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arXiv:1607.01082v2 [math.NT] 9 Aug 2016 ELEMENTARY EVALUATION OF CONVOLUTION SUMS INVOLVING THE SUM OF DIVISORS FUNCTION FOR A CLASS OF POSITIVE INTEGERS EBÉNÉZER NTIENJEM A BSTRACT. We discuss an elementary method for the evaluation of the convolution sums ∑ σ(l)σ(m) for those α, β ∈ N for which gcd (α, β) = 1 and αβ = 2ν 0, where ν ∈ (l,m)∈N20 α l+β m=n {0, 1, 2, 3} and 0 is a squarefree positive integer which is a finite product of distinct odd primes. Modular forms are used to achieve this result. We also generalize the extraction of the convolution sum to all natural numbers. Formulae for the number of representations of a positive integer n by octonary quadratic forms using convolution sums belonging to this class are then determined when ν ≥ 2 or 0 ≡ 0 (mod 3). To achieve this application, we first discuss a method to compute all pairs (a, b), (c, d) ∈ N2 necessary for the determination of such formulae for the number of representations of a positive integer n by octonary quadratic forms when αβ has the above form and ν ≥ 2 or 0 ≡ 0 (mod 3). We illustrate our approach by explicitly evaluating the convolution sum for αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · 7, and then by using them to determine formulae for the number of representations of a positive integer n by octonary quadratic forms. 1. I NTRODUCTION We denote by N, N0 , Z, Q, R and C the sets of positive integers, non-negative integers, integers, rational numbers, real numbers and complex numbers, respectively. Suppose that k ∈ N0 and n ∈ N. The sum of positive divisors of n to the power of k, σk (n), is defined by (1.1) σk (n) = ∑ δk . 0<δ|n It is obvious from the definition that σk (m) = 0 for all m ∈ / N. We write d(n) and σ(n) as a synonym for σ0 (n) and σ1 (n), respectively. Assume that the positive integers α ≤ β are given. Then the convolution sum W(α,β) (n) is defined by (1.2) W(α,β) (n) = ∑ σ(l)σ(m). (l,m)∈N20 α l+β m=n Let Wβ (n) stands for W(1,β) (n). We give the values of (α, β) for those convolution sums W(α,β) (n) which have so far been evaluated in Table 1. From these values of (α, β) the following ones, expressed in the form αβ, do not belong to the class of positive integers that we are handling in this paper: 9, 16, 18, 25, 27, 32, and 36. 2010 Mathematics Subject Classification. 11A25, 11F11, 11F20, 11E20, 11E25, 11F27. Key words and phrases. Sums of Divisors; Convolution Sums; Modular Forms; Dedekind eta function; Eisenstein forms; Cusp Forms; Octonary quadratic Forms; Number of Representations . 1 2 EBÉNÉZER NTIENJEM We evaluate the convolution sum W(α,β) (n) in the case where (1.3) αβ = 2ν 0 with ν ∈ {0, 1, 2, 3}, and 0 odd and squarefree positive integer. The evaluation of the convolution sum for a class of natural numbers and especially for this class is new. We then apply the result for this class to determine the convolution sum for αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · 7. Again, these explicit convolution sums have not been evaluated as yet. Convolution sums are applied to establish explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms (1.4) a (x12 + x22 + x32 + x42 ) + b (x52 + x62 + x72 + x82 ), and (1.5) c (x12 + x1 x2 + x22 + x32 + x3 x4 + x42 ) + d (x52 + x5 x6 + x62 + x72 + x7 x8 + x82 ), respectively, where (a, b), (c, d) ∈ N2 . Known explicit formulae for the number of representations of n by the octonary quadratic form Equation 1.4 are referenced in Table 2 and that for the octonary quadratic form Equation 1.5 in Table 3. Based on the structure of α and β, we provide a method to determine all pairs (a, b) ∈ N2 and (c, d) ∈ N2 that are neccessary for the determination of the formulae for the number of representations of a positive integer by the octonary quadratic forms Equation 1.4 and Equation 1.5. Then we determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms Equation 1.4 and Equation 1.5, whenever αβ has the above form and is such that ν ≥ 2 or 0 ≡ 0 (mod 3). As an example, we determine formulae for the number of representations of a positive integer n by octonary quadratic forms Equation 1.4 and Equation 1.5 using the convolution sums for αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · 7, respectively. This work is structured as follows. In Section 2 we discuss basic knowledge of modular forms, briefly define eta functions and convolution sums. The evaluation of the convolution sum for the above class of positive integers is discussed in Section 3. In Section 4 and Section 5, explicit formulae for the number of representations of n by the octonary forms Equation 1.4 and Equation 1.5 are determined. Examples to illustrate our method are then given in Section 6. We then conclude in Section 7 with a brief outlook. The results of this paper are obtained using Software for symbolic scientific computation. This software is composed of the open source software packages GiNaC, Maxima, REDUCE, SAGE and the commercial software package MAPLE. 2. BASIC K NOWLEDGE The upper half-plane, H = {z ∈ C | Im(z) > 0}, and Γ = SL2 (R) the group of 2 × 2matrices ac db such that a, b, c, d ∈ R and ad − bc = 1 are considered in this paper. Let N ∈ N. Then a b 10 a b Γ(N) = (mod N) c d ∈ SL2 (Z) | c d ≡ 0 1 is a subgroup of Γ and is known as a principal congruence subgroup of level N. If a subgroup H of Γ contains Γ(N), then that subgroup is called a congruence subgroup of level N. The following congruence subgroup of level N is relevant for our purpose a b Γ0 (N) = c d ∈ SL2 (Z) | c ≡ 0 (mod N) . ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 3 Let N ∈ N, Γ0 ⊆ Γ be a congruence subgroup of level N, k ∈ Z, γ ∈ SL2 (Z) and f [γ]k : H ∪ Q ∪ {∞} → C ∪ {∞} be defined by f [γ]k (z) = (cz + d)−k f (γ(z)). The following definition is extracted from N. Koblitz’s book [15, p. 108]. Definition 2.1. Suppose that N ∈ N, k ∈ Z, f is a meromorphic function on H and Γ0 ⊂ Γ is a congruence subgroup of level N. Let furthermore N− 0 = { −n | n ∈ N0 } be the set of all negative non-zero natural numbers. (a) f is a modular function of weight k for Γ0 if (a1) for all γ ∈ Γ0 it holds that f [γ]k = f . 