On the number of representations of integers by quaternary quadratic forms

Huixue Lao (), Zhenxing Xie () and Dan Wang ()
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Huixue Lao: Shandong Normal University
Zhenxing Xie: Shandong Normal University
Dan Wang: Qilu University of Technology (Shandong Academy of Sciences)

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 395-406

Abstract: Abstract Let $$R_1(n), R_2(n)$$ R 1 ( n ) , R 2 ( n ) denote the numbers of representations of a positive integer n by the quaternary quadratic forms $$g_1(x_1,x_2,x_3,x_4)$$ g 1 ( x 1 , x 2 , x 3 , x 4 ) = $$2( x_{1}^{2}+x_1 x_2+ x_{2}^{2})+2x_1x_3 +x_1x_4+ x_2x_3+2x_2x_4+2(x_{3}^{2}+x_3 x_4+x_{4}^{2}), g_2(x_{1},x_2,x_3,x_4)=8( x_{1}^{2}+x_{2}^{2})+x_{3}^{2}+x_{4}^{2}$$ 2 ( x 1 2 + x 1 x 2 + x 2 2 ) + 2 x 1 x 3 + x 1 x 4 + x 2 x 3 + 2 x 2 x 4 + 2 ( x 3 2 + x 3 x 4 + x 4 2 ) , g 2 ( x 1 , x 2 , x 3 , x 4 ) = 8 ( x 1 2 + x 2 2 ) + x 3 2 + x 4 2 , respectively, where $$x_1$$ x 1 , $$x_2$$ x 2 , $$x_3$$ x 3 and $$x_4$$ x 4 are integers. In this paper, we establish the asymptotic formulae for the sums $$\sum \limits _{n\le x}R_i(n)$$ ∑ n ≤ x R i ( n ) and $$\sum \limits _{n\le x}R_i^{2}(n)$$ ∑ n ≤ x R i 2 ( n ) for $$i=1,2$$ i = 1 , 2 .

Keywords: Quaternary quadratic form; Asymptotic formula; Fourier coefficient; Cusp form; 11F30; 11F70 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00064-1

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