# Evaluation of the convolution sums ∑al+bm=nlσ(l) σ(m) with ab ≤ 9

Yoon Kyung Park 1
• 1 Institute of Mathematical Sciences, Ewha Womans University, 52, Seodaemun-gu, Seoul, South Korea
Yoon Kyung Park
• Corresponding author
• Institute of Mathematical Sciences, Ewha Womans University, 52, Seodaemun-gu, Seoul, 03760, South Korea
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## Abstract

The generating functions of divisor functions are quasimodular forms of weight 2 and their products belong to a space of quasimodular forms of higher weight. In this article, we evaluate the convolution sums

$∑al+bm=nlσ(l)σ(m)$

for all positive integers a, b and n with ab ≤ 9 and gcd(a, b) = 1.

## 1 Introduction

Let a and b be positive integers and let ν be a nonnegative integer. Define $Wa,b(ν)$(n) by the convolution sum

$Wa,b(ν)(n):=∑l,m≥1al+bm=nlvσ(l)σ(m),$

where σ(n) := σ1(n) and

$σs(n):=∑d|nds$

for any positive integers s and n. Denote Wa,b(n) = $Wa,b(0)$(n).

This is a specialized form of the convolution sum which Lahiri introduced in :

$S[(r1,…,rt),(s1,…,st),(a1,…,at)](n)=∑m1,…,mt∈Z>0a1m1+⋯+atmt=nm1r1⋯mtrtσs1(m1)⋯σst(mt)$

and clearly,

$Wa,b(ν)(n)=S[(ν,0),(1,1),(a,b)](n).$

The convolution sum Wa,b(n) has been studied since the mid-nineteenth century. The following table contains references for it:

Mathematicians (year)cases
Besge(1862) , Glaisher(1885) , Ramanujan (1916) ab = 1
Huard-Ou-Spearman-Williams (2002) ab = 2, 3, 4
Lemire-Williams (2005) , Cooper-Toh (2009) ab = 5, 7
Alaca-Williams (2007) ab = 6
Williams (2008-9) [9, 10]ab = 8, 9
Royer (2007) ab = 11, 13 and (a, b) = (1, 10), (1, 14)
Alaca-Alaca-Williams (2006-8) [12, 13, 14, 15]ab = 12, 16, 18, 24
Ramakrishnan-Sahu (2013) ab = 15
Cooper-Ye (2014) ab = 20 and (a, b) = (2, 5)
Chan-Cooper (2008) ab = 23
Xia-Tian-Yao (2014) ab = 25
Ntienjem (2015, 2017) [20, 21, 22]ab = 14, 22, 26, 28, 30, 33, 40, 44, 52, 56
Alaca-Kesicioglu (2016) ab = 27, 32
Ye (2015) ab = 36

Although it may be possible to evaluate Wa,b(n) by means of identities of elementary functions for small a and b, when ab grows, it is easier to use quasimodular forms because Wa,b(n) is a coefficient of a certain quasimodular form of level ab and weight 4 and the dimension of the space of quasimodular forms is known.

In this paper we focus on the convolution sum

$Wa,b(1)(n)=∑al+bm=nlσ(l)σ(m)$

for ab ≤ 9 with (a, b) = 1 since $Wda,db(1)(n)=Wa,b(1)(n/d)$ for d | n. The convolution sum $W1,1(1)$ (n) is

$∑l+m=nlσ(l)σ(m)=n245σ3(n)+(1−6n)σ(n)$

since one can obtain $2W1,1(1)(n)=nW1,1(0)(n)$ easily.

Before stating our main theorem we need the functions δN(q) and ΔN(q) defined on the region |q| < 1:

