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North-Western European Journal of Mathematics

W N

M

E J

Elementary Evaluation of Convolution Sums Involving the Divisor Function

for a Class of Levels

Ebénézer Ntienjem1

Received: January 15, 2019/Accepted: July 5, 2019/Online: August 5, 2019

Abstract

We discuss the evaluation of convolution sums involving the divisor function, P

(l,m)∈N2 α l+β m=n

σ(l)σ(m), for the class of levelsαβbelonging to all natural numbers.

The evaluation of convolution sums belonging to this class is achieved by apply- ing modular forms and primitive Dirichlet characters. We illustrate our method with the explicit examples for the levelsαβ= 33, 40, 45, 50, 54, and 56. As a corollary, the known convolution sums for the levelsαβ= 10, 11, 12, 15, 16, 18, 24, 25, 27, 32 and 36 are improved when we revisit their evaluations. If the levelαβ≡0 (mod 4), we determine natural numbersa, band use the evaluated convolution sums together with other known convolution sums to carry out the number of representations ofnby the octonary quadratic formsaP4

i=1 x2i +bP8

i=5 xi2. Similarly, if the levelαβ≡0 (mod 3), we compute natural numbersc, dand make use of the evaluated convolution sums together with other known convo- lution sums to determine the number of representations ofnby the octonary quadratic formscP2

i=1

(x22i1+x2i1x2i+x2i2 ) +d P4 i=3

(x22i1+x2i1x2i+x22i). In addition, we determine formulae for the number of representations of a positive integernwhen (a, b) = (1,1), (1,3), (1,6), (2,3).

Keywords:Sums of Divisors; Dedekind eta function, Convolution Sums, Modular Forms, Dirichlet Characters, Eisenstein forms, Cusp Forms, Octonary quadratic Forms, Number of Representations.

msc: 11A25, 11F11, 11F20, 11E20, 11E25, 11F27.

1Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University

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1 Introduction

In this work, we denote byN,Z,Q,RandCthe sets of natural numbers, integers, rational numbers, real numbers and complex numbers, respectively. Let in addition N0denote the set of natural numbers without zero, i.e.,N0=N\ {0}. Letk∈Nand letn∈N0. The sumσk(n) of thekthpowers of the positive divisors ofnis defined by

σk(n) = X

0<d|n

dk. (1)

We letσ(n) stand forσ1(n). Form<Nwe setσk(m) = 0 and for allk∈N0we set σk(0) = 0.

Letα, β∈N0be such thatαβ. We define the convolution sum,W(α,β)(n), as follows:

W(α,β)(n) = X

(l,m)∈N2 α l+β m=n

σ(l)σ(m). (2)

We write Wβ(n) as a shorthand for W(1,β)(n). If for all (l, m)∈N2 it holds that α l+β m,n, then we setW(α,β)(n) = 0

So far known convolution sums are displayed in Table 1.

Levelαβ References

1 Besge (1885), Glaisher (1862), and Ramanujan (1916) 2, 3, 4 Huard et al. (2002)

5, 7 Cooper and Toh (2009) and Lemire and Williams (2006)

6 Ş. Alaca and Williams (2007)

8, 9 Williams (2005, 2006)

10, 11, 13, 14 Royer (2007)

12, 16, 18, 24 A. Alaca,Ş. Alaca, and Williams (2006, 2007a,b, 2008)

15 Ramakrishnan and Sahu (2013)

10, 20 Cooper and Ye (2014)

23 Chan and Cooper (2008)

25 Xia, Tian, and Yao (2014)

27, 32 Ş. Alaca and Kesicioˇglu (2016)

36 Ye (2015)

14, 26, 28, 30, 22, 44, 52, 48, 64

Ntienjem (2015, 2017a,b)

Table 1 – Known convolution sumsW(α,β)(n) of levelαβ LetNbe a subset ofN0which is defined as follows:

N={2ν℧|ν∈ {0,1,2,3}and℧is a finite product of distinct odd primes}.

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1. Introduction

We evaluate the convolution sumW(α,β)(n) for the class of levelsαβsuch that (1) αβ∈Nand

(2) αβ∈N0\N.

Therefore, we evaluate the convolution sum W(α,β)(n) for the class of levelsαβ belonging to all natural numbers. We use in particular Dirichlet characters and modular forms to evaluate these convolution sums. The evaluation of the convolu- tion sum for these classes of levels is new.

We observe that the levelsαβ= 9, 16, 18, 25, 27, 32, 36, 48, 64 from Table 1 belong to the class of levels mentioned in item (2) while the other levels from Table 1 are handled in item (1).

As an immediate consequence of the generalization of the evaluation of convolu- tion sums, we revisit the evaluation of the convolution sums for the levelsαβ= 9, 10, 11, 12, 15, 16, 18, 24, 25, 27, 32 and 36, which leads to the improvement of the result of the evaluation of the convolution sums forαβ= 10, 11, 12, 15, 16, 18, 24, 25, 27, 32 and 36.

