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

  • Ebénézer Ntienjem EMAIL logo
Published/Copyright: April 21, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

The convolution sum, ∑(l,m)∈N02αl+βm=nσ(l)σ(m), 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), where (a, b) = (1, 11), (1, 13).

Keywords: Sums of Divisors function; Convolution Sums; Dedekind eta function; Modular Forms; Eisenstein Series; Cusp Forms; Octonary quadratic Forms; Number of Representations
MSC 2010: 11A25; 11E20; 11E25; 11F11; 11F20; 11F27

1 Introduction

Let in the sequel ℕ, ℕ0, 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∊ ℕ. Then the sum of positive divisors of n to the power of k, σk(n), is defined by

σk(n)=∑0<d|ndk. (1)

We write σ(n) as a synonym for σ1(n). For m ∉ ℕ we set σk(m) = 0.

Suppose now that α, β ∊ ℕ are such that α≤β. Then the convolution sum, W(α, β) (n), is defined as follows:

W(α,β)(n)=∑(l,m)∈N02αl+βm=nσ(l)σ(m).(2)

We write Wβ(n) as a synonym for W(1,β) (n). Given α, β ∊ ℕ, if for all (l, m) ∊ N02 it holds that α l + β m ≠ n then we set W(α, β) (n) = 0.

For those convolution sums W(α, β)(n) that have so far been evaluated, the levels αβ are given in Table 1.

Table 1

Known convolution sums W(α,β)(n)

Level αβ Authors References
1 M. Besge, J. W. L. Glaisher, S. Ramanujan [1–3]
2, 3, 4 J. G. Huard & Z. M. Ou & B. K. Spearman & K. S. Williams [4]
5, 7 M. Lemire & K. S. Williams, S. Cooper & P. C. Toh [5, 6]
6 S. Alaca & K. S. Williams [7]
8, 9 K. S. Williams [8, 9]
10, 11, 13, 14 E. Royer [10]
12, 16, 18, 24 A. Alaca & S. Alaca & K. S. Williams [11–14]
15 B. Ramakrishman & B. Sahu [15]
20, 10 S. Cooper & D. Ye [16]
23 H. H. Chan & S. Cooper [17]
25 E. X. W. Xia & X. L. Tian & O. X. M. Yao [18]
27, 32 S. Alaca & Y. Kesicioğlu [19]
36 D. Ye [20]
14, 26, 28, 30 E. Ntienjem [21]

We discuss the evaluation of the convolution sums of level αβ = 22, 44 and αβ = 52, i.e., (α, β) = (1,22), (2, 11), (1, 44), (4, 11), (1, 52), (4, 13). Convolution sums of these levels have not been evaluated yet as one can notice from Table 1.

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

a(x12+x22+x32+x42)+b(x52+x62+x72+x82), (3)

and

c(x12+x1x2+x22+x32+x3x4+x42)+d(x52+x5x6+x62+x72+x7x8+x82), (4)

respectively, where a, b, c, d ∊ ℕ.

So far known explicit formulae for the number of representations of n by the octonary form Equation 3 are referenced in Table 2.

Table 2

Known representations of n by the form Equation 3

(a, b) Authors References
(1,2) K. S. Williams [8]
(1,4) A. Alaca & S. Alaca & K. S. Williams [12]
(1,5) S. Cooper & D. Ye [16]
(1,6) B. Ramakrishman & B. Sahu [15]
(1,8) S. Alaca & Y. Kesicioğlu [19]
(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) = (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.

2 Modular forms and convolution sums

Let ℍ be the upper half-plane, that is ℍ = {z ∊ ℂ| Im(z) > 0}, and let G = SL2(ℝ) be the group of 2 × 2-matrices abcd such that a, b, c, d ∊ ℝ and ad−bc = 1 hold. Let furthermore Γ = SL2(ℤ) be the full modular group which is a subgroup of SL2(ℝ). Let N ∊ ℕ. Then

Γ(N)=abcd∈SL2(Z)|abcd≡1001(modN)

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 Γ(N).