Definition 1.1

$Δ3(q):=qE6(q)E6(q3)=∑n=1∞c3(n)qn,Δ4(q):=qE12(q2)=∑n=1∞c4(n)qn,δ5(q):=qE4(q)E4(q5)=∑n=1∞b5(n)qn,Δ5(q):=14δ5(q)5P(q5)−P(q)=∑n=1∞c5(n)qn,δ6(q):=qE2(q)E2(q2)E2(q3)E2(q6)=∑n=1∞b6(n)qn,Δ6(q):=14δ6(q)6P(q6)−3P(q3)+2P(q2)−P(q)=∑n=1∞c6(n)qn,δ7(q):=qE16(q)E8(q7)+13qE12(q)E12(q7)+49q2E8(q)E16(q7)1/3=∑n=1∞b7(n)qn,Δ7,1(q):=qE10(q)E2(q7)=∑n=1∞c7,1(n)qn,Δ7,2(q):=q2E6(q)E6(q7)=∑n=1∞c7,2(n)qn,Δ7,3(q):=q3E2(q)E10(q7)=∑n=1∞c7,3(n)qn,δ8(q):=qE4(q2)E4(q4)=∑n=1∞b8(n)qn,Δ8(q):=δ8(q)(2P(q4)−P(q2))=∑n=1∞c8(n)qn,δ9(q):=qE(q3)8=∑n=1∞b9(n)qn,Δ9(q):=18δ9(q)9P(q9)−P(q)=∑n=1∞c9(n)qn,$

where

$E(q):=∏n=1∞(1−qn),P(q):=1−24∑n=1∞σ(n)qn,Q(q):=1+240∑n=1∞σ3(n)qn,R(q):=1−504∑n=1∞σ5(n)qn.$

The following result is our main theorem.

Theorem 1.2

Let $Wa,b(1)$(n) be the modified convolution sum of divisor functions

$Wa,b(1)(n):=∑al+bm=nlσ(l)σ(m).$

Assume that a and b are positive integers with ab ≤ 9 and gcd(a, b) = 1. Then $Wa,b(1)$(n) can be explicitly expressed as a linear combination of σs(n/d), bd(n/d) and cd(n/d) for positive integers d, dwith dd′ | n for all positive integers n and s = 1, 3 or 5.

The explicit linear combinations for $Wa,b(1)$(n) are given in Theorems 3.1(a < b) and 3.2(a > b) of Section 3.

This paper is organized as follows. We recall in Section 2 the theory of quasimodular forms and find a basis for the space of quasimodular forms which we need in Section 3. In Section 3, we find the identities between the quasimodular forms and the basis (Lemma 3.3) to prove our main theorems (Theorem 3.1 and 3.2) and write $Wa,b(1)$ as the linear combination of divisor functions and the coefficients of certain cuspforms. We use the MAPLE program to find the identities between quasimodular forms.

## 2 Quasimodular forms

For a positive integer N the congruence subgroup Γ0(N) is defined as a subgroup of SL2(ℤ) by

$Γ0(N):=abcd∈SL2(Z):c≡0(modN).$

We use the theory of quasimodular forms to prove our main theorem. One can refer to  for more details. The space $M~k(≤k/2)$0(N)) of quasimodular forms of weight k and depth ≤k/2 on Γ0(N) has the following structure relationship:

$M~k(≤k/2)(Γ0(N))=⊕j=0k/2−1DjMk−2j(Γ0(N))⊕,$

where D is the differential operator defined by $D:=qddq$ and D(∑n≥0 cnqn) = ∑n≥0 ncnqn. Thus, the space $M~6(≤3)$0 (N)) of quasimodular forms of level N of weight 6 and depth ≤ 3 is

$M~6(≤3)(Γ0(N))=M6(Γ0(N))⊕DM4(Γ0(N))⊕D2M2(Γ0(N))⊕$

for any positive integer N. More explicitly, $M~6(≤3)$0(N)) is spanned by the functions D2P(qd), DQ(qd), R(qd), d(qd) and Δd(qd) for positive divisors d and d′ of N with dd′ | N. Hence we get the following basis.

Lemma 2.1

For N = 2, …, 9, let 𝓑N be the set of quasimodular forms defined by

$B2=D2P(q),D2P(q2),DQ(q),DQ(q2),R(q),R(q2),B3=D2P(q),D2P(q3),DQ(q),DQ(q3),R(q),R(q3),Δ3(q),B4=D2P(q),D2P(q2),D2P(q4),DQ(q),DQ(q2),DQ(q4),R(q),R(q2),R(q4),Δ4(q),B5=D2P(q),D2P(q5),DQ(q),DQ(q5),R(q),R(q5),Dδ5(q),Δ5(q),B6=D2P(q),D2P(q2),D2P(q3),D2P(q6),DQ(q),DQ(q2),DQ(q3),DQ(q6),R(q),R(q2),R(q3),R(q6),Dδ6(q),Δ3(q),Δ3(q2),Δ6(q),B7=D2P(q),D2P(q7),DQ(q),DQ(q7),R(q),R(q7),Dδ7(q),Δ7,1(q),Δ7,2(q),Δ7,3(q),B8=D2P(q),D2P(q2),D2P(q4),D2P(q8),DQ(q),DQ(q2),DQ(q4),DQ(q8),R(q),R(q2),R(q4),R(q8),Dδ8(q),Δ4(q),Δ4(q2),Δ8(q),B9=D2P(q),D2P(q3),D2P(q9),DQ(q),DQ(q3),DQ(q9),R(q),R(q3),R(q9),Dδ9(q),Δ3(q),Δ3(q3),Δ9(q).$