2πizn (a2) for any δ ∈ Γ it holds that f [δ]k (z) can be expressed in the form ∑ an e N , n∈Z wherein an 6= 0 for finitely many n ∈ N− 0. (b) f is a modular form of weight k for Γ0 if (b1) f is a modular function of weight k for Γ0 , (b2) f is holomorphic on H, (b3) for all δ ∈ Γ and for all n ∈ N− 0 it holds that an = 0. (c) f is a cusp form of weight k for Γ0 if (c1) f is a modular form of weight k for Γ0 , (c2) for all δ ∈ Γ it holds that a0 = 0. We denote the set of modular forms of weight k for Γ0 by Mk (Γ0 ), the set of cusp forms of weight k for Γ0 by Sk (Γ0 ) and the set of Eisenstein forms by Ek (Γ0 ). The sets Mk (Γ0 ), Sk (Γ0 ) and Ek (Γ0 ) are vector spaces over C. Hence, Mk (Γ0 (N)) is the space of modular forms of weight k for Γ0 (N), Sk (Γ0 (N)) is the space of cusp forms of weight k for Γ0 (N), and Ek (Γ0 (N)) is the space of Eisenstein forms. It is proved in W. A. Stein’s book (online version) [28, p. 81] that Mk (Γ0 (N)) = Ek (Γ0 (N)) ⊕ Sk (Γ0 (N)). As noted in Section 5.3 of [28, p. 86], if the primitive Dirichlet characters are trivial and ∞ 2 ≤ k is even, then Ek (q) = 1 − B2kk ∑ σk−1 (n) qn , where Bk are the Bernoulli numbers. n=1 For the purpose of this paper we only consider trivial primitive Dirichlet characters and 4 ≤ k even. Theorems 5.8 and 5.9 in Section 5.3 of [28, p. 86] also hold for this special case. 2.1. Eta Quotients. The Dedekind eta function η(z) is defined on the upper half-plane H by η(z) = e 2πiz 24 ∞ ∏ (1 − e2πinz ). When we set q = e2πiz , then we have n=1 1 ∞ 1 η(z) = q 24 ∏ (1 − qn ) = q 24 F(q), n=1 ∞ where F(q) = ∏ (1 − qn ). n=1 In this paper we use eta function, eta quotient and eta product interchangeably as synonyms. The Dedekind eta function was systematically applied by M. Newman [21, 22] to construct modular forms for Γ0 (N) and then to determine when a function f (z) was a modular form for Γ0 (N). That lead to conditions (i)-(iv) in the following theorem. The order of vanishing of an eta function at the cusps of Γ0 (N), which is condition (v) or (v0 ) in the following theorem, was determined by G. Ligozat [18]. In L. J. P. Kilford’s book [14, p. 99] and G. Köhler’s book [16, p. 37] the following theorem is proved. We will use that theorem to determine eta quotients, f (z), which belong to Mk (Γ0 (N)), and especially those eta quotients which are in Sk (Γ0 (N)). 4 EBÉNÉZER NTIENJEM Theorem 2.2 (M. Newman and G. Ligozat ). Let N ∈ N and f (z) = ∏ ηrδ (δz) be an eta 1≤δ|N quotient which satisfies the following conditions: (i) (ii) ∑ δ rδ ≡ 0 (mod 24), ∑ 1≤δ|N (iii) ∏ δrδ is a square in Q, (iv) 1≤δ|N 1≤δ|N k = 21 N δ rδ ≡ 0 (mod 24), ∑ rδ is an even integer, 1≤δ|N (v) for each positive divisor d of N it holds that ∑ 1≤δ|N gcd (δ,d)2 rδ δ ≥ 0. Then f (z) ∈ Mk (Γ0 (N)). The eta quotient f (z) belongs to Sk (Γ0 (N)) if (v) is replaced by (v’) for each positive divisor d of N it holds that ∑ 1≤δ|N gcd (δ,d)2 rδ δ > 0. 2.2. Convolution Sums W(α,β) (n). Suppose that α, β ∈ N are such that α ≤ β. The convolution sum, W(α,β) (n), is defined by Equation 1.2. We set W(α,β) (n) = 0 if for all (l, m) ∈ N2 we have α l + β m 6= n. Now, suppose in addition that gcd (α, β) = δ > 1. In this case, there exist α1 , β1 ∈ N such that gcd (α1 , β1 ) = 1, α = δ α1 and β = δ β1 . Then (2.1) W(α,β) (n) = ∑ σ(l)σ(k) = (l,k)∈N20 α l+β k=n ∑ (l,k)∈N20 δ α1 l+δ β1 k=n n σ(l)σ(k) = W(α1 ,β1 ) ( ). δ Therefore, we may simply assume that gcd (α, β) = 1 as does A. Alaca et al. [2]. We apply the formula proved by Besge [8] to Equation 2.1 to deduce that (2.2) n 5 n 1 1 n ∀ α ∈ N W(α,α) (n) = W(1,1) ( ) = σ3 ( ) + ( − n)σ( ). α 12 α 12 2 α α Let q ∈ C be such that |q| < 1. The Eisenstein series L(q) and M(q) are defined as follows: ∞ (2.3) L(q) = E2 (q) = 1 − 24 ∑ σ(n)qn , n=1 ∞ (2.4) M(q) = E4 (q) = 1 + 240 ∑ σ3 (n)qn . n=1 We state two relevant results for the sequel of this work. These two results generalize the extraction of the convolution sum to all natural numbers. Lemma 2.3. Let α, β ∈ N. Then (α L(qα ) − β L(qβ ))2 ∈ M4 (Γ0 (αβ)). Proof. If α = β, then trivially 0 = (α L(qα ) − α L(qα ))2 ∈ M4 (Γ0 (α)) and there is nothing to prove. Therefore, we may suppose that α 6= β > 0 in the sequel. We apply the result proved by W. A. Stein [28, Thrms 5.8,5.9, p. 86] to deduce L(q) − α L(qα ) ∈ M2 (Γ0 (α)) ⊆ M2 (Γ0 (αβ)) and L(q) − β L(qβ ) ∈ M2 (Γ0 (β)) ⊆ M2 (Γ0 (αβ)). Therefore, α L(qα ) − β L(qβ ) = (L(q) − β L(qβ )) − (L(q) − α L(qα )) ∈ M2 (Γ0 (αβ)) and so (α L(qα ) − β L(qβ ))2 ∈ M4 (Γ0 (αβ)). ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 5 Theorem 2.4. Let α, β ∈ N be such that α < β, and α and β are relatively prime. Then ∞ n n (2.5) (α L(qα ) − β L(qβ ))2 = (α − β)2 + ∑ 240 α2 σ3 ( ) + 240 β2 σ3 ( ) α β n=1 n n + 48 α (β − 6n) σ( ) + 48 β (α − 6n) σ( ) − 1152 αβW(α,β) (n) qn . α β Proof. We first observe that (2.6) (α L(qα ) − β L(qβ ))2 α2 L2 (qα ) + β2 L2 (qβ ) − 2 αβ L(qα )L(qβ ). = J. W. L. Glaisher [12] has proved the following identity ∞ 2 (2.7) L (q) = 1 + ∑ 240 σ3 (n) − 288 n σ(n) qn n=1 which we apply to deduce ∞ n n n L2 (qα ) = 1 + ∑ 240 σ3 ( ) − 288 σ( ) qn α α α n=1 (2.8) and ∞ n n n L2 (qβ ) = 1 + ∑ 240 σ3 ( ) − 288 σ( ) qn . β β β n=1 (2.9) Since ∞ n ∞ n ∑ σ( α )qn ∑ σ( β )qn ∞ = n=1 n=1 ∑ ∑ σ(k)σ(l) qn ∞ ∑ W(α,β) (n)qn , = n=1 αk+βl=n n=1 we conclude, when using the accordingly modified Equation 2.3, that ∞ L(qα )L(qβ ) = 1 − 24 (2.10) ∞ n n ∞ ∑ σ( α )qn − 24 ∑ σ( β )qn + 576 ∑ W(α,β) (n)qn . n=1 n=1 n=1 Therefore, ∞ 2 α L(qα ) − β L(qβ ) = (α − β)2 + ∑ n=1 n n 240 α2 σ3 ( ) + 240 β2 σ3 ( ) α β n n + 48 α (β − 6n) σ( ) + 48 β (α − 6n) σ( ) − 1152 αβW(α,β) (n) qn α β as asserted. 