Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Abstract The convolution sum, ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).


Introduction
Let in the sequel N, N 0 , Z, Q, R and C denote the sets of positive integers, non-negative integers, integers, rational numbers, real numbers and complex numbers, respectively.
Suppose that k; n 2 N. Then the sum of positive divisors of n to the power of k, k .n/, is defined by We write .n/ as a synonym for 1 .n/. For m … N we set k .m/ D 0. Suppose now that˛;ˇ2 N are such that˛Äˇ. Then the convolution sum, W .˛;ˇ/ .n/, is defined as follows: We write Wˇ.n/ as a synonym for W .1;ˇ/ .n/. Given˛;ˇ2 N, if for all .l; m/ 2 N 2 0 it holds that˛l Cˇm ¤ n then we set W .˛;ˇ/ .n/ D 0.
For those convolution sums W .˛;ˇ/ .n/ that have so far been evaluated, the levels˛ˇare given in Table 1.
We discuss the evaluation of the convolution sums of level˛ˇD 22; 44 and˛ˇD 52, i.e., .˛;ˇ/ D . As an application, convolution sums are used to determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms and respectively, where a; b; c; d 2 N. So far known explicit formulae for the number of representations of n by the octonary form Equation 3 are referenced in Table 2. (1,7) E. Ntienjem [21] We determine formulae for the number of representations of a positive integer n by the octonary quadratic form Equation 3 for which .a; b/ D .1; 11/; .1; 13/. These formulae for the number of representations are also new according to Table 2. This paper is organized in the following way. In Section 2 we discuss modular forms, briefly define eta functions and convolution sums, and prove the generalization of the extraction of the convolution sum. Our main results on the evaluation of the convolution sums are discussed in Section 3. The determination of formulae for the number of representations of a positive integer n is discussed in Section 4.
Software for symbolic scientific computation is used to obtain the results of this paper. This software comprises the open source software packages GiNaC, Maxima, REDUCE, SAGE and the commercial software package MAPLE.

Modular forms and convolution sums
Let H be the upper half-plane, that is H D fz 2 C j Im.z/ > 0g, and let G D SL 2 .R/ be the group of 2 2-matrices a b c d such that a; b; c; d 2 R and ad bc D 1 hold. Let furthermore D SL 2 .Z/ be the full modular group which is a subgroup of SL 2 .R/. Let N 2 N. Then is a subgroup of G and is called the principal congruence subgroup of level N. A subgroup H of G is called a congruence subgroup of level N if it contains .N /. Relevant for our purposes is the following congruence subgroup: We denote by f OE k the function whose value at z is .cz C d / k f . .z//, i.e., , wherein a n ¤ 0 for finitely many n 2 Z such that n < 0.
for all ı 2 and for all n 2 Z such that n < 0 it holds that a n D 0.  For the purpose of this paper we only consider trivial Dirichlet characters and 2 Ä k even. Theorems 5.8 and 5.9 in Section 5.3 of [23, p. 86] also hold for this special case.

Eta functions
The Dedekind eta function, Á.z/, is defined on the upper half-plane H by Á.z/ D e M. Newman [24,25] systematically used the Dedekind eta function to construct modular forms for 0 .N /. M. Newman determined when a function f .z/ is a modular form for 0 .N / by providing conditions (i)-(iv) in the following theorem. G. Ligozat [26] determined the order of vanishing of an eta function at the cusps of 0 .N /, which is condition (v) or (v 0 ) in Theorem 2.2.

Convolution sums W .˛;ˇ/ .n/
Recall that given˛;ˇ2 N such that˛Äˇ, the convolution sum is defined by Equation 2.
As observed by A. Alaca et al.
[11], we can assume that gcd.˛;ˇ/ D 1. Let q 2 C be such that jqj < 1. Then the Eisenstein series L.q/ and M.q/ are defined as follows: The following two relevant results are essential for the sequel of this work and are a generalization of the extraction of the convolution sum using Eisenstein forms of weight 4 for all pairs .˛;ˇ/ 2 N 2 . Their proofs are given by E. Ntienjem [21].  We observe the following inclusion relations  Let ı 1 2 D.44/ and .r.i; ı 1 // i;ı 1 be the Table 3 of the powers of Á.ı 1 z/. Let ı 2 2 D.52/ and .r.j; ı 2 // j;ı 2 be the Table 4   Proof. We give the proof for the case˛ˇD 44. The case˛ˇD 52 is proved similarly. Suppose now that the cardinality of the set D.44/ is greater than 1 and that M.q t / are linearly independent for all tj44 and t Ä t 1 for a given t 1 with 1 < t 1 < 44. Let C be the proper non-empty subset of D.44/ which contains all positive divisors of 44 less than or equal to t 1 . Note that all positive divisors of t 1 constitute a subset of C . Let us consider the non-empty subset C [ ft 0 g of D.44/, wherein t 0 is the next ascendant element of D.44/ which is greater than t 1 the greatest element of the set C . Then By the induction hypothesis it holds that x t D 0 for all t 2 C . So, we obtain from the above equation that x t 0 D 0 when we compare the coefficient of q t 0 on both sides of the equation. Hence, the solution is x t D 0 for all t such that t is a positive divisor of 44. Therefore, the set B E;44 is linearly independent. Hence, the set B E;44 is a basis of E 4 . 0 .44//.
x i a i .n/ /q n D 0 which gives the following homogeneous system of linear equations A simple computation using software for symbolic scientific computation shows that the determinant of the matrix of this homogeneous system of linear equations is non-zero. So, x i D 0 for all 1 Ä i Ä 15. Hence, the set    This results in a system of linear equations whose unique solution determines the values of the unknown X ı for all ı 2 D.44/ and the values of the unkown Y j for all 1 Ä j Ä 15. Hence, we obtain the stated result.
Our main result of this section is as follows.