Then, 𝓑N is a basis for the space $M~6(≤3)$0(N)).

Proof

Note that the dimension of the space $M~6(≤3)$0(N)) is

$dim⁡M~6(≤3)(Γ0(N))=1+∑j=02dimM6−2j(Γ0(N)).$

By [26, Proposition 6.1], we have

 N 2 3 4 5 6 7 8 9 dim $M~6(≤3)$(Γ0(N)) 6 7 10 8 16 10 16 13

It is clear that P(qd) ∈ $M~2(≤1)$0(N)), Q(qd) ∈ M40 (N)), R(qd) ∈ M60(N)) for a positive divisor d of N. Assume that N is an integer with 2 ≤ N ≤ 9. Let 0 (N)) be the subspace of $M~6(≤3)$0 (N)) spanned by the set

$BN(E):=R(qd),DQ(qd),D2P(qd):0

of dimension 3 ⋅ (∑d|N 1) = 3 σ0 (N). When 0 < dd′ | N, the functions δd(qd) and Δd(qd) defined in Definition 1.1 are modular forms of weight 4 and 6 of Γ0(N), respectively. Moreover, the functions Δ7,j(q) are modular forms of weight 6 of Γ0 (7) for j = 1, 2, 3. In other words,

$δd(qd′)∈M~6(≤3)(Γ0(N)),Δd(qd′)∈M6(Γ0(N)) and Δ7,j(q)∈M6(Γ0(7)),$

where dd′ | N and j = 1, 2, 3. It is easily checked that the set 𝓑N is linearly independent for each N by the help of the q-expansions of the functions in Appendix.

Since

$dimM~6(≤3)(Γ0(N))−dim⁡E~(Γ0(N))=0,1,1,2,4,4,4,4$

for N = 2, 3, 4, 5, 6, 7, 8, 9, respectively, the set 𝓑N is a basis of the space $M~6(≤3)$0 (N)).□

Remark 2.2

Let $WN=(0−1N0).$ In the theory of modular forms, the functions δN(q) and ΔN(q) are newforms with δN | WN = δN and ΔN|WN = −ΔN when N = 2, 3, 4, 5, 6 and 8.

For N = 7, Δ7,1(q), Δ7,2(q) and Δ7,3(q) are echelon forms. Instead of them,

$f7(q):=Δ7,1(q)−49Δ7,3(q),f7,±(q):=Δ7,1(q)+29±572Δ7,2(q)+49Δ7,3(q)$

are normalized newforms of level 7 with f7|W7 = f7 and (f7,±)|W7 = −f7,±. Furthermore, the modular form $−13Δ3(q)−9Δ3(q3)+43Δ9(q)$ is the normalized newform of Γ0(9) with eigenvalue −1 under the action of W9.

## 3 Proofs of main results

By using the theory of quasimodular forms we prove Theorem 3.1 and 3.2. These are the explicit linear combinations of Theorem 1.2.