We illustrate our general approach in case

1. αβ∈Nby evaluating the convolution sum forαβ= 33, 40 and 56.

2. αβ∈N0\Nby evaluating the convolution sum forαβ= 45, 50 and 54.

Again, these convolution sums have not been evaluated as yet.

As an application, convolution sums are used to determine explicit formulae for the number of representations of a positive integernby the octonary quadratic forms

a

4

X

i=1

xi2+b

8

X

i=5

x2i (3)

and

c X2

i=1

(x2i21+x2i1x2i+x2i2 ) +d X4

i=3

(x22i1+x2i1x2i+x22i), (4)

respectively, wherea, b, c, d∈N0.

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So far known explicit formulae for the number of representations ofnby the octonary form (3) are displayed in Table 2.

(a, b) References

(1,2) Williams (2006)

(1,4) A. Alaca,Ş. Alaca, and Williams (2007a)

(1,5) Cooper and Ye (2014)

(1,6) Ramakrishnan and Sahu (2013)

(1,8) Ş. Alaca and Kesicioˇglu (2016)

(1,7), (1,11), (1,13), (1,12), (1,16), (3,4)

Ntienjem (2015, 2017a,b)

Table 2 – Known representations ofnby the form (3)

Similarly, so far known explicit formulae for the number of representations ofnby the octonary form (4) are referenced in Table 3.

(c, d) References

(1,1) Lomadze (1989)

(1,2) Ş. Alaca and Williams (2007)

(1,3) Williams (2005)

(1,4), (1,6), (1,8), (2,3)

A. Alaca,Ş. Alaca, and Williams (2006, 2007a,b) (1,5) Ramakrishnan and Sahu (2013)

(1,9) Ş. Alaca and Kesicioˇglu (2016)

(1,10), (2,5), (1,16) Ntienjem (2015, 2017b) (1,12), (3,4) Ye (2015)

Table 3 – Known representations ofnby the form (4) We first discuss a method to determine all pairs (a, b)∈N2

0and (c, d)∈N2

0that are necessary for the determination of the formulae for the number of representations of a positive integer by the octonary quadratic forms (3) and (4) when the levelαβis contained in the above-mentioned class of levels. We then determine explicit formu- lae for the number of representations of a positive integernby octonary quadratic forms (3) and (4) wheneverαβ≡0 (mod 3) andαβ≡0 (mod 4), respectively.

We next use the convolution sums,W(α,β)(n), where

αβ= 33, 45 and 54 to give examples of explicit formulae for the number of representations of a positive integernby the octonary quadratic forms (3).

αβ= 40 and 56, to provide examples of explicit formulae for the number of representations of a positive integernby the octonary quadratic forms (4).

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2. Essentials to the Understanding of the Problem

Software for symbolic scientific computation is used to obtain the results of this paper. This software comprises the open source software packagesGiNaC,Maxima, Reduce,SAGEand the commercial software packageMAPLE.

2 Essentials to the Understanding of the Problem

2.1 Modular Forms

LetH={z∈C|Im(z)>0}, be the upper half-plane and letΓ =G= SL2(R) ={a b

c d

| a, b, c, d∈Randadbc= 1}be the group of 2×2-matrices. LetN ∈N0. Then

Γ(N) =n a b

c d

∈SL2(Z)|a b

c d

1 0

0 1

(modN)o

is a subgroup ofGand is called theprincipal congruence subgroup of level N. A sub- groupHofGis called acongruence subgroup of level Nif it containsΓ(N).

For our purposes the following congruence subgroup is relevant:

Γ0(N) =n a b

c d

∈SL2(Z)|c≡0 (modN)o .

Let k, N ∈ N and let Γ ⊆ Γ be a congruence subgroup of level N. Let k ∈ Z, γ∈SL2(Z) andf :H∪Q∪ {∞} →C∪ {∞}. We denote byf[γ]k the function whose value atzis (cz+d)kf(γ(z)), i.e.,f[γ]k(z) = (cz+d)kf(γ(z)). The following definition is according to N. Koblitz2.

Definition 1 – LetN ∈N0,k∈Z,f be a meromorphic function onHandΓ⊂Γ a congruence subgroup of levelN.

(a) f is called amodular function of weightkforΓif (a1) for allγ∈Γit holds thatf[γ]k=f.

(a2) for anyδ∈Γ it holds thatf[δ]k(z) has the form P

n∈Z

ane2πiznN andan,0 for finitely manyn∈Z\N.

(b) f is called amodular form of weightkforΓif (b1) f is a modular function of weightkforΓ, (b2) f is holomorphic onH,

(b3) for allδ∈Γ and for alln∈Z\Nit holds thatan= 0.

