Open Math. 2017; 15: 446–458
Open Mathematics
Open Access
Research Article
Ebénézer Ntienjem*
Evaluation of the convolution sum involving
the sum of divisors function for 22, 44 and 52
DOI 10.1515/math-2017-0041
Received August 30, 2016; accepted February 28, 2017.
Abstract: The convolution sum,
P
.l/ .m/, where ˛ˇ D 22, 44, 52, is evaluated for all natural numbers
.l;m/2N2
0
˛ lCˇ mDn
n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of
level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation
of the convolution sums for ˛ˇ D 22. We then use these convolution sums to determine formulae for the number of
representations of a positive integer by the octonary quadratic forms a .x12 Cx22 Cx32 Cx42 /Cb .x52 Cx62 Cx72 Cx82 /,
where .a; b/ D .1; 11/; .1; 13/.
Keywords: Sums of Divisors function, Convolution Sums, Dedekind eta function, Modular Forms, Eisenstein Series,
Cusp Forms, Octonary quadratic Forms, Number of Representations
MSC: 11A25, 11E20, 11E25, 11F11, 11F20, 11F27
1 Introduction
Let in the sequel N, N0 , Z, Q, R and C denote the sets of positive integers, non-negative integers, integers, rational
numbers, real numbers and complex numbers, respectively.
Suppose that k; n 2 N. Then the sum of positive divisors of n to the power of k, k .n/, is defined by
X
k .n/ D
d k:
(1)
0<d jn
We write .n/ as a synonym for 1 .n/. For m … N we set k .m/ D 0.
Suppose now that ˛; ˇ 2 N are such that ˛ ˇ. Then the convolution sum, W.˛;ˇ/ .n/, is defined as follows:
X
W.˛;ˇ/ .n/ D
.l/ .m/:
(2)
.l;m/2N2
0
˛ lCˇ mDn
We write Wˇ .n/ as a synonym for W.1;ˇ/ .n/. Given ˛; ˇ 2 N, if for all .l; m/ 2 N20 it holds that ˛ l C ˇ m ¤ n
then we set W.˛;ˇ/ .n/ D 0.
For those convolution sums W.˛;ˇ/ .n/ that have so far been evaluated, the levels ˛ˇ are given in Table 1.
We discuss the evaluation of the convolution sums of level ˛ˇ D 22; 44 and ˛ˇ D 52, i.e., .˛; ˇ/ D .1; 22/,
.2; 11/, .1; 44/, .4; 11/, .1; 52/, .4; 13/. Convolution sums of these levels have not been evaluated yet as one can
notice from Table 1.
*Corresponding Author: Ebénézer Ntienjem: Centre for Research in Algebra and Number Theory, School of
Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada,
E-mail: [email protected], [email protected]
© 2017 Ntienjem, published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
447
Table 1. Known convolution sums W.˛;ˇ/ .n/
Level ˛ˇ
Authors
References
1
M. Besge, J. W. L. Glaisher, S. Ramanujan
[1–3]
2, 3, 4
J. G. Huard & Z. M. Ou
& B. K. Spearman & K. S. Williams
[4]
5, 7
M. Lemire & K. S. Williams,
S. Cooper & P. C. Toh
[5, 6]
6
S. Alaca & K. S. Williams
[7]
8, 9
K. S. Williams
[8, 9]
10, 11, 13, 14
E. Royer
[10]
12, 16, 18, 24
A. Alaca & S. Alaca & K. S. Williams
[11–14]
15
B. Ramakrishman & B. Sahu
[15]
20, 10
S. Cooper & D. Ye
[16]
23
H. H. Chan & S. Cooper
[17]
25
E. X. W. Xia & X. L. Tian & O. X. M. Yao
[18]
27, 32
L lu
S. Alaca & Y. Kesiciog
[19]
36
D. Ye
[20]
14, 26, 28, 30
E. Ntienjem
[21]
As an application, convolution sums are used to determine explicit formulae for the number of representations of a
positive integer n by the octonary quadratic forms
a .x12 C x22 C x32 C x42 / C b .x52 C x62 C x72 C x82 /;
(3)
c .x12 C x1 x2 C x22 C x32 C x3 x4 C x42 / C d .x52 C x5 x6 C x62 C x72 C x7 x8 C x82 /;
(4)
and
respectively, where a; b; c; d 2 N.
So far known explicit formulae for the number of representations of n by the octonary form Equation 3 are
referenced in Table 2.