3. E VALUATING W(α,β) (n) FOR A CLASS OF NATURAL NUMBERS αβ Suppose that α and β are positive integers which satisfy the following two conditions: (i) gcd (α, β) = 1 (ii) Equation 1.3. We derive the formula for the convolution sum W(α,β) (n) for all such α and β. Let in the sequel D(αβ) denote the set of all positive divisors of αβ. 6 EBÉNÉZER NTIENJEM 3.1. Bases for E4 (Γ0 (αβ)) and S4 (Γ0 (αβ)). The existence of a basis of the space of cusp forms of weight k ≥ 2 for Γ0 (αβ) when αβ is not a perfect square is discussed by A. Pizer [24]. We recall that the Dirichlet character χ is assumed to be trivial, that is χ = 1. According to the dimension formulae in T. Miyake’s book [20, Thrm 2.5.2, p. 60] or [28, Prop. 6.1, p. 91], • and in addition due to the special form of αβ, we deduce that dim(E4 (Γ0 (αβ))) = (3.1) ∑ ϕ(gcd(δ, δ|αβ αβ )) = ∑ 1 = σ0 (αβ) = d(αβ), δ δ|αβ where ϕ is the Euler’s totient function. • we may assume that dim(S4 (Γ0 (αβ))) = mS ∈ N. To determine as many elements of S4 (Γ0 (αβ)) for an explicitly given αβ as possible, we use an exhaustive search when we apply Theorem 2.2. Theorem 3.1. (a) The set BE = {M(qt ) | t ∈ D(αβ) } is a basis of E4 (Γ0 (αβ)). (b) Let 1 ≤ i ≤ mS be positive integers, δ ∈ D(αβ) and (r(i, δ))i,δ be a table of the powers of η(δz). Let furthermore Bαβ,i (q) = ∏ ηr(i,δ) (δz) be selected elements of δ|αβ S4 (Γ0 (αβ)). Then the set BS = { Bαβ,i (q) | 1 ≤ i ≤ mS } is a basis of S4 (Γ0 (αβ)). (c) The set BM = BE ∪ BS constitutes a basis of M4 (Γ0 (αβ)). Remark 3.2. (r1) For each 1 ≤ i ≤ mS the eta quotient Bαβ,i (q) can be expressed in ∞ the form ∑ bαβ,i (n)qn , where for each n ≥ 1 the bαβ,i (n) are integers. n=1 (r2) If we divide the summation that results from Theorem 2.2 (v0 ) when we have set d = N by 24, then we obtain the smallest positive integer degree of qn in Bαβ,i (q). Proof. (a) By Theorem 5.8 in Section 5.3 of W. A. Stein [28, p. 86] each M(qt ) is in M4 (Γ0 (t)), where t ∈ D(αβ). Since E4 (Γ0 (αβ)) has a finite dimension, it suffices to show that M(qt ) with t ∈ D(αβ) are linearly independent. Suppose that xt ∈ C with t ∈ D(αβ). Then n t x M(q ) = x + 240 x σ ( ) ∑t ∑t ∑ ∑ t 3 t qn = 0. n≥1 t|αβ t|αβ t|αβ We set the coefficients of qn on both side for n ∈ D(αβ) equal to obtain the following homogeneous system of d(αβ) equations in d(αβ) unknowns: t ∑ σ3 ( u )xu = 0, t ∈ D(αβ). u|αβ Since σ3 ( ut ) = 0 whenever u - t, the matrix of this homogeneous system of equations is triangular with positive integer values on the diagonal. Hence, the solution of the homogeneous system of d(αβ) equations is xt = 0 for all t ∈ D(αβ). Therefore, the set BE is linearly independent and hence is a basis of E4 (Γ0 (αβ)). (b) Since Bαβ,i (q) with 1 ≤ i ≤ mS are obtained from an exhaustive search using Theorem 2.2 (i) − (v0 ), it holds that each Bαβ,i (q) is in the space S4 (Γ0 (αβ)). Since the dimension of S4 (Γ0 (αβ)) is mS ∈ N, it is sufficient to show that the set {Bαβ,i (q) | 1 ≤ i ≤ mS } is linearly independent. Suppose that xi ∈ C and ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 7 mS ∑ xi Bαβ,i (q) = 0. Then i=1 mS mS ∞ ∑ xi Bαβ,i (q) = i=1 ∑ ( ∑ xi bαβ,i (n) )qn = 0 n=1 i=1 which gives the following homogeneous system of mS equations in mS unknowns mS 1 ≤ n ≤ mS . ∑ bαβ,i (n) xi = 0, (3.2) i=1 By Remark 3.2 (r2) we may consider without lost of generality two cases. Case 1: For each 1 ≤ i ≤ mS the smallest degree of qn in Bαβ,i (q) is i. It is then obvious that the mS × mS matrix which corresponds to this homogeneous system of equations is triangular with 1’s on the diagonal. Hence, the determinant of that matrix is 1 and so xi = 0 for all 1 ≤ i ≤ mS . Case 2: The set BS does contain a subset, say B0S = { Bαβ,i (q) | 1 ≤ i ≤ u } for some 1 ≤ u < ms , for which the smallest degree of qn in Bαβ,i (q) is i. Since the smallest degree of qn is 1, B0S is not empty. Let us consider BS as an ordered set of the form B0S ∪ B00S , where B00S = { Bαβ,i (q) | u < i ≤ mS }. Let A = (bαβ,i (n)) be the mS ×mS matrix in Equation 3.2. In this matrix i indicates the i-th column and n indicates the n-th row. Note that applying case 1 the subset B0S is liniearly independent since the determinant of the corresponding u × u matrix is 1. If det(A) 6= 0, then xi = 0 for all 1 ≤ i ≤ mS . Suppose now that det(A) = 0. Then for some u < k ≤ mS there exists Bk (q) which is causing the system of equations to be inconsistent. We substitute Bk (q) with, say B0k (q), which does not occur in BS and compute the determinant of the new matrix A. Since there are finitely many Bk (q) with u < k ≤ mS that may cause the system of equations to be inconsistent and finitely many elements of S4 (Γ0 (αβ)) \ BS , the procedure will terminate with a consistent system of equations. So, xi = 0 for all 1 ≤ i ≤ mS . Therefore, the set {Bαβ,i (q) | 1 ≤ i ≤ mS } is linearly independent and so is a basis of S4 (Γ0 (αβ)). (c) Since M4 (Γ0 (αβ)) = E4 (Γ0 (αβ)) ⊕ S4 (Γ0 (αβ)), the result follows from (a) and (b). 3.2. Evaluating the convolution sum W(α,β) (n). Lemma 3.3. Let α, β ∈ N be such that gcd (α, β) = 1. Let furthermore BM = BE ∪ BS be a basis of M(Γ0 (αβ)). Then there exist Xδ ∈ C and Y j ∈ C with δ ∈ D(αβ) and 1 ≤ j ≤ mS such that (3.3) α β 2 (α L(q ) − β L(q )) = ∞ ∑ Xδ + ∑ δ|αβ n=1 n 240 ∑ σ3 ( ) Xδ + δ δ|αβ mS ∑ j=1 b j (n)Y j qn . 