(c) f is called acusp form of weightkforΓif (c1) f is a modular form of weightkforΓ, (c2) for allδ∈Γ it holds thata0= 0.

2Koblitz, 1993,Introduction to Elliptic Curves and Modular Forms, p. 108.

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Let us denote byMk) the set of modular forms of weightkforΓ, bySk) the set of cusp forms of weightkforΓand byEk) the set of Eisenstein forms. The sets Mk), Sk) andEk) are vector spaces overC. Therefore,Mk0(N)) is the space of modular forms of weightkforΓ0(N),Sk0(N)) is the space of cusp forms of weight kforΓ0(N), andEk0(N)) is the space of Eisenstein forms. 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.,Mk0(N)) =Ek0(N))⊕Sk0(N)), is well- known; see for example the online version of the book3.

We assume in this paper that 2 ≤k is even and that χandψ are primitive Dirichlet characters with conductorsLandR, respectively. It has been noted by W. A. Stein4that

Ek,χ,ψ(q) =C0+

X

n=1

X

d|n

ψ(d)χ n

d

dk1

!

qn, (5)

where C0=





0 ifL >1

Bk,χ

2k ifL= 1

andBk,χare the generalized Bernoulli numbers. Theorems 5.8 and 5.9 in Section 5.3 in Stein (2011, p. 86) are then applicable.

If the primitive Dirichlet charactersχandψare trivial, then their conductors LandRare one, respectively. Therefore, (5) can be normalized and then given as follows: Ek(q) = 1−2k

Bk

P

n=1

σk1(n)qn. This will be the case whenever the levelαβ belongs toN.

2.2 Eta Quotients

On the upper half-planeH, the Dedekindη-function,η(z), is defined byη(z) = e2πiz24

Q

n=1

(1−e2πinz). Let us setq=e2πiz. Then it follows that

η(z) =q241

Y

n=1

(1−qn) =q241F(q), whereF(q) =

Y

n=1

(1−qn).

Letj, κ∈Nandej∈Z. According to G. Köhler5aneta productoreta quotient,

3Stein, 2011,Modular Forms, A Computational Approach, p. 81.

4Ibid., p. 86.

5Köhler, 2011,Eta Products and Theta Series Identities, p. 31.

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2. Essentials to the Understanding of the Problem

f(z), is a finite product of eta functions of the form

κ

Y

j=1

η(jz)ej. (6)

Based on this definition of an eta quotient there exists N ∈ N such that N = lcm{j | 1≤jκ}. We call such anN thelevelof an eta product. Therefore, an eta quotient will simply be understood as

Y

j|N

η(jz)ej.

Ifk=12

κ

P

j=1

ej, then an eta quotientf(z) behaves like a modular form of weightkon Γ0(N) with some multiplier system.

We will use eta function, eta quotient and eta product interchangeably as syn- onyms.

The Dedekindη-function is applied by M. Newman6to systematically construct modular forms forΓ0(N). Newman then establishes conditions (i)-(iv) in the follow- ing theorem. G. Ligozat7determined the order of vanishing of anη-function at all cusps ofΓ0(N), which is condition (v) or (v) in the following theorem.

The following theorem is formulated by L. J. P. Kilford8and G. Köhler9; it will be used to exhaustively determineη-quotients,f(z), which belong toMk0(N)), and especially thoseη-quotients which are inSk0(N)).

Theorem 1 (M. Newman and G. Ligozat) – LetN ∈N0,D(N)be the set of all posi- tive divisors ofN,δD(N)andrδ∈Z. Let furthermoref(z) = Q

δD(N)

ηrδ(δz)be an eta quotient. If the following four conditions are satisfied

(i) P

δD(N)

δ rδ≡0 (mod 24), (ii) P

δD(N) N

δ rδ≡0 (mod 24), (iii) Q

δD(N)

δrδis a square inQ,

6Newman, 1957, “Construction and Application of a Class of Modular Functions”;

Newman, 1959, “Construction and Application of a Class of Modular Functions II”.

7Ligozat, 1975, “Courbes Modulaires de Genre 1”.

8Kilford, 2008,Modular forms: A classical and computational introduction, p. 99.

9Köhler, 2011,Eta Products and Theta Series Identities, Cor. 2.3, p. 37.

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(iv) 0< P

δD(N)

rδ ≡0 (mod 4),

(v) ∀dD(N)it holds P

δD(N) gcd (δ,d)2

δ rδ≥0, thenf(z)∈Mk0(N)), wherek=12 P

δD(N)

rδ.

Moreover, theη-quotientf(z)is an element ofSk0(N))if(v)is replaced by (v) ∀dD(N)it holds P

δD(N) gcd (δ,d)2

δ rδ>0.

