EVALUATION OF FOUR CONVOLUTION SUMS AND REPRESENTATION OF INTEGERS BY CERTAIN QUADRATIC FORMS IN TWELVE VARIABLES

In this paper the convolution sums ∑6i+ j=n σ(l)σ3(m), ∑2i+3 j=n σ(l)σ3(m), ∑i+6 j=n σ(l)σ3(m) and ∑3i+2 j=n σ(l)σ3(m) are evaluated for all n ∈N, and then their evaluations are used to determine the representation number formulae N(1,1,1,1,1,2;n),N(1,1,1,1,2,2;n) and N(1,1,1,2,2,2;n) where N(a1, ...,a6;n) denote the representation numbers of n by the form a1(x 1 + x1x2 + x 2 2)+ a2(x 2 3 + x3x4 + x 2 4)+ a3(x 2 5 + x5x6 + x 2 6)+ a4(x 2 7 + x7x8 + x2 8)+a5(x 2 9 + x9x10 + x 2 10)+a6(x 2 11 + x11x12 + x 2 12).


INTRODUCTION
Let N, Z, R, and C denote the set of positive integers, integers, real numbers and complex numbers respectively and let N 0 = N ∪ {0}. Throughout this paper q ∈ C is taken to satisfy |q| < 1. For k, n ∈ N we set (1) where d runs through the positive divisors of n. If n / ∈ N we set σ k (n) = 0. We write σ (n) for σ 1 (n). For a, b, r, s, n ∈ N, we define the convolution sum W r,s a,b (n) by σ (n).
The following two sums are given by Huard, Ou, Spearman and Williams [2].
In this paper, motivated from the work of Yao and Xia[4] and using the (p, k)-parametrization of Eisenstein series and theta functions given by Alaca, Alaca and Williams we determine the convolution sums W 1,3 6,1 (n), W 1,3 2,3 (n), W 1,3 1,6 (n) and W 1,3 3,2 (n). We've used 4 Eisenstein series N(q), N(q 2 ), N(q 3 ), N(q 6 ) and 3 eta products which form a basis of the Modular space M 6 (Γ 0 (6)) (with dimension 7) and express the products 6L(q 6 ) − L(q) M (q), Series (all of which clearly belongs to space) as a linear combination the mentioned Eisenstein series and eta products. One of the eta products which is used here appeared in literature [4].
Today, some other authors use the theory of quasimodular forms with softwares Magma or the database LMFDP of L functions and modular forms to obtain formulae for convolution sums.

PROOF OF THEOREM 1
In his second notebook [18] Ramanujan gives the definitions of Eisenstein series L(q), M(q) and N(q) by It can be easily seen that The Jacobi theta function ϕ(q) is defined by see for example ( [20], p.92).
By (23) and (24), we see that Appealing to (16) and (25), we obtain For n ∈ N, equating the coefficients of q n on both sides of (65) and using (2) we have u 4 (n) .
The following two formulae was proved by Köklüce [22].
Proof. We just prove (ii) in detail. The remaining can be proved in a similar way.