8 EBÉNÉZER NTIENJEM Proof. That (αL(qα ) − βL(qβ ))2 ∈ M4 (Γ0 (αβ)) follows from Lemma 2.3. Hence, by Theorem 3.1 (c), there exist Xδ ,Y j ∈ C, 1 ≤ j ≤ mS and δ ∈ D(αβ), such that (αL(qα ) − βL(qβ ))2 = mS ∑ Xδ M(qδ ) + ∑ Y j B j (q) j=1 δ|αβ ∞ = (3.4) ∑ Xδ + ∑ n=1 δ|αβ mS n 240 ∑ σ3 ( ) Xδ + ∑ b j (n)Y j qn . δ j=1 δ|αβ We compare the coefficients of qn on the right hand side of Equation 3.4 with that on the right hand side of Equation 2.5, which yields mS ∞ ∞ n n n n b (n)Y q = ) X + 240 σ ( j j 3 ∑ 240 α2 σ3 ( α ) + 240 β2 σ3 ( β ) ∑ ∑ δ δ ∑ j=1 n=1 n=1 δ|αβ n n + 48 α (β − 6 n) σ( ) + 48 β (α − 6 n) σ( ) − 1152 α βW(α,β) (n) qn . α β We then take the coefficients of qn for which n is in D(αβ) and 1 ≤ n ≤ mS , but as many as the unknowns Xδ and Y j . This results in a system of d(αβ) + mS linear equations whose unique solution determines the values of the unknown Xδ for all δ ∈ D(αβ) and the values of the unknown Y j for all 1 ≤ j ≤ mS . Hence, we obtain the stated result. In the following theorem, let Xδ and Y j stand for their values obtained in the previous theorem. Theorem 3.4. Let n be a positive integer. Then W(α,β) (n) = − + 5 24 α β ∑ δ|αβ δ6=α,β n 5 n Xδ σ3 ( ) + (α2 − Xα ) σ3 ( ) δ 24 α β α mS 5 n 1 (β2 − Xβ ) σ3 ( ) − ∑ Y j b j (n) 24 α β β j=1 1152 α β +( 1 1 n 1 1 n − n)σ( ) + ( − n)σ( ). 24 4β α 24 4α β Proof. We set the right hand side of Equation 3.3 with that of Equation 2.5 equal, which yields 1152 α βW(α,β) (n) = −240 n mS n ∑ σ3 ( δ ) Xδ − ∑ b j (n)Y j + 240 α2 σ3 ( α ) δ|αβ j=1 n n n + 240 β2 σ3 ( ) + 48 α (β − 6 n) σ( ) + 48 β (α − 6 n) σ( ). β α β Then we solve for W(α,β) (n) to obtain the stated result. Remark 3.5. Observe that the following part of Theorem 3.4 ( 1 1 n 1 1 n − n)σ( ) + ( − n)σ( ) 24 4β α 24 4α β depends only on n, α and β and not on the basis of the modular space M4 (Γ0 (αβ)). ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 9 4. N UMBER OF R EPRESENTATIONS OF A POSITIVE I NTEGER n BY THE O CTONARY Q UADRATIC F ORM E QUATION 1.4 In this section, we restrict the general form of αβ to 2ν 0 where ν ∈ {2, 3} and 0 is odd squarefree finite product of distinct odd primes. 4.1. Determination of (a, b) ∈ N2 . We carry out a method to determine all pairs (a, b) ∈ N2 necessary for the determination of N(a,b) (n) for a given αβ ∈ N which belongs to the above class. ν−2 0, P = {p = 2ν−2 } ∪ S {p | p is a prime divisor of 0 } and Let Λ = αβ 4 0 j j 4 = 2 j>1 P(P4 ) be the power set of P4 . Then for each Q ∈ P(P4 ) we define µ(Q) = ∏ p. We p∈Q set µ(Q) = 1 if Q is an empty set. Let now Ω4 = {(µ(Q1 ), µ(Q2 )) | there exist Q1 , Q2 ∈ P(P4 ) such that gcd (µ(Q1 ), µ(Q2 )) = 1 and µ(Q1 ) µ(Q2 ) = Λ }. Observe that Ω4 6= 0/ since (1, Λ) ∈ Ω4 . To illustrate our method, suppose that αβ = 23 · 3 · 5. Then Λ = 2 · 3 · 5, P4 = {2, 3, 5} and Ω4 = {(1, 30), (2, 15), (3, 10), (5, 6)}. Proposition 4.1. Suppose that αβ has the above restricted form and suppose that Ω4 is defined as above. Then for all n ∈ N the set Ω4 contains all pairs (a, b) ∈ N2 such that N(a,b) (n) can be obtained by applying W(α,β) (n) and some other evaluated convolution sums. Proof. We prove this by induction on the structure of α β. Suppose that αβ = 2ν p2 , where ν ∈ {2, 3} and p2 is an odd prime. Then by the above definitions we have Λ = 2ν−2 p2 , P4 = { 2ν−2 , p2 }, / {2ν−2 }, {p2 }, { 2ν−2 , p2 } }, P(P4 ) = { 0, and Ω4 = { (1, 2ν−2 p2 ), (2ν−2 , p2 ) }. We show that Ω4 is the largest such set. Assume now that there exist another set, say Ω04 , which results from the above definitions. Then there are two cases. Case Ω04 ⊆ Ω4 : There is nothing to show. So, we are done. Case Ω4 ⊂ Ω04 : Let (e, f ) ∈ Ω04 \ Ω4 . Since e f = 2ν−2 p2 and gcd (e, f ) = 1, we must have either (e, f ) = (1, 2ν−2 p2 ) or (e, f ) = (2ν−2 , p2 ). So, (e, f ) ∈ Ω4 . Hence, Ω4 = Ω04 . Suppose now that αβ = 2ν p2 p3 , where ν ∈ {2, 3} and p2 , p3 are distinct odd primes. Then by the induction hypothesis and by the above definitions we have essentially Ω4 = { (1, 2ν−2 p2 p3 ), (2ν−2 , p2 p3 ), (2ν−2 p2 , p3 ), (2ν−2 p3 , p2 ) }. Again, we show that Ω4 is the largest such set. Suppose that there exist another set, say Ω04 , which results from the above definitions. Two cases arise. Case Ω04 ⊆ Ω4 : There is nothing to prove. So, we are done. Case Ω4 ⊂ Ω04 : Let (e, f ) ∈ Ω04 \ Ω4 . Since e f = 2ν−2 p2 p3 and gcd (e, f ) = 1, we must have (e, f ) = (1, 2ν−2 p2 p3 ) or (e, f ) = (2ν−2 , p2 p3 ) or (e, f ) = (2ν−2 p2 , p3 ) or (e, f ) = (2ν−2 p3 , p2 ). So, (e, f ) ∈ Ω4 . Hence, Ω4 = Ω04 . 10 EBÉNÉZER NTIENJEM 4.2. Number of Representations by the Octonary Quadratic Form Equation 1.4. As an immediate application of Theorem 3.4 a formula for the number of representations of a positive integer n by the octonary quadratic form Equation 1.4 is determined for each (a, b) ∈ Ω4 . Let n ∈ N0 and let the number of representations of n by the quaternary quadratic form x12 + x22 + x32 + x42 be r4 (n) = card({(x1 , x2 , x3 , x4 ) ∈ Z4 | n = x12 + x22 + x32 + x42 }). It follows from the definition that r4 (0) = 1. For all n ∈ N, the following Jacobi’s identity is proved in K. S. Williams’ book [31, Thrm 9.5, p. 83] n r4 (n) = 8σ(n) − 32σ( ). 4 (4.1) Now, let the number of representations of n by the octonary quadratic form Equation 1.4 be N(a,b) (n) = card({(x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ) ∈ Z8 | n = a (x12 + x22 + x32 + x42 ) + b (x52 + x62 + x72 + x82 )}), where a, b ∈ N. Let 1 < λ ∈ N and τ : N 7→ N be an injective function such that τ(n) = λ · n for each n ∈ N. We then derive the following result: Theorem 4.2. Let n ∈ N and let (a, b) ∈ Ω4 . Then n n n n n N(a,b) (n) = 8σ( ) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(a,b) (n) + 1024W(a,b) ( ) a 4a b 4b 4 − 256 W(4a,b) (n) +W(a,4b) (n) . Proof. We have N(a,b) (n) = ∑ n n r4 (l)r4 (m) = r4 ( )r4 (0) + r4 (0)r4 ( ) + a b 2 (l,m)∈N0 a l+b m=n ∑ r4 (l)r4 (m). (l,m)∈N2 a l+b m=n We make use of Equation 4.1 to obtain n n n n N(a,b) (n) = 8σ( ) − 32σ( ) + 8σ( ) − 32σ( ) a 4a b 4b + ∑ m l (8σ(l) − 32σ( ))(8σ(m) − 32σ( )). 4 4 2 (l,m)∈N al+bm=n We know that l m l (8σ(l) − 32σ( ))(8σ(m) − 32σ( )) = 64σ(l)σ(m) − 256σ( )σ(m) 4 4 4 m l m − 256σ(l)σ( ) + 1024σ( )σ( ). 4 4 4 In the sequel of this proof, we assume that the evaluation of W(a,b) (n) = ∑ (l,m)∈N2 al+bm=n σ(l)σ(m), ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 11 W(4a,b) (n) and W(a,4b) (n) are known. We set λ = 4 in the sequel. When we use the function τ with l as argument we derive l W(4a,b) (n) = ∑ σ( )σ(m) = ∑ σ(l)σ(m). 