2.3 Convolution Sums W

(α,β)

(n)

Givenα, β∈N0such thatαβ, let the convolution sum be defined by (2).

Suppose in addition that gcd (α, β) =δ >1 for someδ∈N0. Then there exist α1, β1∈N0such that gcd (α1, β1) = 1, α=δ α1andβ=δ β1. Hence,

W(α,β)(n) = X

(l,k)∈N2 α l+β k=n0

σ(l)σ(k) = X

(l,k)∈N2 δ α1l+δ β10k=n

σ(l)σ(k) =W11)

n δ

. (7)

Therefore, we may simply assume that gcd (α, β) = 1 as does A. Alaca et al.10. The formula proved by M. Besge11, J. W. L. Glaisher12 and S. Ramanujan13 is applied to (7) to deduce that

α∈N0 W(α,α)(n) =W(1,1) n

α

= 5 12σ3

n α

+ 1

12− 1 2αn

σ

n α

. (8)

Letq∈Cbe such that|q|<1. Let furthermoreχandψbe primitive Dirichlet characters with conductorsLandR, respectively.

We assume that the primitive Dirichlet charactersχandψ 1. are trivial wheneverαβ∈Nholds.

2. are such thatχ=ψand thatχis a Legendre-Jacobi-Kronecker symbol other- wise.

10A. Alaca,Ş. Alaca, and Williams, 2006, “Evaluation of the convolution sumsP

l+12m=nσ(l)σ(m) andP

3l+4m=nσ(l)σ(m)”.

11Besge, 1885, “Extrait d’une lettre de M Besge à M Liouville”.

12Glaisher, 1862, “On the square of the series in which the coefficients are the sums of the divisors of the exponents”.

13Ramanujan, 1916, “On certain arithmetical functions”.

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2. Essentials to the Understanding of the Problem

Then the following Eisenstein series hold:

L(q) =E2(q) = 1−24

X

n=1

σ(n)qn, (9)

M(q) =E4(q) = 1 + 240

X

n=1

σ3(n)qn, (10)

Mχ(qλ) =E4(qλ)⊗χ

=χ(λ) C0+

X

n=1

χ(n)σ3(n)qλ n

!

=χ(λ)C0+

X

n=1

χ(λ n)σ3(n)qλ n

=χ(λ)C0+

X

n=1

χ(n)σ3 n

λ

qn, (11)

whereλ∈N0, C0=





0 ifL >1

B4,χ

8 ifL= 1

andB4,χare the specially generalized Bernoulli numbers.

Note thatM(q) is a special case of (5) or (11) and hold ifαβ∈N. We state two relevant results for the sequel of this work.

Lemma 1 – Letα, β∈N0. Then (α L(qα)−β L(qβ))2M40(αβ)).

Proof. Ifα=β, then trivially 0 = (α L(qα)−α L(qα))2M40(α)) and there is nothing to prove. Therefore, we may suppose thatα,βin the sequel. We apply the result proved by W. A. Stein14 to deduce L(q)α L(qα)∈ M20(α))⊆ M20(αβ)) and L(q)β L(qβ)∈M20(β))⊆M20(αβ)). Therefore,

α L(qα)−β L(qβ) = (L(q)−β L(qβ))−(L(q)−α L(qα))∈M20(αβ))

and so (α L(qα)−β L(qβ))2M40(αβ)). □

14Stein, 2011,Modular Forms, A Computational Approach, Thrms 5.8, 5.9, p. 86.

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Theorem 2 – Letα, β, N∈N0be such thatN =αβ, α < β, andαandβare relatively prime. The caseα=βis discussed above. Then

(α L(qα)−β L(qβ))2= (α−β)2+

X

n=1

240α2σ3 n

α

+ 240β2σ3 n β

!

+ 48α(β−6n)σ n

α

+ 48β(α−6n)σ n β

!

−1152αβ W(α,β)(n)

!

qn. (12) Proof. We first observe that

(α L(qα)−β L(qβ))2=α2L2(qα) +β2L2(qβ)−2αβ L(qα)L(qβ). (13) J. W. L. Glaisher15has proved the following identity

L2(q) = 1 +

X

n=1

240σ3(n)−288n σ(n)

!

qn (14)

which we apply to deduce L2(qα) = 1 +

X

n=1

240σ3

n α

−288n ασ

n α

qn (15)

and

L2(qβ) = 1 +

X

n=1

240σ3

n β

!

−288n βσ n

β

!!

qn. (16)

Since

X

n=1

σ n

α

qn

! X

n=1

σ n β

! qn

!

=

X

n=1

X

αk+βl=n

σ(k)σ(l)

! qn=

X

n=1

W(α,β)(n)qn,

we conclude, when using an accordingly modified version of (9), that L(qα)L(qβ) = 1−24

X

n=1

σ n

α

qn−24

X

n=1

σ n β

!

qn+ 576

X

n=1

W(α,β)(n)qn. (17) Therefore,

α L(qα)−β L(qβ)2

= (α−β)2+

X

n=1

240α2σ3

n α

+ 240β2σ3 n β

!