Relevant for our purposes is the following congruence subgroup:

Γ0(N)=abcd∈SL2(Z)|c≡0(modN).

Let k, N ∊ ℕ and let Γ′ ⊆ Γ be a congruence subgroup of level N ∊ ℕ. Let k ∊ ℤ, γ∊ SL2(ℤ) and f : ℍ∪ ℚ∪{∞}→ ℂ∪ {∞}. We denote by f[γ]k the function whose value at z is (cz+d)−k f(γ(z)), i.e., f[γ]k(z) = (cz+d)−k f(γ(z)). The following definition is based on the textbook by N. Koblitz [22, p. 108].

Definition 2.1

Let N ∊ ℕ, k ∊ ℤ, f be a meromorphic function on ℍ and Γ′ ⊂ Γ a congruence subgroup of level N.

  1. f is called a modular function of weight k for Γ′ if

    (a1)for all γ∊ Γ′ it holds that f[γ]k = f.

    (a2)for any δ ∊ Γ it holds that f[δ]k(z) can be expressed in the form ∑n∈Zane2πiznN, wherein an ≠ 0 for finitely many n ∊ ℤ such that n < 0.

  2. f is called a modular form of weight k for Γ′ if

    (b1)f is a modular function of weight k for Γ′,

    (b2)f is holomorphic on ℍ,

    (b3)for all δ ∊ Γ and for all n ∊ ℤ such that n < 0 it holds that an = 0.

  3. f is called a cusp form of weight k for Γ′ if

    (c1)f is a modular form of weight k for Γ′,

    (c2)for all δ ∊ Γ it holds that a0 = 0.

For k, N ∊ ℕ, let 𝔐k(Γ0(N)) be the space of modular forms of weight k for Γ0(N), 𝔖k(Γ0(N)) be the subspace of cusp forms of weight k for Γ0(N), and 𝔈k(Γ0(N)) be the subspace of Eisenstein forms of weight k for Γ0(N). Then the decomposition of the space of modular forms as a direct sum of the space generated by the Eisenstein series and the space of cusp forms, i.e., 𝔐k(Γ0(N)) = 𝔈k(Γ0(N))⊕ 𝔖k(Γ0(N)), is well-known; see for example W. A. Stein’s book (online version) [23, p. 81].

As noted in Section 5.3 of W. A. Stein’s book [23, p. 86] if the primitive Dirichlet characters are trivial and 2 ≤ k is even, then Ek(q) = 1−2kBk∑n=1∞σk−1(n)qn, where Bk are the Bernoulli numbers.

For the purpose of this paper we only consider trivial Dirichlet characters and 2 ≤ k even. Theorems 5.8 and 5.9 in Section 5.3 of [23, p. 86] also hold for this special case.

2.1 Eta functions

The Dedekind eta function, η(z), is defined on the upper half-plane ℍ by η(z)=e2πiz24∏n=1∞(1−e2πinz). We set q = e2π iz. Then η(z)=q124∏n=1∞(1−qn)=q124F(q), where F(q)=∏n=1∞(1−qn).

M. Newman [24, 25] systematically used the Dedekind eta function to construct modular forms for Γ0(N). M. Newman determined when a function f(z) is a modular form for Γ0(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 Γ0(N), which is condition (v) or (v′) in Theorem 2.2.

The following theorem is proved in L. J. P. Kilford’s book [27, p. 99] and G. Köhler’s book [28, p. 37]; we will apply 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)).

Theorem 2.2

(M. Newman and G. Ligozat). Let N ∊ ℕ, D(N) be the set of all positive divisors of N, δ∊ D(N) and rδ∊ ℤ. Let furthermore f(z) = ∏δ∈D(N)ηrδ(δz) be an η-quotient. If the following five conditions are satisfied

  1. ∑ δ ∈ D ( N ) δ r δ ≡ 0 ( m o d 24 ) ,

  2. ∑ δ ∈ D ( N ) N δ r δ ≡ 0 ( m o d 24 ) ≡(mod 24),

  3. ∏δ∈D(N)δrδ is a square in ℚ,

  4. 0<∑δ∈D(N)rδ ≡ 0 (mod 4)

  5. for each d ∊ D(N) it holds that ∑δ∈D(N)gcd(δ,d)2δrδ≥0,

    then f(z)∊ 𝔐k(Γ0(N)), where k=12∑δ∈D(N)rδ.