4 2 2 (l,m)∈N al+bm=n (l,m)∈N 4a l+bm=n When we apply the function τ with m as argument we infer m W(a,4b) (n) = ∑ σ(l)σ( ) = ∑ σ(l)σ(m). 4 2 2 (l,m)∈N al+4b m=n (l,m)∈N al+bm=n We simultaneously apply the function τ with l and m as arguments, respectively, to conclude l n m ∑ σ( 4 )σ( 4 ) = ∑ σ(l)σ(m) = W(a,b) ( 4 ). 2 2 (l,m)∈N al+bm=n (l,m)∈N al+bm= n4 We finally put all these evaluations together to obtain the stated result for N(a,b) (n). 5. N UMBER OF R EPRESENTATIONS OF A P OSITIVE I NTEGER n BY THE O CTONARY Q UADRATIC F ORM E QUATION 1.5 We restrict the general form of αβ to 2ν 0, where 0 ≡ 0 (mod 3). 5.1. Determination of (c, d) ∈ N2 . The following method determine all pairs (c, d) ∈ N2 necessary for the determination of R(c,d) (n) for a given αβ ∈ N belonging to the above class. The following method is quasi similar to the one used in Subsection 4.1. S 2ν 0 ν {p j | p j is a prime divisor of 0 }. Let Let ∆ = αβ 3 = 3 . Let P3 = {p0 = 2 } ∪ j>2 P(P3 ) be the power set of P3 . Then for each Q ∈ P(P3 ) we define µ(Q) = ∏ p. We set p∈Q µ(Q) = 1 if Q is an empty set. Let now Ω3 be defined in a similar way as Ω4 in Subsection 4.1, however with ∆ instead of Λ, i.e., Ω3 = {(µ(Q1 ), µ(Q2 )) | there exist Q1 , Q2 ∈ P(P3 ) such that gcd (µ(Q1 ), µ(Q2 )) = 1 and µ(Q1 ) µ(Q2 ) = ∆ }. Note that Ω3 6= 0/ since (1, ∆) ∈ Ω3 . As an example, suppose again that αβ = 23 · 3 · 5. Then ∆ = 23 · 5, P3 = {23 , 5} and Ω3 = {(1, 40), (5, 8)}. Proposition 5.1. Suppose that αβ has the above restricted form and Suppose that Ω3 be defined as above. Then for all n ∈ N the set Ω3 contains all pairs (c, d) ∈ N2 such that R(c,d) (n) can be obtained by applying W(α,β) (n) and some other evaluated convolution sums. Proof. Simlar to the proof of Proposition 4.1. 5.2. Number of Representations by the Octonary Quadratic Form Equation 1.5. We apply Theorem 3.4 to determine a formula for the number of representations of a positive integer n by the octonary quadratic form Equation 1.5 for each (c, d) ∈ Ω3 . Let n ∈ N0 and let s4 (n) denote the number of representations of n by the quaternary quadratic form x12 + x1 x2 + x22 + x32 + x3 x4 + x42 , that is, s4 (n) = card({(x1 , x2 , x3 , x4 ) ∈ Z4 | n = x12 + x1 x2 + x22 + x32 + x3 x4 + x42 }). 12 EBÉNÉZER NTIENJEM It is obvious that s4 (0) = 1. J. G. Huard et al. [13], G. A. Lomadze [19] and K. S. Williams [31, Thrm 17.3, p. 225] have proved that for all n ∈ N n (5.1) s4 (n) = 12σ(n) − 36σ( ). 3 Now, let the number of representations of n by the octonary quadratic form Equation 1.5 be R(c,d) (n) = card({(x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ) ∈ Z8 | n = c (x12 + x1 x2 + x22 + x32 + x3 x4 + x42 ) + d (x52 + x5 x6 + x62 + x72 + x7 x8 + x82 )}). Let λ and τ be defined as in Section 4.2. We infer the following result: Theorem 5.2. Let n ∈ N and (c, d) ∈ Ω3 . Then n n n n n R(c,d) (n) =12σ( ) − 36σ( ) + 12σ( ) − 36σ( ) + 144W(c,d) (n) + 1296W(c,d) ( ) c 3c d 3d 3 − 432 W(3c,d) (n) +W(c,3d) (n) . Proof. It holds that R(c,d) (n) = ∑ n n s4 (l)s4 (m) = s4 ( )s4 (0) + s4 (0)s4 ( ) + c d 2 (l,m)∈N0 cl+dm=n ∑ s4 (l)s4 (m). (l,m)∈N2 cl+dm=n We apply Equation 5.1 to derive n n n n R(c,d) (n) = 12σ( ) − 36σ( ) + 12σ( ) − 36σ( ) c 3c d 3d + ∑ l m (12σ(l) − 36σ( ))(12σ(m) − 36σ( )). 3 3 2 (l,m)∈N cl+dm=n We know that m l l (12σ(l) − 36σ( ))(12σ(m) − 36σ( )) = 144σ(l)σ(m) − 432σ( )σ(m) 3 3 3 m l m − 432σ(l)σ( ) + 1296σ( )σ( ). 3 3 3 We assume that the evaluation of W(c,d) (n) = ∑ σ(l)σ(m), (l,m)∈N2 cl+dm=n W(c,3d) (n) and W(3c,d) (n) are known. We set λ = 3 in the sequel. We apply the function τ to m to derive m ∑ σ(l)σ( 3 ) = ∑ σ(l)σ(m) = W(c,3d) (n). 2 2 (l,m)∈N cl+dm=n (l,m)∈N cl+3d m=n We make use of the function τ with l as argument to conclude l ∑ σ(m)σ( 3 ) = ∑ σ(l)σ(m) = W(3c,d) (n). 2 2 (l,m)∈N cl+dm=n (l,m)∈N 3c l+dm=n ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS 13 We simultaneously apply apply the function τ to l and to m as arguments, respectively, to infer m n l ∑ σ( 3 )σ( 3 ) = ∑ σ(l)σ(m) = W(c,d) ( 3 ). 2 2 (l,m)∈N cl+dm=n (l,m)∈N cl+dm= n3 Finally, we bring all these evaluations together to obtain the stated result for R(c,d) (n). 6. E VALUATION OF THE CONVOLUTION SUMS WHEN αβ = 33, 40, 56 In this section, we give explicit formulae for the convolution sum W(α,β) (n) when αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · 7. We then make use of these convolution sums among others to determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms Equation 1.4 and Equation 1.5, respectively. We notice that (6.1) M4 (Γ0 (11)) ⊂ M4 (Γ0 (33)) (6.2) M4 (Γ0 (5)) ⊂ M4 (Γ0 (10)) ⊂ M4 (Γ0 (20)) ⊂ M4 (Γ0 (40)) (6.3) M4 (Γ0 (8)) ⊂ M4 (Γ0 (40)). (6.4) M4 (Γ0 (7)) ⊂ M4 (Γ0 (14)) ⊂ M4 (Γ0 (28)) ⊂ M4 (Γ0 (56)) (6.5) M4 (Γ0 (8)) ⊂ M4 (Γ0 (56)). This implies the same inclusion relation for the bases, the space of Eisenstein forms of weight 4 and the spaces of cusp forms of weight 4. 6.1. Bases of E4 (Γ0 (αβ)) and S4 (Γ0 (αβ)) for αβ = 33, 40, 56. We apply the dimension formulae in T. Miyake’s book [20, Thrm 2.5.2, p. 60] or [28, Prop. 6.1, p. 91] to deduce that dim(S4 (Γ0 (33)) = 10, dim(S4 (Γ0 (40)) = 14 and dim(S4 (Γ0 (56)) = 20. We use Equation 3.1 to infer that dim(E4 (Γ0 (33))) = 4 and dim(E4 (Γ0 (40))) = dim(E4 (Γ0 (56))) = 8. We use the observation made in the third paragraph of Subsection 3.1 to explicitly determine as many elements of BE,33 , BE,40 and BE,56 as possible. Then we apply Remark 3.2 (r2) when selecting basis elements of a given space of cusp forms as stated in the proof of Theorem 3.1 (b). Corollary 6.1. (a) The sets BE,33 = { M(qt ) | t|33 }, BE,40 = { M(qt ) | t|40 } and BE,56 = { M(qt ) | t|56 } are bases of E4 (Γ0 (33)), E4 (Γ0 (40)) and E4 (Γ0 (56)), respectively. (b) Let 1 ≤ i ≤ 10, 1 ≤ j ≤ 14, 1 ≤ k ≤ 20 be positive integers. Let δ1 ∈ D(33) and (r(i, δ1 ))i,δ1 be the Table 4 of the powers of η(δ1 z). Let δ2 ∈ D(40) and (r( j, δ2 )) j,δ2 be the Table 5 of the powers of η(δ2 z). Let δ3 ∈ D(56) and (r(k, δ3 ))k,δ3 be the Table 6 of the powers of η(δ3 z). Let furthermore B33,i (q) = ∏ ηr(i,δ1 ) (δ1 z), B40, j (q) = δ1 |33 B56,k (q) = ∏ ηr( j,δ2 ) (δ2 z), δ2 |40 r(k,δ3 ) ∏η (δ3 z) δ3 |56 be selected elements of S4 (Γ0 (33)), S4 (Γ0 (40)) and S4 (Γ0 (56)), respectively. 