+ 48α(β−6n)σ n

α

+ 48β(α−6n)σ n β

!

−1152αβ W(α,β)(n)

! qn

as asserted. □

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3. EvaluatingW(α,β)(n), whereαβ∈N0

3 Evaluating W

(α,β)

(n), where αβ ∈ N

0

We carry out an explicit formula for the convolution sumW(α,β)(n).

3.1 Bases of E

4

0

(αβ)) and S

4

0

(αβ))

LetD(αβ) denote the set of all positive divisors ofαβ.

We apply the dimension formulae as discussed in T. Miyake16or W. A. Stein17 to conclude that

• for the space of Eisenstein forms, dim(E40(αβ))) =X

d|αβ

ϕ

gcd

d,αβ d

=mE, (18)

wheremE∈N0andϕis the Euler’s totient function.

We observe that, ifαβ∈N, then dim(E40(αβ))) =X

δ|αβ

ϕ

gcd

δ,αβ δ

=X

δ|αβ

1 =σ0(αβ) =d(αβ). (19)

• for the space of cusp forms, dim(S40(αβ))) =mS, wheremS∈N.

We use Theorem 1 (i)–(v) to exhaustively determine as many elements of the space S40(αβ)) as possible. From these elements of the spaceS40(αβ)) we select relevant ones for the purpose of the determination of a basis of this space. The proof of the following theorem provides a method to effectively determine such a basis.

The so-determined basis of the vector space of cusp forms is in general not unique. However, due to the change of basis which is an automorphism, it is sufficient to only consider this basis for our purpose.

LetCdenote a set of primitive Dirichlet charactersχ(n) = mn

, wherem, n∈Z withmfixed and mn

is the Legendre-Jacobi-Kronecker symbol. Let furthermore Dχ(αβ)⊆ D(αβ) denote the subset ofD(αβ) associated with the Dirichlet charac- ter χ. Then it is obvious that not every primitive Dirichlet character is a good candidate which, applied in (11), will make it constitute a basis element of the space of Eisenstein series for a given level that belongs toN0\N. For example, if the levels are 25 and 32, the primitive Dirichlet characters 5nand n4will not permit one to build a basis ofE40(25)) andE40(32)), respectively.

15Glaisher, 1862, “On the square of the series in which the coefficients are the sums of the divisors of the exponents”.

16Miyake, 1989,Modular Forms, Thrm 2.5.2, p. 60.

17Stein, 2011,Modular Forms, A Computational Approach, Prop. 6.1, p. 91.

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Leti, κbe natural numbers. It is standard to express a natural number in the for Qκ

i=1

piei, wherepi is a prime number andei is inN0, modulo a permutation of the primespi. Therefore, we will also in the following use this form to express a level αβ∈N0\N.

Definition 2 – Leti, κ∈N0andn∈N. LetC∈Zbe fixed. Suppose that the level αβ∈N0\Nis fixed and of the form Qκ

i=1

peii, wherepi is a prime number andei is inN0. We say that the primitive Dirichlet characterχ(n) = CnannihilatesE40(αβ)) or is anannihilatorofE40(αβ)) if for some 1≤jκwe have 1< pejj ∈N0\Nand Mχ(qδ) vanishes for all 1< δpositive divisor ofpjej.

A setCof primitive Dirichlet charactersannihilatesE40(αβ)) or is anannihilator ofE40(αβ)) if eachχ(n)∈ Cis an annihilator ofE40(αβ)).

To illustrate the above definition, suppose thatαβ= 2×32and the primitive Dirichlet characters isχ(n) = n3

. ThenC=−3 so that|C|is a positive divisor of 9 = 32. For all 1< δ∈ D(9) ={1,3,9}one easily verify when applying (11) that

Mχ(qδ) =

X

n=1

χ(n)σ3 n

δ

qn= 0, since it holds that

χ(n) = −3 n

!

=









−1 ifn≡2 (mod 3), 0 if gcd(3, n),1, 1 ifn≡1 (mod 3) and

σ3 n

δ

=





0 if nδ<N0, nonzero gcd(3, n),1.

Hence, the primitive Dirichlet characterχ(n) = n3

is an annihilator ofE40(2×32)).

The following theorem provides a strong criterion for the selection of a primitive Dirichlet character for a given levelαβ∈N0\N.

