Download Elementary Evaluation of Convolution Sums

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
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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)).
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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.
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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 .
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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.
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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 ,