    Moreover, the η-quotient f(z) belongs to 𝔖k(Γ0(N)) if (v) is replaced by

    (v’) for each d ∊ D(N) it holds that ∑δ∈D(N)gcd(δ,d)2δrδ>0.

2.2 Convolution sums W(α, β)(n)

Recall that given α, β ∊ ℕ such that α ≤ β, the convolution sum is defined by Equation 2.

As observed by A. Alaca et al. [11], we can assume that gcd(α, β) = 1. Let q ∊ ℂ be such that |q| < 1. Then the Eisenstein series L(q) and M(q) are defined as follows:

L(q)=E2(q)=1−24∑n=1∞σ(n)qn, (5)

M(q)=E4(q)=1+240∑n=1∞σ3(n)qn. (6)

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 (α, β) ∊ ℕ2. Their proofs are given by E. Ntienjem [21].

Lemma 2.3

Let α, β ∊ ℕ. Then

(αL(qα)−βL(qβ))2∈M4(Γ0(αβ)).

Theorem 2.4

Let α, β ∊ ℕ be such that α and β are relatively prime and α < β. Then

( α L ( q α ) − β L ( q β ) ) 2 = ( α − β ) 2 + ∑ n = 1 ∞ 240 α 2 σ 3 ( n α ) + 240 β 2 σ 3 ( n β ) + 48 α ( β − 6 n ) σ ( n α ) + 48 β ( α − 6 n ) σ ( n β ) − 1152 α β W ( α , β ) ( n ) q n . (7)

3 Evaluation of the convolution sums W(α,β)(n), where αβ = 22, 44, 52

In this section, we give explicit formulae for the convolution sums W(1,22)(n), W(2,11)(n), W(1,44)(n), W(4,11)(n), W(1,52)(n) and W(4,13)(n).

3.1 Bases for 𝔈4(Γ0(αβ)) and 𝔖4(Γ0(αβ)) with αβ = 44, 52

We observe the following inclusion relations

M4(Γ0(11))⊂M4(Γ0(22))⊂M4(Γ0(44)) (8)

M4(Γ0(13))⊂M4(Γ0(26))⊂M4(Γ0(52)). (9)

Therefore, it suffices to correspondingly determine the basis of the spaces 𝔐4(Γ0(44)) and 𝔐4(Γ0(52)), respectively.

We use the dimension formulae for the space of Eisenstein forms and the space of cusp forms in T. Miyake’s book [29, Thrm 2.5.2, p. 60] or W. A. Stein’s book [23, Prop. 6.1, p. 91] to deduce that dim(𝔈4(Γ0(44))) = dim(𝔈4(Γ0(52))) = 6, dim(𝔖4(Γ0(44)) = 15 and dim(𝔖4(Γ0(52)) = 18.

Let D(44) = {1, 2, 4, 11, 22, 44} and D(52) = {1, 2, 4, 13, 26, 52} be the sets of all positive divisors of 44 and 52, respectively.

Theorem 3.1

  1. The sets 𝔅E,44 = {M(qt)| t ∈ D(44)} and 𝔅E,52 = {M(qt)|t ∈ D(52)} are bases of 𝔈4(Γ0(44)) and 𝔈4(Γ0(52)), respectively

  2. Let 1 ≤ i ≤15 and 1 ≤ j ≤ 18 be positive integers.

    Let δ1 ∈ D(44) and (r(i, δ1))i, δ1 be the Table 3 of the powers of η(δ1z).

    Let δ2 ∈ D(52) and (r(j, δ2))j, δ2 be the Table 4 of the powers of η(δ2z).