Theorem 3 – Let i, κbe in N0. LetC ∈ Zbe fixed. Let χ(n) = Cnbe a primitive Dirichlet character with conductor|C|>1and let the levelαβ∈N0\Nbe fixed and of the form Qκ

i=1

peii, wherepiis a prime number andei is inN0. Suppose furthermore that pejj ∈N0\Nis a positive divisor ofαβfor some1≤jκ. If the conductor ofχ(n)is a positive divisor ofpj and hence of the levelαβ, thenχ(n) = Cn

is an annihilator of E40(αβ)).

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3. EvaluatingW(α,β)(n), whereαβ∈N0

Proof. Suppose thatαβ∈N0\Nis fixed and of the form Qκ

i=1

peii, wherepi is a prime number andei is inN0. As an immediate consequence of the structure ofαβthere exists 1≤jκsuch thatpejj ∈N0\Nis a positive divisor ofαβ. Now, it is well- known that for each 1≤fej it holds thatpfj is a positive divisor ofpejj. Since the conductor ofχ(n) is a positive divisor of the levelpjej, there exists 1≤fej such thatpfj =|C|. On the other hand, it holds that

χ(n) = C n

!

=





0 if gcd(|C|, n),1,

nonzero otherwise. (20)

For each 1< δ∈ D(pejj) it holds that gcd(|C|, δ),1; hence, Cn

= 0 for each 1< δ∈ D(pejj) such that gcd(|C|, δ),1. Since the conductor ofχ(n) is greater than one, it follows thatC0= 0. Then we apply (11) we obtain

Mχ(qδ) =

X

n=1

χ(n)σ3 n

δ

qn.

Since it also holds that σ3

n δ

=





0 if nδ<N0,

nonzero gcd(|C|, n),1, (21)

we obtain the stated result by simply putting altogether; that isMχ(qδ) = 0 for all

1< δ∈ D(pjej). □

Ifαβ∈Nholds, then the primitive Dirichlet characters are trivial. Therefore, the setCis empty. Hence, the case whereαβ∈Nholds is a special case of the following theorem.

Theorem 4 – (a) LetCbe a set of primitive Dirichlet characters such that for each χ(n)∈ C it holds that χ(n) is not an annihilator ofE40(αβ)). Then the set BE={M(qt)|t∈ D(αβ)} ∪S

χ∈C

{Mχ(qt)|tDχ(αβ)}is a basis ofE40(αβ)).

(b) Let1≤imS be positive integers,δD(αβ)and(r(i, δ))i,δ be a table of the powers ofη(δ z). Let furthermoreBαβ,i(q) = Q

δ|αβ

ηr(i,δ)(δ z)be selected elements of S40(αβ)). Then the setBS={Bαβ,i(q)|1≤imS}is a basis ofS40(αβ)).

(c) The setBM=BE∪ BSconstitutes a basis ofM40(αβ)).

(14)

Remark 1 – (r1) EachBαβ,i(q) can be expressed in the form

P

n=1

bαβ,i(n)qn, where 1≤imSand for eachn≥1 the coefficientbαβ,i(n) is an integer.

(r2) If we divide the sum that results from Theorem 1 (v), whend=N, by 24, then we obtain the smallest positive degree ofqinBαβ,i(q).

The existence of a basis of the space of cusp forms in terms of theta series has been proved by A. Pizer18 for all square-free levels. M. Newman and G. Ligozat, see Theorem 1 (i)–(v), have determined a method to find as many theta series belonging to a space of cusp forms as possible. Therefore, the proof of this theorem is essentially restricted to show that the selected elements of the space of modular forms of the given level are linearly independent.

Proof. We only consider the case whereαβis inN0\Nsince the caseαβ∈Nis proved similarly using the fact thatC=∅.

(a) W. A. Stein19 has shown that for eacht positive divisor ofαβ it holds that M(qt) is inM40(t)). SinceM40(t)) is a vector space and the setCof primitive Dirichlet characters does not annihilatesE40(αβ)), it also holds for each primitive Dirichlet characterχ(n) = Cn∈ CandtDχ(αβ) that 0,Mχ(qt) is inM40(t)). Since the dimension ofE40(αβ)) is finite, it suffices to show that BEis linearly independent. Suppose that for eachχ∈ C, sDχ(αβ) we have z(χ)s∈Cand that for eachtD(αβ) we havext∈C. Then

X

t|αβ

xtM(qt) +X

χ∈C

X

sDχ(αβ)

z(χ)sMχ(qs)

!

= 0.

We recall thatχis a Legendre-Jacobi-Kronecker symbol; therefore, for all 0,a∈Zit holds that a0= 0. Since the primitive Dirichlet charactersχandψ are not trivial and have the conductorsLandRwhich we may assume greater than one, we can deduce thatC0= 0 in (11). Then we equate the coefficients ofqn fornD(αβ)∪ S

χ∈C

{s|sDχ(αβ)}to obtain the homogeneous system of linear equations inmEunknowns:

X

u|αβ

σ3 t

u

xu+X

χ∈C

X

vDχ(αβ)

χ(t)σ3 t

v

Z(χ)v= 0, tD(αβ).