Theorem 3.1

Let n be a positive integer. Then

1. $∑l+2m=nlσ(l)σ(m)=n24σ3(n)+n6σ3(n2)+n−2n224σ(n)−n212σ(n2),$
2. $∑l+3m=nlσ(l)σ(m)=n48σ3(n)+3n16σ3(n3)+3n−4n272σ(n)−n212σ(n3)−1144c3(n),$
3. $∑l+4m=nlσ(l)σ(m)=n96σ3(n)+n32σ3(n2)+n6σ3(n4)+n−n224σ(n)−n212σ(n4)−196c4(n),$
4. $∑l+5m=nlσ(l)σ(m)=5n624σ3(n)+125n624σ3(n5)+5n−4n2120σ(n)−n212σ(n5)−n260b5(n)−180c5(n),$
5. $∑l+6m=nlσ(l)σ(m)=n240σ3(n)+n60σ3(n2)+3n80σ3(n3)+3n20σ3(n6)+3n−2n272σ(n)−n212σ(n6)−n240b6(n)−1144c3(n)−118c3(n2)−1144c6(n),$
6. $∑2l+3m=nlσ(l)σ(m)=n480σ3(n)+n120σ3(n2)+3n160σ3(n3)+3n40σ3(n6)+3n−4n2144σ(n2)−n248σ(n3)−n480b6(n)−1288c3(n)−136c3(n2)+1288c6(n),$
7. $∑l+7m=nlσ(l)σ(m)=n240σ3(n)+49n240σ3(n7)+7n−4n2168σ(n)−n212σ(n7)−n140b7(n)−5336c7,1(n)−1784c7,2(n)−3548c7,3(n),$
8. $∑l+8m=nlσ(l)σ(m)=n384σ3(n)+n128σ3(n2)+n32σ3(n4)+(n6)σ3(n8)+2n−n248σ(n)−n212σ(n8)−n128b8(n)−1128c4(n)−116c4(n2)−1128c8(n),$
9. $∑l+9m=nlσ(l)σ(m)=n432σ3(n)+n54σ3(n3)+3n16σ3(n9)+9n−4n2216σ(n)−n212σ(n9)−n108b9(n)+1432c3(n)+116c3(n3)−154c9(n).$

For ν > 0, $Wa,b(ν)$ is not symmetric on (a, b) and we obtain results for $Wa,b(1)$ (a > b) in the following theorem.

Theorem 3.2

Let n be a positive integer. Then

1. $∑2l+m=nlσ(l)σ(m)=n48σ3(n)+n12σ3(n2)−n248σ(n)+n−4n248σ(n2),$
2. $∑3l+m=nlσ(l)σ(m)=n144σ3(n)+n16σ3(n3)−n2108σ(n)+n−4n272σ(n3)+1432c3(n),$
3. $∑4l+m=nlσ(l)σ(m)=n384σ3(n)+n128σ3(n2)+n24σ3(n4)−n2192σ(n)+n−4n296σ(n4)+1384c4(n),$
4. $∑5l+m=nlσ(l)σ(m)=n624σ3(n)+25n624σ3(n5)−n2300σ(n)+n−4n2120σ(n5)−n1300b5(n)+1400c5(n),$
5. $∑6l+m=nlσ(l)σ(m)=n1440σ3(n)+n360σ3(n2)+n160σ3(n3)+n40σ3(n6)−n2432σ(n)+n−4n2144σ(n6)−n1440b6(n)+1864c3(n)+1108c3(n2)+1864c6(n),$
6. $∑3l+2m=nlσ(l)σ(m)=n720σ3(n)+n180σ3(n2)+n80σ3(n3)+n20σ3(n6)−n2108σ(n2)+n−2n272σ(n3)−n720b6(n)+1432c3(n)+154c3(n2)−1432c6(n),$
7. $∑7l+m=nlσ(l)σ(m)=n1680σ3(n)+7n240σ3(n7)−n2588σ(n)+n−4n2168σ(n7)−n980b7(n)+52352c7,1(n)+17588c7,2(n)+548c7,3(n),$
8. $∑8l+m=nlσ(l)σ(m)=n3072σ3(n)+n1024σ3(n2)+n256σ3(n4)+n48σ3(n8)−n2768σ(n)+n−4n2192σ(n8)−n1024b8(n)+11024c4(n)+1128c4(n2)+11024c8(n),$
9. $∑9l+m=nlσ(l)σ(m)=n3888σ3(n)+n486σ3(n3)+n48σ3(n9)−n2972σ(n)+n−4n2216σ(n9)−n972b9(n)−13888c3(n)−1144c3(n3)+1486c9(n).$

Lemma 3.3 gives nine identities involving the functions in Definition 1.1. Using it, Theorems 3.1 and 3.2 are obtained by equating coefficients of qn.