    Table 3

    Exponents of η-functions being basis elements of 𝔖4(Γ0(44))

    1 2 4 11 22 44
    1 6 −2 0 6 −2 0
    2 4 0 0 4 0 0
    3 2 2 0 2 2 0
    4 0 4 0 0 4 0
    5 −2 6 0 −2 6 0
    6 0 2 2 0 2 2
    7 0 −3 5 0 5 1
    8 0 0 4 0 0 4
    9 3 0 1 −1 0 5
    10 0 −2 6 0 −2 6
    11 1 −3 4 −3 5 4
    12 2 0 0 2 −4 8
    13 0 2 0 0 −2 8
    14 −3 9 0 1 1 0
    15 0 0 2 0 −4 10
    Table 4

    Exponents of η-functions being basis elements of 𝔖4(Γ0(52))

    1 2 4 13 26 52
    1 1 5 0 3 −1 0
    2 3 3 0 1 1 0
    3 1 3 0 3 1 0
    4 3 1 0 1 3 0
    5 1 1 0 3 3 0
    6 3 −1 0 1 5 0
    7 1 −1 0 3 5 0
    8 0 3 1 0 1 3
    9 2 1 1 −2 3 3
    10 0 1 1 0 3 3
    11 2 −1 1 −2 5 3
    12 0 3 −1 0 1 5
    13 2 1 −1 −2 3 5
    14 0 1 −1 0 3 5
    15 −1 5 0 5 −1 0
    16 0 −1 5 0 5 −1
    17 7 −3 0 −3 7 0
    18 0 7 −3 0 −3 7

    Let furthermore Ai(q)=∏δ1∈D(44)ηr(i,δ1)(δ1z)andBj(q)=∏δ2∈D(52)ηr(j,δ2)(δ2z) be selected elements of 𝔖4(Γ0(44)) and 𝔖4(Γ0(52)), respectively.

    Then the sets 𝔅S,44 = {Ai(q)| 1 ≤i≤15} and 𝔅S,52 = {Bj(q)| 1 ≤ j ≤ 18} are bases of 𝔖4(Γ0(44)) and 𝔖4(Γ0(52)), repectively.

  3. The sets 𝔅M,44 = 𝔅E,44 ∪ 𝔅S,44 and 𝔅M,52 = 𝔅E,52∪ 𝔅S,52 constitute bases of 𝔐4(Γ0(44)) and 𝔐4(Γ0(52)), respectively.

For 1 ≤ i ≤ 15 and 1 ≤ j ≤ 18 let in the sequel Ai(q) be expressed in the form ∑n=1∞ai(n)qn and Bj(q) be expressed in the form ∑n=1∞bj(n)qn.

Proof

We give the proof for the case αβ = 44. The case αβ = 52 is proved similarly.

  1. By Theorem 5.8 in Section 5.3 of W. A. Stein [23, p. 86] M(qt) is in 𝔐4(Γ0(t)) for each t which is an element of D(44). Since 𝔈4(Γ0(44)) has a finite dimension, it suffices to show that M(qt) with t ∈ D(44) are linearly independent. Suppose that xt ∈ ℂ with t ∈ D(44). We prove this by induction on the elements of the set D(44) which is assumed to be ascendantly ordered.

    The case t = 1 ∈ D(44) is obvious since comparing the coefficients of qt on both sides of the equation xtM(qt) = 0 clearly gives xt = 0.

    Suppose now that the cardinality of the set D(44) is greater than 1 and that M(qt) are linearly independent for all t|44 and t ≤ t1 for a given t1 with 1 < t1 < 44. Let C be the proper non-empty subset of D(44) which contains all positive divisors of 44 less than or equal to t1. Note that all positive divisors of t1 constitute a subset of C. Let us consider the non-empty subset C ∪ {t′} of D(44), wherein t′ is the next ascendant element of D(44) which is greater than t1 the greatest element of the set C. Then

    ∑t∈C∪{t′}xtM(qt)=∑t∈CxtM(qt)+xt′M(qt′)=0.

    By the induction hypothesis it holds that xt = 0 for all t ∈ C. So, we obtain from the above equation that xt′ = 0 when we compare the coefficient of qt′ on both sides of the equation.