The determinant of the matrix of this homogeneous system of linear equations is not zero. Hence, the unique solution isxt=z(χ)s= 0 for alltD(αβ) and for allχ∈ C, sDχ(αβ). So, the setBEis linearly independent and hence is a basis ofE40(αβ)).

18Pizer, 1976, “The representability of modular forms by theta series”.

(15)

3. EvaluatingW(α,β)(n), whereαβ∈N0

(b) We show that each Bαβ,i(q), where 1 ≤imS, is in the spaceS40(αβ)).

This is obviously the case sinceBαβ,i(q),1≤imS,are obtained using an exhaustive search which applies items (i)–(v′) in Theorem 1.

Since the dimension ofS40(αβ)) is finite, it suffices to show that the setBS is linearly independent. Suppose that xi ∈Cand

mS

P

i=1

xiBαβ,i(q) = 0. Then

mS

P

i=1

xiBαβ,i(q) =

P

n=1

(

mS

P

i=1

xibαβ,i(n) )qn= 0 which gives the homogeneous system ofmS linear equations inmSunknowns:

mS

X

i=1

bαβ,i(n)xi = 0, 1≤nmS. (22) Two cases arise:

The smallest degree ofBαβ,i(q)isifor each1≤imS Then the square ma- trix which corresponds to this homogeneous system ofmS linear equa- tions is triangular with 1’s on the diagonal. Hence, the determinant of that matrix is 1 and so the unique solution isxi = 0 for all 1≤imS. The smallest degree ofBαβ,i(q)isifor1≤i < mS Letn be the largest posi-

tive integer such that 1≤in< mS. LetB

S ={Bαβ,i(q)|1≤in}and B′′

S ={Bαβ,i(q)|n< imS}. ThenBS=B

S∪ B′′

S and we may considerBS as an ordered set. By the case above, the setB

Sis linearly independent.

Hence, the linear independence of the setBS depends on that of the setB′′

S. Let A= (bαβ,i(n)) be themS×mS matrix in (22). If det(A),0, thenxi= 0 for all 1≤imS and we are done. Suppose that det(A) = 0.

Then for somen < kmS there existsBαβ,k(q) which is causing the system of equations to be inconsistent. We substituteBαβ,k(q) with, say Bαβ,k(q), which does not occur inBS and compute the determinant of the new matrixA. Since there are finitely manyBαβ,k(q) withn< kmS that may cause the system of linear equations to be inconsistent and finitely many elements ofS40(αβ))\ BS, the procedure will terminate with a consistent system of linear equations. Hence, we will find a set of linearly independent elements ofS40(αβ)).

Therefore, the set{Bαβ,i(q)|1≤imS}is linearly independent and hence is a basis ofS40(αβ)).

(c) SinceM40(αβ)) =E40(αβ))⊕S40(αβ)), the result follows from (a) and (b).

19Stein, 2011,Modular Forms, A Computational Approach, Thrms 5.8 and 5.9, p. 86.

(16)

IfC=∅, that means that the primitive Dirichlet character is trivial, then the formulation and the proof of Theorem 4 is straightforward.

If the levelαβbelongs to the classN, then Theorem 4 (a) is provable by induction on the set of positive divisors ofαβ; see for example E. Ntienjem20. Note that each positive divisor ofαβis in the classNwhenever the levelαβbelongs to the classN. This nice property does not hold in general if the levelαβbelongs to the classN\N. For example 45 is an element of the classN\N; however, 15 which is a positive divisor of 45 does not belong toN\N.

The proof of Theorem 4 (b) provides us with an effective method to determine a basis of the space of cusp forms of levelαβwheneverαβbelongs toN0.

3.2 Evaluating the Convolution Sum W

(α,β)

(n)

We recall that it is sufficient to assume that the primitive Dirichlet characterχis not trivial since the caseχtrivial can be concluded as an immediate corollary.

Lemma 2 – Letα, β∈Nbe such thatgcd(α, β) = 1. Let furthermoreBM=BE∪ BSbe a basis ofM40(αβ)). Then there existXδ, Z(χ)s, Yj∈C, where1jmS, χ∈ C, sDχ(αβ)andδ|αβ, such that

(α L(qα)−β L(qβ))2=X

δ|αβ

Xδ+X

χ∈C

X

sDχ(αβ)

C0Z(χ)s

+

X

n=1

240X

δ|αβ

σ3

n δ

Xδ+X

χ∈C

X

sDχ(αβ)

σ3

n s

χ(n)Z(χ)s+

mS

X

j=1

bj(n)Yj

! qn.