Lemma 3.3

Since DP(qa) P(qb) is an element of $M~6(≤3)$0(ab)), we have the following identities:

1. $DP(q)P(q2)=2D2P(q)+8D2P(q2)+110DQ(q)+45DQ(q2),$
2. $DP(q)P(q3)=43D2P(q)+18D2P(q3)+120DQ(q)+2720DQ(q3)−4Δ3(q),$
3. $DP(q)P(q4)=D2P(q)+32D2P(q4)+140DQ(q)+320DQ(q2)+85DQ(q4)−6Δ4(q),$
4. $DP(q)P(q5)=45D2P(q)+50D2P(q5)+152DQ(q)+12552DQ(q5)−14465Dδ5(q)−365Δ5(q),$
5. $DP(q)P(q6)=23D2P(q)+72D2P(q6)+1100DQ(q)+225DQ(q2)+27100DQ(q3)+5425DQ(q6)−125Dδ6(q)−4Δ3(q)−32Δ3(q2)−4Δ6(q),$
6. $DP(q2)P(q3)=83D2P(q2)+92D2P(q3)+1200DQ(q)+125DQ(q2)+27200DQ(q3)+2725DQ(q6)−65Dδ6(q)−2Δ3(q)−16Δ3(q2)+2Δ6(q),$
7. $DP(q)P(q7)=47D2P(q)+98D2P(q7)+1100DQ(q)+343100DQ(q7)−14435Dδ7(q)−607Δ7,1(q)−8167Δ7,2(q)−420Δ7,3(q),$
8. $DP(q)P(q8)=12D2P(q)+128D2P(q8)+1160DQ(q)+380DQ(q2)+310DQ(q4)+165DQ(q8)−92Δ4(q)−36Δ4(q2)−92Δ8(q),$
9. $DP(q)P(q9)=49D2P(q)+162D2P(q9)+1180DQ(q)+215DQ(q3)+8120DQ(q9)−163Dδ9(q)+43Δ3(q)+36Δ3(q3)−323Δ9(q).$

Proof

By comparing the coefficients of DP(qa)P(qb) with ones of the basis of $M~6(≤3)$0(ab))((a, b) = (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (1, 7), (1, 8), (1, 9)) we gave in Lemma 2.1 we can complete our statement.□

Proof of Theorem 3.1

It is easy to see that

$DP(qa)P(qb)=∑n=1∞−24naσ(na)+576Wa,b(1)(n)qn.$

Since the right hand sides of (1)-(9) are written as a linear combination of the functions in Definition 1.1, our theorem is proved.□

Lemma 3.4

For any relatively prime positive integers a, b, let

$S[(ν,μ),(1,1),(a,b)](n)=∑l,m∈Z>0al+bm=nlνmμσ(l)σ(m).$

Then

$S[(0,0),(1,1),(a,b)](n)=n−ν∑t=0ννtaν−tbtS[(ν−t,t),(1,1),(a,b)](n).$

In particular, we have

$aWa,b(1)(n)+bWb,a(1)(n)=nWa,b(0)(n).$

Proof

The proof is easy using the binomial theorem:

$nνS[(0,0),(1,1),(a,b)](n)=∑l,m∈Z>0al+bm=n(al+bm)νσ(l)σ(m)=∑l,m∈Z>0al+bm=n∑t=0ννtaν−tbtlν−tmtσ(l)σ(m)=∑t=0ννtaν−tbtS[(ν−t,t),(1,1),(a,b)](n).$

If ν = 1, then we have that

$aWa,b(1)(n)+bWb,a(1)(n)=nWa,b(0)(n).$

Hence we need the evaluation of the convolution sums Wa,b(n) which already occur in the literature in order to deduce Theorem 3.2.

Lemma 3.5

1. $∑l+2m=nσ(l)σ(m)=112σ3(n)+13σ3(n2)+1−3n24σ(n)+1−6n24σ(n2),$
2. $∑l+3m=nσ(l)σ(m)=124σ3(n)+38σ3(n3)+1−2n24σ(n)+1−6n24σ(n3),$
3. $∑l+4m=nσ(l)σ(m)=148σ3(n)+116σ3(n2)+13σ3(n4)+2−3n48σ(n)+1−6n24σ(n4),$
4. $∑l+5m=nσ(l)σ(m)=5312σ3(n)+125312σ3(n5)+5−6n120σ(n)+1−6n24σ(n5)−1130b5(n),$
5. $∑l+6m=nσ(l)σ(m)=1120σ3(n)+130σ3(n2)+340σ3(n3)+310σ3(n6)+1−n24σ(n)+1−6n24σ(n6)−1120b6(n),$
6. $∑2l+3m=nσ(l)σ(m)=1120σ3(n)+130σ3(n2)+340σ3(n3)+310σ3(n6)+1−2n24σ(n2)+1−3n24σ(n3)−1120b6(n).$
7. $∑l+7m=nσ(l)σ(m)=1120σ3(n)+49120σ3(n7)+7−6n168σ(n)+1−6n24σ(n7)−170b7(n),$
8. $∑l+8m=nσ(l)σ(m)=1192σ3(n)+164σ3(n2)+116σ3(n4)+13σ3(n8)+4−3n96σ(n)+1−6n24σ(n8)−164b8(n),$
9. $∑l+9m=nσ(l)σ(m)=1216σ3(n)+127σ3(n3)+38σ3(n9)+3−2n72σ(n)+1−6n24σ(n9)−154b9(n).$