14 EBÉNÉZER NTIENJEM Then the sets BS,33 = { B33,i (q) | 1 ≤ i ≤ 10 }, BS,40 = { B40, j (q) | 1 ≤ j ≤ 14 }, BS,56 = { B56,k (q) | 1 ≤ k ≤ 20 } are bases of S4 (Γ0 (33)), S4 (Γ0 (40)) and S4 (Γ0 (56)), repectively. (c) The sets BM,33 = BE,33 ∪ BS,33 , BM,40 = BE,40 ∪ BS,40 and BM,56 = BE,56 ∪ BS,56 constitute bases of M4 (Γ0 (33)), M4 (Γ0 (40)) and M4 (Γ0 (56)), respectively. By Remark 3.2 (r1), B33,i (q), B40, j (q) and B56,k (q) can be expressed in the form ∞ ∞ ∞ n=1 n=1 ∑ b33,i (n)qn , ∑ b40, j (n)qn and ∑ b56,k (n)qn , respectively. n=1 We observe that • by Equation 6.1 B33,2 (q) is contained in S4 (Γ0 (11)) and is the only one. In addition, B33,6 (q) = B33,2 (q2 ). Hence, b33,6 (n) = b33,2 ( 2n ). We also note that B33,8 (q), B33,9 (q) and B33,10 (q) all belong to S2 (Γ0 (33)) since Theorem 2.2 does not seem to produce sufficient elements for the space S4 (Γ0 (33)). • the basis elements of S4 (Γ0 (40)) have been determined almost with respect to the inclusion relation Equation 6.2, except that B40,5 (q) results from the basis element of S4 (Γ0 (8)) according to Equation 6.3. • there is no element of S4 (Γ0 (7)) which occurs as a basis element of S4 (Γ0 (56)). This indicates that an element of S4 (Γ0 (7)) cannot be determined when using Theorem 2.2. Other than that, the inclusion relation Equation 6.4 and Equation 6.5 preserve the bases. Proof. It follows immediately from Theorem 3.1. In case (a): the result is obtained when we set n = 1, 3, 11, 33, n = 1, 2, 4, 5, 8, 10, 20, 40 and n = 1, 2, 4, 7, 8, 14, 28, 56, respectively. In case (b): the linear independence of the sets BS,33 and BS,56 is proved by applying case 2 in the proof of Theorem 3.1 and by taking n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and n = 1, 2, 3, . . . , 13, 14, respectively. Finally BS,40 is linearly independent by case 1 in the proof of Theorem 3.1 and by taking n = 1, 2, 3, . . . , 19, 20. Therefore, we obtain the stated result. 6.2. Evaluation of W(α,β) (n) when αβ = 33, 40, 56. Corollary 6.2. We have (6.6) 33 2 ∞ (L(q) − 33 L(q )) = 1024 + ∑ n=1 3696 n 676632 n 14904 σ3 (n) + σ3 ( ) − σ3 ( ) 61 61 3 61 11 2854104 n 12180 181500 984510 − σ3 ( ) + b33,1 (n) + b33,2 (n) + b33,3 (n) 61 33 61 61 61 2882334 4920480 5073354 2295810 + b33,4 (n) + b33,5 (n) + b33,6 (n) + b33,7 (n) 61 61 61 61 + 3960 b33,8 (n) + 3762 b33,9 (n) + 1188 b33,10 (n) qn , ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS ∞ (6.7) (3 L(q3 ) − 11 L(q11 ))2 = 100 + ∑ n=1 n n 100080 σ3 ( ) + 7392 σ3 ( ) 61 3 11 49760832 n 1260 3780 7560 + σ3 ( ) − b33,3 (n) − b33,4 (n) − b33,5 (n) 61 33 61 61 61 3780 15444 b33,6 (n) − b33,7 (n) + 396 b33,9 (n) qn , − 61 61 (6.8) 40 ∞ 2 (L(q) − 40 L(q )) = 1521 + ∑ n=1 26800 43520 n σ3 (n) + σ3 ( ) 117 117 2 245120 n 26800 n 1766400 n 127760 n + σ3 ( ) − σ3 ( ) − σ3 ( ) − σ3 ( ) 39 4 117 5 13 8 117 10 n 6558720 n 192224 439744 357440 σ3 ( ) + σ3 ( ) + b40,1 (n) + b40,2 (n) − 39 20 13 40 117 117 304832 1061120 41840 + b40,3 (n) + b40,4 (n) + b40,5 (n) − 15360 b40,6 (n) 39 39 3 24320 1688320 128000 − b40,7 (n) + b40,8 (n) + 116800 b40,9 (n) − b40,10 (n) 3 39 3 485120 1130240 121280 − b40,11 (n) − b40,12 (n) − b40,13 (n) − 69120 b40,14 (n) qn , 3 3 3 5 8 2 ∞ (6.9) (5 L(q ) − 8 L(q )) = 9 + ∑ n=1 76000 n 5920 σ3 (n) − σ3 ( ) 117 117 2 16960 n 668000 n 721920 n 8240 n − σ3 ( ) + σ3 ( ) + σ3 ( ) − σ3 ( ) 39 4 117 5 13 8 117 10 n n 22720 95360 721920 5920 σ3 ( ) − σ3 ( ) − b40,1 (n) + b40,2 (n) − 39 20 13 40 117 117 59200 12800 38800 − b40,3 (n) + b40,4 (n) − b40,5 (n) + 7680 b40,6 (n) 39 39 3 47360 505088 12800 − b40,7 (n) − b40,8 (n) − 67520 b40,9 (n) − b40,10 (n) 3 39 3 113920 298240 63040 + b40,11 (n) + b40,12 (n) + b40,13 (n) + 69120 b40,14 (n) qn , 3 3 3 15 16 EBÉNÉZER NTIENJEM ∞ (6.10) (L(q) − 56 L(q56 ))2 = 3025 + ∑ n=1 n 31584 n 1284 σ3 (n) − 420 σ3 ( ) + σ3 ( ) 5 2 5 4 1764 n 32256 n n 51744 n 3687936 n − σ3 ( ) − σ3 ( ) − 588 σ3 ( ) − σ3 ( ) + σ3 ( ) 5 7 5 8 14 5 28 5 56 92604 1140216 11916 b56,1 (n) + b56,2 (n) + 29568 b56,3 (n) + b56,4 (n) + 5 5 5 2557632 − 411936 b56,5 (n) + b56,6 (n) + 223608 b56,7 (n) + 3998400 b56,8 (n) 5 145152 + 4042752 b56,9 (n) + b56,10 (n) − 8064 b56,11 (n) − 48384 b56,12 (n) 5 532224 225792 + b56,14 (n) + 161280 b56,15 (n) − b56,16 (n) + 129024 b56,17 (n) 5 5 2515968 b56,18 (n) + 1354752 b56,19 (n) − 225792 b56,20 (n) qn , + 5 7 8 2 ∞ (6.11) (7 L(q ) − 8 L(q )) = 1 + ∑ − n=1 308 1876 n 40096 n σ3 (n) + σ3 ( ) − σ3 ( ) 25 25 2 25 4 285908 n 409088 n 27076 n 60704 n + σ3 ( ) σ3 ( ) − σ3 ( ) − σ3 ( ) 25 7 25 8 25 14 25 28 562688 n 308 2436 11648 − σ3 ( ) + b56,1 (n) + b56,2 (n) + b56,3 (n) 25 56 25 25 25 121352 101472 288064 87864 b56,4 (n) − b56,5 (n) + b56,6 (n) − b56,7 (n) + 25 25 25 25 2190912 1821312 201984 + b56,8 (n) + b56,9 (n) + b56,10 (n) + 29568 b56,11 (n) 25 25 25 284928 1512192 b56,12 (n) − 59136 b56,13 (n) − b56,14 (n) + 59136 b56,15 (n) 5 25 1616896 6724096 b56,16 (n) + 118272 b56,17 (n) + b56,18 (n) + 25 25 + 430080 b56,19 (n) + 53760 b56,20 (n) qn . Proof. These identities follow immediately on taking (α, β) = (1, 33), (3, 11), (1, 40), (5, 8), (1, 56), (7, 8) in Lemma 3.3. In case αβ = 40 we take all n in {1, 2, . . . , 20, 40, 80} to obtain a system of 22 linear equations with unknowns Xδ and Y j , where δ ∈ D(40) and 1 ≤ j ≤ 14. We are now prepared to state and to prove our main result of this section. Corollary 6.3. Let n be a positive integer. Then 1 