{(a,b) = (1,7), (7,1), (1,8), (8,1), (1,9), (9,1)}$$ ( a, b ) = ( 1, 7 ), ( 7, 1 ), ( 1, 8 ), ( 8, 1 ), ( 1, 9 ), ( 9, 1 ) and representations by certain quadratic forms in twelve variables

Şaban Alaca () and Yavuz Kesicioğlu ()
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Şaban Alaca: Carleton University
Yavuz Kesicioğlu: Recep Tayyip Erdogan University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 99-112

Abstract: Abstract For a positive integer n we evaluate the convolution sum $$\displaystyle {\sum _{al+bm=n} \hspace{-3mm} \sigma (l)\sigma _ 3 (m) }$$ ∑ a l + b m = n σ ( l ) σ 3 ( m ) for $$(a,b) = (1,7)$$ ( a , b ) = ( 1 , 7 ) , (7, 1), (1, 8), (8, 1), (1, 9) and (9, 1). We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the forms $$ x_1^2 + x_2^2 + x_3^2 + x_4^2 + 2(x_5^2 + x_6^2 + x_7^2 + x_8^2 + x_9^2 + x_{10}^2 + x_{11}^2 + x_{12}^2), ~x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 + 2(x_9^2 + x_{10}^2 + x_{11}^2 + x_{12}^2), ~x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2 + 3(x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2 + x_9^2 + x_9x_{10} + x_{10}^2 + x_{11}^2 + x_{11}x_{12} + x_{12}^2)$$ x 1 2 + x 2 2 + x 3 2 + x 4 2 + 2 ( x 5 2 + x 6 2 + x 7 2 + x 8 2 + x 9 2 + x 10 2 + x 11 2 + x 12 2 ) , x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 + 2 ( x 9 2 + x 10 2 + x 11 2 + x 12 2 ) , x 1 2 + x 1 x 2 + x 2 2 + x 3 2 + x 3 x 4 + x 4 2 + 3 ( x 5 2 + x 5 x 6 + x 6 2 + x 7 2 + x 7 x 8 + x 8 2 + x 9 2 + x 9 x 10 + x 10 2 + x 11 2 + x 11 x 12 + x 12 2 ) , $$x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2 + x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2 + 3(x_9^2 + x_9x_{10} + x_{10}^2 + x_{11}^2 + x_{11}x_{12} + x_{12}^2)$$ x 1 2 + x 1 x 2 + x 2 2 + x 3 2 + x 3 x 4 + x 4 2 + x 5 2 + x 5 x 6 + x 6 2 + x 7 2 + x 7 x 8 + x 8 2 + 3 ( x 9 2 + x 9 x 10 + x 10 2 + x 11 2 + x 11 x 12 + x 12 2 ) . We use a modular form approach.

Keywords: Convolution sums; Sum of divisors function; Eisenstein series; Modular forms; Cusp forms; Dedekind eta function; Quadratic forms; Representations; 11A25; 11E20; 11E25; 11F11; 11F20; 11F27 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00459-2

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