Proof

The evaluations of Wa,b(n) = ∑al+bm=n σ(l) σ(m) are in the following references: (1)-(3) in , (4) in (4) is in [6, 7] and (5) and (6) are in .□

Proof of Theorem 3.2

All formulas are linear combinations for $Wb,a(1)$(n) satisfying a < b, (a, b) = 1 and ab ≤ 9.

$Wa,b(0)(n) and Wa,b(1)(n)$ are obtained in Lemma 3.5 and Theorem 3.1, respectively. Additionally, by Lemma 3.4, that is,

$Wb,a(1)(n)=nbWa,b(0)(n)−abWa,b(1)(n)$

we prove our theorem.□

Acknowledgement

The author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03029519).

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Alaca A., Alaca Ş. and Williams K. S., Evaluation of the convolution sums ∑l+24m=n σ(l) σ(m) and ∑3l+8m=n σ(l) σ(m), Math. J. Okayama Univ. 49 (2007) 93-111.

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Alaca A., Alaca Ş. and Williams K. S., The convolution sum ∑m<n/16 σ(m) σ (n−16m), Canad. Math. Bull. 51 (2008) 3-14.

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Ramakrishnan B. and Sahu B., Evaluation of the convolution sums ∑l+15m=n σ(l) σ(m) and ∑3l+5m=n σ(l) σ(m) and an application, Int. J. Number Theory 9 (3) (2013) 799-809.

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• Export Citation
• 

Cooper S. and Ye D., Evaluation of the convolution sums ∑l+20m=n σ(l) σ(m), ∑4l+5m=n σ(l) σ(m) and ∑2l+5m=n σ(l) σ(m), Int. J. Number Theory 10 (6) (2014) 1385-1394.

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• Export Citation
• 

Chan H. H. and Cooper S., Powers of theta functions, Pacific J. Math. 235 (2008) 1-14.

• Crossref
• Export Citation
• 

Xia E. X. W., Tian X. L. and Yao O. X. M., Evaluation of the convolution sums ∑i+25j=n σ(i)σ(j), Int. J. Number Theory 10 (6) (2014) 1421-1430.

• Crossref
• Export Citation
• 

Ntienjem E., Evaluation of the convolution sums ∑αl+βm=n σ(l) σ(m), where (α, β) is in {(1, 14), (2, 7), (1, 26), (2, 13), (1, 28), (4, 7), (1, 30), (5, 6)}, M. Sc. thesis, Carleton University, Ottawa, Ontario, Canada, 2015.

• 

Ntienjem E., Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52. Open Math. 15 (2017), 446-458.

• Crossref
• Export Citation
• 

Ntienjem E., Elementary evaluation of convolution sums involving the sum of divisors function for a class of positive integers, preprint.

• 

Alaca Ş. and Kesicioğlu Y., Evaluation of the convolution sums ∑l+27m=n σ(l) σ(m) and ∑l+32m=n σ(l) σ(m), Int. J. Number Theory 12 (1) (2016) 1-13.

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• 

Ye D., Evaluation of the convolution sums ∑l+36m=n σ(l) σ(m), ∑4l+9m=n σ(l) σ(m), Int. J. Number Theory 11 (1) (2015) 171-183.

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• 

Kaneko M. and Zagier D., A generalized Jacobi theta function and quasimodular forms, in: The Moduli Spaces of Curves, vol. 129, Birkhäuser, Boston, MA, 1995, 165-172.

• 

Stein W., Modular Forms: a Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, 2007.

