On the number of representations of an integer by certain quadratic forms in sixteen variables

Abstract

We evaluate the convolution sums ∑l,m∈ℕ,l+2m=n σ3(l)σ3(m), ∑l,m∈ℕ,l+3m=n σ3(l) × σ3(m), ∑l,m∈ℕ,2l+3m=n σ3(l)σ3(m) and ∑l,m∈ℕ,l+6m=n σ3(l)σ3(m) for all n ∈ ℕ using the theory of modular forms and use these convolution sums to determine the number of representations of a positive integer n by the quadratic forms Q8 ⊕ Q8 and Q8 ⊕ 2Q8, where the quadratic form Q8 is given by [Formula: see text]

On the number of representations of certain quadratic forms in 20 and 24 variables

“…However, the other formulas mentioned above are in terms of divisor functions and Fourier coefficients of certain cusp forms. Like in the works of [17] and [13], by comparing the formulas of Lomadze with our results, we also obtain identities connecting the Fourier coefficients of certain cusp forms in terms of finite sums (see Corollary 2.5).…”

“…be the number of representations of a positive integer n by the quadratic form F k . For k = 2, 4, 6, 8 formulas for s 2k are known due to the works of J. Liouville [9], J. G. Huard et al [5], O. X. M. Yao and E. X. W. Xia [17] and the authors [13]. In [10], G. A. Lomadze gave formulas for s 2k (n) for 2 n 17, which involves the divisor functions and certain finite sums which involve the solution set of the representation of same quadratic forms of lower variables.…”

Representation numbers by sums of quadratic forms $x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}$ in sixteen variables

“…Formulae for many other convolution sums of this type have also been given by the authors in that study. Ramakrishnan and Sahu [8] have recently evaluated the following four formulae containing formulae for W 3;3 1;2 .n/ as well. To prove the theorem we need the representation number formulae for the following octonary quadratic forms, is given by Lomadze [6].…”

“…Ramakrishnan and Sahu [8] have recently proved formulae for R.1 8 I n/ and R.1 4 ; 2 4 I n/ for all n 2 N. In the present paper, motivated from the work of Ramakrishnan and Sahu we derive formulae for R.1 1 ; 2 7 I n/, R.1 3 ; 2 5 I n/, R.1 5 ; 2 3 I n/ and R.1 7 ; 2 1 I n/ by using the method of Alaca, Alaca and Williams (see for example [1]). These formulae are given in terms of the function 7 .n/ and the numbers 8;2 .n/ and 8;6 .n/ which are introduced in [8]. Köklüce [4] has derived formulae for the quadratic forms in twelve and sixteen variables which are sums of quadratic forms with discriminant 23 by using the method developed by Lomadze [7].…”

On the number of representations of certain quadratic forms and a formula for the Ramanujan Tau function

“…al. [3], O. X. M. Yao and E. X. W. Xia [16] and the first two authors [10,11]. Most of these works make use of the convolution sums of the divisor functions to evaluate the formulas.…”