(23) Proof. That (αL(qα)−βL(qβ))2M40(αβ)) follows from Lemma 1. Hence, by Theorem 4 (c), there exist Xδ, Z(χ)s, Yj ∈ C,1jmS, χ∈ C, sDχ(αβ) andδis a divisor ofαβ, such that

(αL(qα)−βL(qβ))2=X

δ|αβ

XδM(qδ) +X

χ∈C

X

sDχ(αβ)

Z(χ)sMχ(qs) +

mS

X

j=1

YjBj(q)

=X

δ|αβ

Xδ+X

χ∈C

X

sDχ(αβ)

C0Z(χ)s+

X

n=1

240X

δ|αβ

σ3 n

δ

Xδ

+X

χ∈C

X

sDχ(αβ)

χ(n)σ3

n s

Z(χ)s+

mS

X

j=1

bj(n)Yj

! qn.

20Ntienjem, 2017a, “Evaluation of the Convolution Sum involving the Sum of Divisors Function for 22, 44 and 52”.

(17)

3. EvaluatingW(α,β)(n), whereαβ∈N0

We equate the right hand side of (23) with that of (12) to obtain

X

n=1

240X

δ|αβ

Xδσ3 n

δ

+X

χ∈C

X

sDχ(αβ)

χ(n)σ3 n

s

Z(χ)s

! +

mS

X

j=1

Yjbj(n)

! qn

=

X

n=1

240α2σ3 n

α

+ 240β2σ3 n β

!

+ 48α(β−6n)σ n

α

+ 48β(α−6n)σ n β

!

−1152αβ W(α,β)(n)

! qn.

We then take the coefficients ofqn such thatnis inD(αβ) and 1nmS, but as many as the unknown,X1, . . . , Xαβ,Z(χ)sfor allχ∈ C, sDχ(αβ), andY1, . . . , YmS, to obtain a system ofmE+mSlinear equations whose unique solution determines the values of the unknowns. Hence, we obtain the result. □ For the following theorem, let for the sake of simplicityXδ, Z(χ)sandYjstand for their values obtained in the previous lemma.

Theorem 5 – Letnbe a positive integer. Then W(α,β)(n) =− 5

24αβ X

δ|αβ δ,α, β

σ3 n

δ

Xδ− 1 1152αβ

X

χ∈C

X

sDχ(αβ)

Z(χ)sσ3 n

s

+ 5

24α β2Xα)σ3 n

α

+ 5

24α β2Xβ3 n β

!

mS

X

j=1

1

1152αβbj(n)Yj+ 1 24− 1

n

! σ

n α

+ 1

24− 1 4αn

σ n

β

! .

Proof. We equate the right hand side of (23) with that of (12) to yield 1152αβ W(α,β)(n) =−240X

δ|αβ

σ3 n

δ

Xδ−X

χ∈C

X

sDχ(αβ)

Z(χ)sσ3 n

s

mS

X

j=1

bj(n)Yj+ 240α2σ3

n α

+ 240β2σ3 n β

!

+ 48α(β−6n)σ n

α

+ 48β(α−6n)σ n β

! .

We then solve forW(α,β)(n) to obtain the stated result. □

(18)

Remark 2 – (a) We observe that the following part of Theorem 5 depends only onn,αandβbut not on the basis of the modular spaceM40(αβ)):

1 24− 1

n

! σ

n α

+ 1

24− 1 4αn

σ n

β

! .

(b) For allχ∈ Cand for allsDχ(αβ) the value ofZ(χ)sappears to be zero in all explicit examples evaluated as yet. Will the value ofZ(χ)salways vanish for allαβbelonging toN0\N?

We now have the prerequisite to determine a formula for the number of repre- sentations of a positive integernby an octonary quadratic form.

4 Number of Representations of a Positive Integer for this Class of Levels

We discuss in this section the determination of formulae for the number of represen- tations of a positive integer by the octonary quadratic forms (3) and (4), respectively.

4.1 Representations of a Positive Integer by the Octonary Quadratic Form (3)

We determine formulae for the number of representations of a positive integer by the octonary quadratic forms (3).

Formulae for the Number of Representations by (3)

Letn∈Nand let the number of representations ofnby the quaternary quadratic form P4

i=1

x2i be defined byr4(n) = card({(x1, x2, x3, x4)∈Z4 |n= P4

i=1

xi2}). It follows from the definition thatr4(0) = 1. Jacobi’s identityr4(n), n∈N0, is

r4(n) = 8σ(n)−32σ n

4

, (24)

a proof of which is given in K. S. Williams21.

Now, let the number of representations ofnby the octonary quadratic form (3) be

N(a,b)(n) = card (

(x1, x2, x3, x4, x5, x6, x7, x8)∈Z8

n=a X4

i=1

x2i +b X8

i=5

xi2 )!

, wherea, b∈N0.

21Williams, 2011,Number Theory in the Spirit of Liouville, Thrm 9.5, p. 83.

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