    Hence, the solution is xt = 0 for all t such that t is a positive divisor of 44. Therefore, the set 𝔅E,44 is linearly independent. Hence, the set 𝔅E,44 is a basis of 𝔈4(Γ0(44)).

  2. The Ai(q) with 1 ≤ i ≤ 15 are obtained from an exhaustive search using Theorem 2.2 (i) − (v′). Hence, each Ai(q) is an element of the space 𝔖4(Γ0(44)).

    Since the dimension of 𝔖4(Γ0(44)) is 15, it suffices to show that the set {Ai(q)| 1 ≤ i ≤ 15} is linearly independent. Suppose that xi∈Cand∑i=115xiAi(q)=0. Then

    ∑i=115xiAi(q)=∑n=1∞(∑i=115xiai(n))qn=0

    which gives the following homogeneous system of linear equations

    ∑i=115ai(n)xi=0,1≤n≤15. (10)

    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, xi = 0 for all 1 ≤ i ≤ 15. Hence, the set {Ai(q)| 1 ≤ i ≤ 15} is linearly independent and therefore a basis of 𝔖4(Γ0(44)).

  3. Since 𝔐4(Γ0(44)) = 𝔈4(Γ0(44)) ⊕ 𝔖4(Γ0(44)), the result follows from (a) and (b).     □

    According to Equation 8 the basis elements Ai(q), where 1 ≤ i ≤ 5, are contained in 𝔖4(Γ0(22)). The basis element A2(q) is the only element of the space 𝔖4(Γ0(11)) that we are able to generate with the help of Theorem 2.2. Even though the basis element A14(q) looks like an element of 𝔖4(Γ0(22)), it cannot be generated at level 22 using Theorem 2.2.

To evaluate the convolution sums W(1,22)(n) and W(2,11)(n), we determine two additional basis elements of 𝔖4(Γ0(22)) which are

A6′(q)=η(2z)η3(11z)η5(22z)η(z)=∑n=1∞a6′(n)qn,A7′(q)=η9(2z)η7(11z)η5(z)η3(22z)=∑n=1∞a7′(n)qn.

Due to Equation 9 the basis elements Bj(q), where 1 ≤ j ≤ 7 and j = 15, 17, belong to 𝔖4(Γ0(26)). We are unable to generate any elements of the space 𝔖4(Γ0(13)) using Theorem 2.2. We note that B2j(q) = Bj(q2), where 4 ≤ j ≤ 7, B16(q) = B15(q2) and B18(q) = B17(q2). Therefore, one can easily replicate the evaluation of the convolution sums W(1,26)(n) and W(2,13)(n) shown by E. Ntienjem [21].

3.2 Evaluation of W(α,β)(n) where αβ = 22, 44, 52

Lemma 3.2

We have

(L(q2)−22L(q22))2=441+∑n=1∞(−331261σ3(n)+1267261σ3(n2)−11088061σ3(n11)+655776061σ3(n22)+1209661a1(n)+4579261a2(n)+1987261a3(n)+7372861a4(n)−5068861a5(n)+22176a6′(n)+864a7′(n))qn, (11)

(2L(q2)−11L(q11))2=81+∑n=1∞(1584061σ3(n)+3744061σ3(n2)+162676861σ3(n11)−49420861σ3(n22)+3686461a1(n)+35740861a2(n)+116035261a3(n)+153907261a4(n)+83404861a5(n)−22176a6′(n)−864a7′(n))qn, (12)

(L(q)−44L(q44))2=1849+∑n=1∞(12446461σ3(n)−57766233640565σ3(n2)+689863685795σ3(n4)−17424061σ3(n11)+620642885795σ3(n22)+25256901125795σ3(n44)+144061a1(n)−829278725795a2(n)−8873455685795a3(n)−16764295685795a4(n)−28040071685795a5(n)+37533807365795a6(n)−1335628819a7(n)+42266096645795a8(n)−63360019a9(n)−5273326081159a10(n)+767923219a11(n)−1523174495a12(n)−13107916895a13(n)+31795219a14(n)−1259596895a15(n))qn, (13)