7 n 2563 n 71201 n W(1,33) (n) = − σ3 (n) − σ3 ( ) + σ3 ( ) + σ3 ( ) 8784 4392 3 8784 11 8784 33 1 1 1 1 n 1015 1375 +( − n)σ(n) + ( − n)σ( ) − b33,1 (n) − b33,2 (n) 24 132 24 4 33 193248 17568 54695 480389 5695 8541 − b33,3 (n) − b33,4 (n) − b33,5 (n) − b33,6 (n) 128832 386496 2684 3904 3865 5 19 1 (6.12) − b33,7 (n) − b33,8 (n) − b33,9 (n) − b33,10 (n), 3904 48 192 32 ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS W(3,11) (n) = (6.13) W(1,40) (n) = (6.14) 5 n 41 n 23561 n 1 1 1 σ3 ( ) + σ3 ( ) − σ3 ( ) + ( − n)σ( n) 366 3 72 11 1098 33 24 44 3 1 n 35 35 35 1 b33,3 (n) + b33,4 (n) + b33,5 (n) + ( − n)σ( ) + 24 12 11 64416 21472 10736 13 35 1 + b33,6 (n) + b33,7 (n) − b33,9 (n), 1952 21472 96 17 n 383 n 335 n 115 n 1 σ3 (n) − σ3 ( ) − σ3 ( ) + σ3 ( ) + σ3 ( ) 4212 2106 2 2808 4 67392 5 39 8 1597 n 1117 n 34 n 1 1 + σ3 ( ) + σ3 ( ) − σ3 ( ) + ( − n)σ(n) 67392 10 5616 20 13 40 24 160 1 1 n 6007 6871 4763 + ( − n)σ( ) − b40,1 (n) − b40,2 (n) − b40,3 (n) 24 4 40 168480 84240 28080 523 1 19 829 b40,4 (n) − b40,5 (n) + b40,6 (n) + b40,7 (n) − 1404 1728 3 108 1319 365 25 379 − b40,8 (n) − b40,9 (n) + b40,10 (n) + b40,11 (n) 1404 144 27 108 379 3 883 b40,12 (n) + b40,13 (n) + b40,14 (n), + 108 432 2 475 n 53 n 425 n 34 n 37 σ3 (n) + σ3 ( ) + σ3 ( ) + σ3 ( ) − σ3 ( ) 33696 33696 2 5616 4 67392 5 39 8 103 n 149 n 47 n 1 1 n + σ3 ( ) + σ3 ( ) + σ3 ( ) + ( − n)σ( ) 67392 10 2808 20 39 40 24 32 5 1 1 n 37 71 185 + ( − n)σ( ) + b40,1 (n) − b40,2 (n) + b40,3 (n) 24 20 8 33696 16848 5616 485 1 37 1973 5 b40,4 (n) + b40,5 (n) − b40,6 (n) + b40,7 (n) + b40,8 (n) − 702 1728 6 108 7020 211 5 89 233 + b40,9 (n) + b40,10 (n) − b40,11 (n) − b40,12 (n) 144 54 108 108 197 3 − b40,13 (n) − b40,14 (n), 432 2 W(5,8) (n) = − (6.15) 1 5 n 47 n 7 n 1 n σ3 (n) + σ3 ( ) − σ3 ( ) + σ3 ( ) + σ3 ( ) 3840 768 2 480 4 1280 7 10 8 7 n 77 n 7 n 1 1 + σ3 ( ) + σ3 ( ) + σ3 ( ) + ( − n)σ(n) 768 14 480 28 30 56 24 224 1 1 n 331 7717 11 + ( − n)σ( ) − b56,1 (n) − b56,2 (n) − b56,3 (n) 24 4 56 8960 26880 24 613 1903 1331 6787 − b56,4 (n) + b56,5 (n) − b56,6 (n) − b56,7 (n) 1920 96 240 384 2975 188 9 1 3 − b56,8 (n) − b56,9 (n) − b56,10 (n) + b56,11 (n) + b56,12 (n) 48 3 20 8 4 33 5 7 39 − b56,14 (n) − b56,15 (n) b56,16 (n) − 2 b56,17 (n) − b56,18 (n) 20 2 10 5 7 − 21 b56,19 (n) + b56,20 (n), 2 W(1,56) (n) = − (6.16) 17 18 EBÉNÉZER NTIENJEM W(7,8) (n) = (6.17) 11 67 n 179 n 289 n 7 n σ3 (n) − σ3 ( ) + σ3 ( ) + σ3 ( ) − σ3 ( ) 57600 57600 2 7200 4 57600 7 450 8 n 271 n 157 n 1 1 n 967 σ3 ( ) + σ3 ( ) + σ3 ( ) + ( − n)σ( ) + 57600 14 7200 28 450 56 24 32 7 1 1 n 11 29 13 + ( − n)σ( ) − b56,1 (n) − b56,2 (n) − b56,3 (n) 24 28 8 57600 19200 1800 151 643 523 2167 b56,4 (n) + b56,5 (n) − b56,6 (n) + b56,7 (n) − 28800 2400 3600 9600 11411 1581 263 11 − b56,8 (n) − b56,9 (n) − b56,10 (n) − b56,11 (n) 8400 1400 2100 24 53 11 1969 11 − b56,12 (n) + b56,13 (n) + b56,14 (n) − b56,15 (n) 60 12 2100 12 11 1579 20 13133 b56,16 (n) − b56,17 (n) − b56,18 (n) − b56,19 (n) − 3150 6 1575 3 5 − b56,20 (n). 6 Proof. These identities follow from Theorem 3.4 when we set (α, β) = (1, 33), (3, 11), (1, 40), (5, 8), (1, 56), (7, 8). 6.3. Formlae for the Number of Representations of a positive Integer n. We determine formulae for the number of representations of a positive integer n by the Octonary Quadratic Form Equation 1.5 when we mainly apply the evaluation of the convolution sums W(1,33) (n) and W(3,11) (n). In order to do that, we recall that 33 = 3 · 11 is of the restricted form in Section 5. Hence, from Proposition 5.1 we derive that Ω3 = {(1, 11)}. We then deduce the following result: Corollary 6.4. Let n ∈ N. Then n n n R(1,11) (n) =12σ(n) − 36σ( ) + 12σ( ) − 36σ( ) + 144W(1,11) (n) 3 11 33 n + 1296W(1,11) ( ) − 432 W(3,11) (n) +W(1,33) (n) . 3 Proof. It follows immediately from Theorem 4.2 with (c, d) = (1, 11). One can then make use of the result of E. Royer [27, Thrm 1.3], Equation 6.12 and Equation 6.13 to simplify the formula. Next, we give formulae for the number of representations of a positive integer n by the Octonary Quadratic Form Equation 1.4 by mainly applying the evaluation of the convolution sums W(1,40) (n), W(1,56) (n), W(5,8) (n) and W(7,8) (n). To achieve that, we recall that 40 = 23 ·5 and 56 = 23 ·7 are of the restricted form in Section 4. Therefore, we apply Proposition 4.1 to conclude that Ω4 = {(1, 10), (2, 5)} in case αβ = 40 and Ω4 = {(1, 14), (2, 7)} in case αβ = 56. Instead of using Royer’s formula [27, Thrm 1.1] for W(1,10) (n), we rather apply the following equivalent one which is an immediate corollary of Theorem 3.4 since αβ = 10 = 2 · 5 and because of Equation 6.2. Since B40,2 (q) = B40,1 (q2 ), we conclude that b40,2 (n) = b40,1 ( 2n ). ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS W(1,10) (n) = ∑ σ(l)σ(m) = (l,m)∈N2 l+10 m=n 19 1 1 n 25 n 25 n σ3 (n) + σ3 ( ) + σ3 ( ) + σ3 ( ) 312 78 2 312 5 78 10 1 1 1 1 n 1 − n)σ(n) + ( − n)σ( ) + b40,1 (n) 24 40 24 4 10 1040 1 n 1 − b40,1 ( ) − b40,3 (n). (6.18) 78 2 48 In the following corollary, the number of representations of a positive integer n, N(a,b) (n) for (a, b) = (1, 1), (1, 3), (2, 3), (1, 9), are applications of the evaluation of the convolution sums W(1,4) (n) by J. G. Huard et al. [13], W(1,12) (n), W(3,4) (n), W(1,24) (n) and W(3,8) (n) by A. Alaca et al. [2, 4], and W(1,36) (n) and W(4,9) (n) by D. Ye [33]. These numbers of representations of a positive integer n are discovered due to Proposition 4.1. +( Corollary 6.5. Let n ∈ N. Then n n N(1,1) (n) = 16 σ(n) − 64 σ( ) + 64W(1,1) (n) + 1024W(1,1) ( ) − 512W(1,4) (n) 4 4 n n = 16 σ3 (n) − 32 σ3 ( ) + 256 σ3 ( ), 2 4 n n n n N(1,3) (n) = 8σ(n) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(1,3) (n) + 1024W(1,3) ( ) 4 3 12 4 − 256 W(3,4) (n) +W(1,12) (n) , n n n n n N(2,3) (n) = 8 σ( ) − 32 σ( ) + 8 σ( ) − 32 σ( ) + 64W(1,3) (n) + 1024W(1,3) ( ) 2 8 3 12 4 − 256 W(3,8) (n) +W(1,12) (n) , n n n n N(1,9) (n) = 8 σ(n) − 32 σ( ) + 8 σ( ) − 32 σ( ) + 64W(1,9) (n) + 1024W(1,9) ( ) 4 9 36 4 − 256 W(4,9) (n) +W(1,36) (n) , n n n n N(1,10) (n) = 8σ(n) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(1,10) (n) + 1024W(1,10) ( ) 4 10 40 4 n − 256 W(2,5) ( ) +W(1,40) (n) , 2 n n n n n N(2,5) (n) = 8σ( ) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(2,5) (n) + 1024W(2,5) ( ) 2 8 5 20 4 n − 256 W(5,8) (n) +W(1,10) ( ) , 2 n n n n N(1,14) (n) = 8σ(n) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(1,14) (n) + 1024W(1,14) ( ) 4 14 56 4 n − 256 W(2,7) ( ) +W(1,56) (n) , 2 20 EBÉNÉZER NTIENJEM n n n n n N(2,7) (n) = 8σ( ) − 32σ( ) + 8σ( ) − 32σ( ) + 64W(2,7) (n) + 1024W(2,7) ( ) 2 8 7 28 4 n − 256 W(7,8) (n) +W(1,14) ( ) . 