Appendix. q-expansions of the function defined in Definition 1.1

We give the coefficients of the functions defined in Definition 1.1 up to q30:

$Δ3(q)=q−6q2+9q3+4q4+6q5−54q6−40q7+168q8+81q9−36q10−564q11+36q12+638q13+240q14+54q15−1136q16+882q17−486q18−556q19+24q20−360q21+3384q22−840q23+1512q24−3089q25−3828q26+729q27−160q28+4638q29−324q30+O(q31),Δ4(q)=q−12q3+54q5−88q7−99q9+540q11−418q13−648q15+594q17+836q19+1056q21−4104q23−209q25+4104q27−594q29+O(q31),δ5(q)=q−4q2+2q3+8q4−5q5−8q6+6q7−23q9+20q10+32q11+16q12−38q13−24q14−10q15−64q16+26q17+92q18+100q19−40q20+12q21−128q22−78q23+25q25+152q26−100q27+48q28−50q29+40q30+O(q31),Δ5(q)=q+2q2−4q3−28q4+25q5−8q6+192q7−120q8−227q9+50q10−148q11+112q12+286q13+384q14−100q15+656q16−1678q17−454q18+1060q19−700q20−768q21−296q22+2976q23+480q24+625q25+572q26+1880q27−5376q28−3410q29−200q30+O(q31),δ6(q)=q−2q2−3q3+4q4+6q5+6q6−16q7−8q8+9q9−12q10+12q11−12q12+38q13+32q14−18q15+16q16−126q17−18q18+20q19+24q20+48q21−24q22+168q23+24q24−89q25−76q26−27q27−64q28+30q29+36q30+O(q31),Δ6(q)=q+4q2−9q3+16q4−66q5−36q6+176q7+64q8+81q9−264q10−60q11−144q12−658q13+704q14+594q15+256q16−414q17+324q18+956q19−1056q20−1584q21−240q22+600q23−576q24+1231q25−2632q26−729q27+2816q28+5574q29+2376q30+O(q31),δ7(q)=q−q2−2q3−7q4+16q5+2q6−7q7+15q8−23q9−16q10−8q11+14q12+28q13+7q14−32q15+41q16+54q17+23q18−110q19−112q20+14q21+8q22+48q23−30q24+131q25−28q26+100q27+49q28−110q29+32q30+O(q31),Δ7,1(q)=q−10q2+35q3−30q4−105q5+238q6−262q8−145q9+70q10+1114q11−560q12−1071q13−196q15+2502q16+140q17−2078q18−735q19−868q20+2401q21+1012q22−2684q23+2100q24+501q25−1638q26+2786q27−4802q28+1556q29−392q30+O(q31),Δ7,2(q)=q2−6q3+9q4+10q5−30q6+11q8+36q9+36q10−124q11−42q12+126q13+49q14+24q15−243q16−76q17+441q18−18q19−56q20−294q21−360q22+568q23−6q24−180q25+392q26−324q27+441q28+252q29−720q30+O(q31),Δ7,3(q)=q3−2q4−q5+2q6+q7+2q8−2q9−10q10+18q11+8q12−19q13−10q14−20q15+22q16+38q17−52q18−13q19+60q20+35q21+68q22−92q23−60q24+10q25−62q26−26q27−30q28−38q29+152q30+O(q31),δ8(q)=q−4q3−2q5+24q7−11q9−44q11+22q13+8q15+50q17+44q19−96q21−56q23−121q25+152q27+198q29+O(q31),Δ8(q)=q+20q3−74q5−24q7+157q9+124q11+478q13−1480q15−1198q17+3044q19−480q21+184q23+2351q25−1720q27−3282q29+O(q31),δ9(q)=q−8q4+20q7−70q13+64q16+56q19−125q25−160q28+O(q31),Δ9(q)=q+3q2+9q3+4q4−3q5−54q6−40q7−84q8+81q9−36q10+282q11+36q12+638q13−120q14+54q15−1136q16−441q17−486q18−556q19−12q20−360q21+3384q22+420q23+1512q24−3089q25+1914q26+729q27−160q28−2319q29−324q30+O(q31).$

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Ramanujan S., On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916) 159-184.

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Huard J. G., Ou Z. M., Spearman B. K. and Williams K. S., Elementary evaluation of certain convolution sums involving divisor functions, in: Number Theory for the Millennium, II, A.K. Peters, Natick, MA. 2002, 229-274.