(4L(q4)−11L(q11))2=49+∑n=1∞(−11088061σ3(n)+801218885795σ3(n2)−483386885795σ3(n4)+181790461σ3(n11)−984804485795σ3(n22)−273208325795σ3(n44)+11088061a1(n)+1748574725795a2(n)+11694271685795a3(n)+21141895685795a4(n)+30255137285795a5(n)−35110805765795a6(n)+1331827219a7(n)−36417623045795a8(n)+63360019a9(n)+6639137281159a10(n)−767923219a11(n)+1523174495a12(n)+13107916895a13(n)−31795219a14(n)+1259596895a15(n))qn, (14)

(L(q)−52L(q52))2=2601+∑n=1∞(61090081243σ3(n)−4565040848166064597σ3(n2)+25459241σ3(n4)−73619521243σ3(n13)−48295288273446064597σ3(n26)+43473830441σ3(n52)−30661441243b1(n)+4981571790486064597b2(n)+9273270707046064597b3(n)−4425775005606064597b4(n)−85304136696486064597b5(n)−101616997322886064597b6(n)−103883663521243b7(n)+104083241b8(n)+7488b9(n)+329100929664147917b10(n)+27456b11(n)−152492885101446064597b12(n)+17472b13(n)+4700966441b14(n)−25166713896551327b15(n)+41670318268806064597b16(n)−1264250239206064597b17(n)+86860841b18(n))qn, (15)

(4L(q4)−13L(q13))2=81+∑n=1∞(30661441243σ3(n)−2400612306726064597σ3(n2)+13939241σ3(n4)+457986721243σ3(n13)−539220318246064597σ3(n26)+2029017641σ3(n52)−30661441243b1(n)+2127358198806064597b2(n)+2518488510246064597b3(n)−4005610378086064597b4(n)−51524598204006064597b5(n)−54087483121926064597b6(n)−54893553121243b7(n)+15033641b8(n)−7488b9(n)+151016538432147917b10(n)−27456b11(n)−82248324316806064597b12(n)−17472b13(n)−54489641b14(n)−11115614088551327b15(n)+20569536096006064597b16(n)−647456933286064597b17(n)−230441b18(n))qn. (16)

Proof

We just prove the case (4L(q4) − 11L(q11))2. The other cases are proved similarly.

It follows from Lemma 2.3 that (4L(q4) − 11L(q11))2 ∈ 𝔐4(Γ0(44)). Hence, by Theorem 3.1 (c), there exist Xδ, Yj ∈ ℂ, 1 ≤ j ≤ 15 and δ ∈ D(44), such that

(4L(q4)−11L(q11))2=∑δ∈D(44)XδM(qδ)+∑j=115YjAj(q)=∑δ∈D(44)Xδ+∑n=1∞(240∑δ∈D(44)σ3(nδ)Xδ+∑j=1mSaj(n)Yj)qn. (17)

We equate the right hand side of Equation 17 with that of Equation 7 when setting (α, β) = (4,11) to obtain

∑n=1∞(240∑δ∈D(44)σ3(nδ)Xδ+∑j=115aj(n)Yj)qn=∑n=1∞(3840σ3(n4)+29040σ3(n11)+192(11−6n)σ(n4)+528(4−6n)σ(n11)−50688W(4,11)(n))qn.

We now take the coefficients of qn for which n is in

{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,22,44}.

This results in a system of linear equations whose unique solution determines the values of the unknown Xδ for all δ ∈ D(44) and the values of the unkown Yj for all 1 ≤ j ≤ 15. Hence, we obtain the stated result.     □

Our main result of this section is as follows.