2 Proof. These formulae follow immediately from Theorem 4.2 when we set (a, b) = (1, 1), (1, 3), (2, 3), (1, 9), (1, 10), (2, 5), (1, 14), (2, 7), respectively. One can then use the result of • S. Cooper and D. Ye [11, Thrm 2.1], Equation 6.18, Equation 6.14 and Equation 6.15 for the sake of simplification in case of N10 and N(2,5) . • E. Royer [27, Thrms 1.7], A. Alaca et al. [1, Thrm 3.3], Equation 6.16 and Equation 6.17 to simplify the formulae in case of N(1,14) and N(2,7) . 7. C ONCLUDING R EMARK To evaluate the convolution sum for αβ that belongs to this class of positive integers, it now suffices to determine a basis of the space of cusp forms of weight 4 for Γ0 (αβ). It is straightforward from Theorem 3.1 (a), that a basis of the space of Eisenstein forms of weight 4 for Γ0 (αβ) is already given. The determination of a basis of the space of cusp forms when αβ is large is tedious. A future work is to carry out an effective and efficient method to build a basis of a space of cusp forms of weight 4 for Γ0 (αβ) when αβ is large. A natural number can be expressed as a product of distinct primes to the power of some positive integers. The form for αβ that we have considered falls under such an expression; however that form does not cover all natural numbers. The consideration of the class of natural numbers which is not discussed in this paper is a work in progress. C OMPETING INTERESTS The authors declare that they have no competing interests. ACKNOWLEDGMENTS I am indebtedly thankful to Prof. Emeritus Kenneth S. Williams for fruitful comments and suggestions on a draft of this paper. R EFERENCES [1] A. Alaca, Ş. Alaca, and E. Ntienjem. Evaluation of the Convolution Sum involving the Sum of Divisors Function for 14, 22 and 26. ArXiv e-prints, apr 2016. [2] A. Alaca, Ş. Alaca, and K. S. Williams. Evaluation of the convolution sums ∑ σ(l)σ(m) and l+12m=n ∑ σ(l)σ(m). 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Cambridge University Press, Cambridge, 2011. [32] E. X. W. Xia, X. L. Tian, and O. X. M. Yao. Evaluation of the convolution sum ∑ σ(l)σ(m). Int J l+25m=n Number Theory, 10(6):1421–1430, 2014. [33] D. Ye. Evaluation of the convolution sums ∑ σ(l)σ(m) and l+36m=n 11(1):171–183, 2015. TABLES ∑ 4l+9m=n σ(l)σ(m). Int J Number Theory, 22 EBÉNÉZER NTIENJEM (α, β) (1,1) (1,2),(1,3),(1,4) (1,5),(1,7) (1,6),(2,3) (1,8), (1,9) (1,10),(1,11),(1,13), (1,14) (1,12),(1,16),(1,18), (1,24),(2,9),(3,4), (3,8) (1,15),(3,5) (1,20),(2,5),(4,5) (1,23) (1,25) Authors M. Besge, J. W. L. Glaisher, S. Ramanujan J. G. Huard & Z. M. Ou & B. K. Spearman & K. S. Williams M. Lemire & K. S. Williams, S. Cooper & P. C. Toh Ş. Alaca & K. S. Williams K. S. Williams References E. Royer [27] A. Alaca & Ş. Alaca & K. S. Williams B. Ramakrishnan & B. Sahu S. Cooper & D. Ye H. H. Chan & S. Cooper E. X. W. Xia & X. L. Tian & O. X. M. Yao Ş. Alaca & Y. Kesicioǧlu D. Ye [2, 3, 4, 5] (1,27),(1,32) (1,36),(4,9) (1,22),(1,26),(2,7), (2,11),(2,13) A. Alaca & Ş. Alaca & E. Ntienjem (1,44),(1,52), (4,11),(4,13) E. Ntienjem Table 1: Known convolution sums W(α,β) (n) [8, 12, 26] [13] [17, 10] [7] [30, 29] [25] [11] [9] [32] [6] [33] [1] [23] (a, b) Authors References (1,2) K. S. Williams [30] (1,4) A. Alaca & Ş. Alaca & K. S. Williams [3] (1,5) S. Cooper & D. Ye [11] (1,6) B. Ramakrishnan & B. Sahu [25] (1,8) Ş. Alaca & Y. Kesicioǧlu [6] (1,11),(1,13) E. Ntienjem [23] Table 2: Known representations of n by the form Equation 1.4 (c, d) (1,1) (1,2) (1,3) (1,4),(1,6), (1,8),(2,3) (1,5) (1,9) Authors G. A. Lomadze Ş. Alaca & K. S. Williams K. S. Williams References [19] [7] [29] A. Alaca & Ş. Alaca & K. S. Williams B. Ramakrishnan & B. Sahu Ş. Alaca & Y. Kesicioǧlu [2, 3, 4] [25] [6] ELEMENTARY EVALUATION OF CONVOLUTION SUMS FOR A CLASS OF POSITIVE INTEGERS (1,12),(3,4) D. Ye [33] Table 3: Known representations of n by the form Equation 1.5 1 3 11 33 1 5 -1 5 -1 2 4 0 4 0 3 3 1 3 1 4 2 2 2 2 5 1 3 1 3 6 0 4 0 4 7 -1 5 -1 5 8 1 1 1 1 9 0 2 0 2 10 2 0 2 0 Table 4: Power of η-quotients being basis elements of S4 (Γ0 (33)) 1 2 4 5 8 10 20 40 1 4 0 0 4 0 0 0 0 2 0 4 0 0 0 4 0 0 3 2 0 0 -2 0 8 0 0 4 0 0 4 0 0 0 4 0 5 0 0 0 0 0 4 4 0 6 0 2 0 0 0 -2 8 0 7 2 -2 0 -2 0 2 8 0 8 0 0 0 0 4 0 0 4 9 0 0 0 0 2 4 -4 6 10 2 -2 2 2 -2 0 0 6 11 1 0 0 -1 1 2 -2 7 12 0 0 2 0 0 0 -2 8 13 0 4 0 0 -2 0 -4 10 14 0 2 -2 0 0 -2 2 8 Table 5: Power of η-quotients being basis elements of S4 (Γ0 (40)) 1 2 3 4 5 6 7 8 1 5 2 6 0 0 0 0 1 2 4 -1 0 2 0 -2 0 2 2 0 2 6 -2 4 -2 1 0 7 5 2 -2 0 0 0 0 1 8 0 0 0 0 0 0 0 0 14 -1 2 6 2 4 -2 0 -3 28 0 0 0 2 2 6 6 8 56 0 0 0 0 0 0 0 0 23 24 EBÉNÉZER NTIENJEM 9 0 1 1 0 0 -3 9 0 10 0 0 0 0 2 0 4 2 11 0 -2 8 0 -2 2 -4 6 12 0 0 6 0 -2 0 -2 6 13 0 0 3 0 -1 4 -5 7 14 0 0 4 0 -2 0 0 6 15 0 2 2 0 -2 -2 2 6 16 0 1 1 0 0 1 -3 8 17 0 3 -1 0 0 -1 -1 8 18 0 0 1 0 1 0 -3 9 19 0 1 0 0 -1 -3 4 7 20 -2 5 -3 2 0 -5 7 4 Table 6: Power of η-quotients being basis elements of S4 (Γ0 (56)) C ENTRE FOR R ESEARCH IN A LGEBRA AND N UMBER T HEORY, S CHOOL OF M ATHEMATICS AND S TA C ARLETON U NIVERSITY, OTTAWA , O NTARIO , C ANADA , K1S 5B6 E-mail address: [email protected];[email protected] TISTICS ,