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Lemire M. and Williams K. S., Evaluation of two convolution sums involving the sum of divisor functions, Bull. Aust. Math. Soc. 73 (2005) 107-115.

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Cooper S. and Toh P.C., Quintic and septic Eisenstein series, Ramanujan J. 19 (2009) 163-181.

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Alaca Ş. and Williams K. S., Evaluation of the convolution sums ∑l+6m=n σ(l) σ(m) and ∑2l+3m=n σ(l)σ(m), J. Number Theory 124 (2007) 491-510.

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• 

Williams K. S., The convolution sum ∑m<n/9 σ(m) σ(n − 9m), Int. J. Number Theory 1 (2) (2005) 193-205.

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Williams K. S., The convolution sum ∑m<n/8 σ(m) σ(n−8m), Pacific J. Math. 228 (2006) 387-396.

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• 

Royer E., Evaluating the convolution sums of the divisor function by quasimodular forms, Int. J. Number Theory 3 (2) (2007) 231-261.

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• 

Alaca A., Alaca Ş. and Williams K. S., Evaluation of the convolution sums ∑l+12m=n σ(l) σ(m) and ∑3l+4m=n σ(l) σ(m), Adv. Theor. Appl. Math. 1 (2006) 27-48.

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Alaca A., Alaca Ş. and Williams K. S., Evaluation of the convolution sums ∑l+18m=n σ(l) σ(m) and ∑2l+9m=n σ(l) σ(m), Int. Math. Forum 2 (2007) 45-68.

• 

Alaca A., Alaca Ş. and Williams K. S., Evaluation of the convolution sums ∑l+24m=n σ(l) σ(m) and ∑3l+8m=n σ(l) σ(m), Math. J. Okayama Univ. 49 (2007) 93-111.

• 

Alaca A., Alaca Ş. and Williams K. S., The convolution sum ∑m<n/16 σ(m) σ (n−16m), Canad. Math. Bull. 51 (2008) 3-14.

• Crossref
• Export Citation
• 

Ramakrishnan B. and Sahu B., Evaluation of the convolution sums ∑l+15m=n σ(l) σ(m) and ∑3l+5m=n σ(l) σ(m) and an application, Int. J. Number Theory 9 (3) (2013) 799-809.

• Crossref
• Export Citation
• 

Cooper S. and Ye D., Evaluation of the convolution sums ∑l+20m=n σ(l) σ(m), ∑4l+5m=n σ(l) σ(m) and ∑2l+5m=n σ(l) σ(m), Int. J. Number Theory 10 (6) (2014) 1385-1394.

• Crossref
• Export Citation
• 

Chan H. H. and Cooper S., Powers of theta functions, Pacific J. Math. 235 (2008) 1-14.

• Crossref
• Export Citation
• 

Xia E. X. W., Tian X. L. and Yao O. X. M., Evaluation of the convolution sums ∑i+25j=n σ(i)σ(j), Int. J. Number Theory 10 (6) (2014) 1421-1430.

• Crossref
• Export Citation
• 

Ntienjem E., Evaluation of the convolution sums ∑αl+βm=n σ(l) σ(m), where (α, β) is in {(1, 14), (2, 7), (1, 26), (2, 13), (1, 28), (4, 7), (1, 30), (5, 6)}, M. Sc. thesis, Carleton University, Ottawa, Ontario, Canada, 2015.

• 

Ntienjem E., Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52. Open Math. 15 (2017), 446-458.

• Crossref
• Export Citation
• 

Ntienjem E., Elementary evaluation of convolution sums involving the sum of divisors function for a class of positive integers, preprint.

• 

Alaca Ş. and Kesicioğlu Y., Evaluation of the convolution sums ∑l+27m=n σ(l) σ(m) and ∑l+32m=n σ(l) σ(m), Int. J. Number Theory 12 (1) (2016) 1-13.

• Crossref
• Export Citation
• 

Ye D., Evaluation of the convolution sums ∑l+36m=n σ(l) σ(m), ∑4l+9m=n σ(l) σ(m), Int. J. Number Theory 11 (1) (2015) 171-183.

• Crossref
• Export Citation
• 

Kaneko M. and Zagier D., A generalized Jacobi theta function and quasimodular forms, in: The Moduli Spaces of Curves, vol. 129, Birkhäuser, Boston, MA, 1995, 165-172.

• 

Stein W., Modular Forms: a Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, 2007.

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