Theorem 3.3

Let n be a positive integer Then

W(1,22)(n)=171464σ3(n)−1122σ3(n2)+35488σ3(n11)+125366σ3(n22)+(124−188n)σ(n)+(124−14n)σ(n22)−212684a1(n)−1595368a2(n)−695368a3(n)−32671a4(n)+261a5(n)−78a6′(n)−388a7′(n), (18)

W(2,11)(n)=−5488σ3(n)+5366σ3(n2)+1371464σ3(n11)+39122σ3(n22)+(124−144n)σ(n2)+(124−18n)σ(n11)−16671a1(n)−12415368a2(n)−40295368a3(n)−668671a4(n)−362671a5(n)+78a6′(n)+388a7′(n), (19)

W(1,44)(n)=−13366σ3(n)+5014431784860σ3(n2)−13615795σ3(n4)+55976σ3(n11)−1959192720σ3(n22)+987817385σ3(n44)+(124−1176n)σ(n)+(124−14n)σ(n44)−510736a1(n)+35993127490a2(n)+30810611019920a3(n)+6614711590a4(n)+1217017127490a5(n)−3258143254980a6(n)+52738a7(n)−91723363745a8(n)+2538a9(n)+208072318a10(n)−30338a11(n)+601190a12(n)+258695a13(n)−69209a14(n)+497190a15(n), (20)

W(4,11)(n)=35976σ3(n)−2529192720σ3(n2)+417817385σ3(n4)−11732σ3(n11)+1554346360σ3(n22)+5395795σ3(n44)+(124−144n)σ(n4)+(124−116n)σ(n11)−35976a1(n)−75893127490a2(n)−40605111019920a3(n)−917617127490a4(n)−1313157127490a5(n)+3047813254980a6(n)−105176a7(n)+79031363745a8(n)−2538a9(n)−28815725498a10(n)+30338a11(n)−601190a12(n)−258695a13(n)+69209a14(n)−497190a15(n), (21)

W(1,52)(n)=−971243σ3(n)+731577059582201312σ3(n2)−17164σ3(n4)+589959664σ3(n13)+7739629531582201312σ3(n26)−81757492σ3(n52)+(124−1208n)σ(n)+(124−14n)σ(n52)+31939775632b1(n)−69188497095045744704b2(n)−193193139737568617056b3(n)+236419605194067104b4(n)+4556844909194067104b5(n)+135706460148516776b6(n)+554934139776b7(n)−139328b8(n)−18b9(n)-659256671775004b10(n)−1124b11(n)+203649686348516776(n)b12(n)−724b13(n)−3139164b14(n)+349537693458704064b15(n)−55649463548516776b16+67534735194067104b17(n)−2982b18(n), (22)

W(4,13)(n)=−31939775632σ3(n)+50012756397568617056σ3(n2)+476396σ3(n4)+24049387816σ3(n13)+11233756637568617056σ3(n26)−176132132σ3(n52)+(124−152n)σ(n4)+(124−116n)σ(n13)+31939b775632b1(n)−29546641655045744704b2(n)−52468510637568617056b3(n)+83450216217568617056b4(n)+357809709752522872352b5(n)+4695094021315359044b6(n)+38120523517088b7(n)−2614264b8(n)+18b9(n)−78654447146150104b10(n)+1124b11(n)+428376689151892154264b12(n)+724b13(n)+4732132b14(n)+154383529458704064b15(n)−5356650025946077132b16(n)+13488686117568617056b17(n)+11066b18(n). (23)

Proof

We prove the case W(4,13)(n) as the other cases are proved similarly.

We equate the right hand side of Equation 16 with that of Equation 7 when setting (α, β) = (4,13), namely

∑n=1∞(3840σ3(n4)+40560σ3(n13)+192(13−6n)σ(n4)+624(4−6n)σ(n13)−59904W(4,13)(n))qn=∑n=1∞(30661441243σ3(n)−2400612306726064597σ3(n2)+13939241σ3(n4)+457986721243σ3(n13)−539220318246064597σ3(n26)+2029017641σ3(n52)−30661441243b1(n)+2127358198806064597b2(n)+2518488510246064597b3(n)−4005610378086064597b4(n)−51524598204006064597b5(n)−54087483121926064597b6(n)−54893553121243b7(n)+15033641b8(n)−7488b9(n)+151016538432147917b10(n)−27456b11(n)−82248324316806064597b12(n)−17472b13(n)−54489641b14(n)−11115614088551327b15(n)+20569536096006064597b16(n)−647456933286064597b17(n)−230441b18(n))qn.

We then solve for W(4,13)(n) to obtain the stated result.     □

4 Number of representations of a positive integer n by the qctonary quadratic form using W(α,β)(n) when αβ = 44, 52

Let n ∈ ℕ0 and the number of representations of n by the quaternary quadratic form x12+x22+x32+x42 be denoted by r4(n). That means,

r4(n)=card({(x1,x2,x3,x4)∈Z4|m=x12+x22+x32+x42}).

We set r4(0) = 1. For all n ∈ ℕ, the following Jacobi’s identity is proved in K. S. Williams’ book [30, Thrm 9.5, p. 83]

r4(n)=8σ(n)−32σ(n4). (24)

Let furthermore the number of representations of n by the octonary quadratic form

a(x12+x22+x32+x42)+b(x52+x62+x72+x82)

be denoted by N(a,b)(n). That means,

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)}).

We infer the following result:

Theorem 4.1

Let n ∈ ℕ and (a, b) = (1, 11), (1, 13). Then

N(1,11)(n)=8σ(n)−32σ(n4)+8σ(n11)−32σ(n44)+64W(1,11)(n)+1024W(1,11)(n4)−256(W(4,11)(n)+W(1,44)(n)),N(1,13)(n)=8σ(n)−32σ(n4)+8σ(n13)−32σ(n52)+64W(1,13)(n)+1024W(1,13)(n4)−256(W(4,13)(n)+W(1,52)(n)).

Proof

We only prove N(1, 11)(n) since that for N(1, 13)(n) is done similarly.

It holds that

N(1,11)(n)=∑(l,m)∈N02l+11m=nr4(l)r4(m)=r4(n)r4(0)+r4(0)r4(n11)+∑(l,m)∈N2l+11m=nr4(l)r4(m).

We make use of Equation 24 to derive

N(1,11)(n)=8σ(n)−32σ(n4)+8σ(n11)−32σ(n52)+∑(l,m)∈N2l+11m=n(8σ(l)−32σ(l4))(8σ(m)−32σ(m4)).

We observe that

(8σ(l)−32σ(l4))(8σ(m)−32σ(m4))=64σ(l)σ(m)−256σ(l4)σ(m)−256σ(l)σ(m4)+1024σ(l4)σ(m4).

The evaluation of

W(1,11)(n)=∑(l,m)∈N2l+11m=nσ(l)σ(m)

is shown by E. Royer [10, Thrm 1.3]. We map l to 4l to infer

W(4,11)(n)=∑(l,m)∈N2l+11m=nσ(l4)σ(m)=∑(l,m)∈N24l+11m=nσ(l)σ(m).

The evaluation of W(4,11)(n) is given in Equation 21. We next map m to 4m to conclude

W(1,44)(n)=∑(l,m)∈N2l+11m=nσ(l)σ(m4)=∑(l,m)∈N2l+44m=nσ(l)σ(m).

The evaluation of W(1,44)(n) is provided by Equation 20. We simultaneously map l to 4l and m to 4m to deduce

∑(l,m)∈N2l+11m=nσ(l4)σ(m4)=∑(l,m)∈N2l+11m=n4σ(l)σ(m)=W(1,11)(n4).

Again, E. Royer [10, Thrm 1.3] has proved the evaluation of W(1, 11)(n).

We then put these evaluations together to obtain the stated result for N(1, 11)(n).     □

Acknowledgement

I am indebtedly thankful to the anonynuous referee for fruitful comments and suggestions on a draft of this paper.

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Received: 2016-8-30
Accepted: 2017-2-28
Published Online: 2017-4-21

© 2017 Ntienjem

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0041
Keywords for this article
Sums of Divisors function; Convolution Sums; Dedekind eta function; Modular Forms; Eisenstein Series; Cusp Forms; Octonary quadratic Forms; Number of Representations
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