Table 2. Known representations of n by the form Equation 3
.a; b/
Authors
References
(1,2)
K. S. Williams
[8]
(1,4)
A. Alaca & S. Alaca & K. S. Williams
[12]
(1,5)
S. Cooper & D. Ye
[16]
(1,6)
B. Ramakrishman & B. Sahu
[15]
(1,8)
L lu
S. Alaca & Y. Kesiciog
[19]
(1,7)
E. Ntienjem
[21]
We determine formulae for the number of representations of a positive integer n by the octonary quadratic form
Equation 3 for which .a; b/ D .1; 11/; .1; 13/. These formulae for the number of representations are also new
according to Table 2.
This paper is organized in the following way. In Section 2 we discuss modular forms, briefly define eta functions
and convolution sums, and prove the generalization of the extraction of the convolution sum. Our main results on
the evaluation of the convolution sums are discussed in Section 3. The determination of formulae for the number of
representations of a positive integer n is discussed in Section 4.
Software for symbolic scientific computation is used to obtain the results of this paper. This software comprises
the open source software packages GiNaC, Maxima, REDUCE, SAGE and the commercial software package
MAPLE.
448
E. Ntienjem
2 Modular forms and convolution sums
Let H be the upper half-plane, that is H D fz 2 C j Im.z/ > 0g, and let G D SL2 .R/ be the group of 2 2-matrices
a b such that a; b; c; d 2 R and ad
bc D 1 hold. Let furthermore € D SL2 .Z/ be the full modular group which
c d
is a subgroup of SL2 .R/. Let N 2 N. Then
˚ a b
€.N / D
2 SL2 .Z/ j ac db 10 01
.mod N /
c d
is a subgroup of G and is called the principal congruence subgroup of level N. A subgroup H of G is called a
congruence subgroup of level N if it contains €.N /.
Relevant for our purposes is the following congruence subgroup:
˚ a b
€0 .N / D
2 SL2 .Z/ j c 0 .mod N / :
c d
Let k; N 2 N and let € 0 € be a congruence subgroup of level N 2 N. Let k 2 Z; 2 SL2 .Z/ and f W
H [ Q [ f1g ! C [ f1g. We denote by f Œk the function whose value at z is .cz C d / k f ..z//, i.e.,
f Œk .z/ D .cz C d / k f ..z//. The following definition is based on the textbook by N. Koblitz [22, p. 108].
Definition 2.1. Let N 2 N, k 2 Z, f be a meromorphic function on H and € 0 € a congruence subgroup of
level N .
(a) f is called a modular function of weight k for € 0 if
(a1) for all 2 € 0 it holds that f Œk D f .
P
2 izn
(a2) for any ı 2 € it holds that f Œık .z/ can be expressed in the form
an e N , wherein an ¤ 0 for finitely
n2Z
many n 2 Z such that n < 0.
(b) f is called a modular form of weight k for € 0 if
(b1) f is a modular function of weight k for € 0 ,
(b2) f is holomorphic on H,
(b3) for all ı 2 € and for all n 2 Z such that n < 0 it holds that an D 0.
(c) f is called a cusp form of weight k for € 0 if
(c1) f is a modular form of weight k for € 0 ,
(c2) for all ı 2 € it holds that a0 D 0.
For k; N 2 N, let Mk .€0 .N // be the space of modular forms of weight k for €0 .N /, Sk .€0 .N // be the subspace
of cusp forms of weight k for €0 .N /, and Ek .€0 .N // be the subspace of Eisenstein forms of weight k for €0 .N /.
Then the decomposition of the space of modular forms as a direct sum of the space generated by the Eisenstein
series and the space of cusp forms, i.e., Mk .€0 .N // D Ek .€0 .N // ˚ Sk .€0 .N //, is well-known; see for example
W. A. Stein’s book (online version) [23, p. 81].
As noted in Section 5.3 of W. A. Stein’s book [23, p. 86] if the primitive Dirichlet characters are trivial and
1
2k P
2 k is even, then Ek .q/ D 1 B
k 1 .n/ q n , where Bk are the Bernoulli numbers.
k
nD1
For the purpose of this paper we only consider trivial Dirichlet characters and 2 k even. Theorems 5.8 and
5.9 in Section 5.3 of [23, p. 86] also hold for this special case.
2.1 Eta functions
The Dedekind eta function, .z/, is defined on the upper half-plane H by .z/ D e
2 iz
24
1
Q
.1
e 2 i nz /. We set
nD1
1
q D e 2 iz . Then .z/ D q 24
1
Q
nD1
.1
1
q n / D q 24 F .q/, where F .q/ D
1
Q
.1
q n /.
nD1
M. Newman [24, 25] systematically used the Dedekind eta function to construct modular forms for €0 .N /.
M. Newman determined when a function f .z/ is a modular form for €0 .N / by providing conditions (i)-(iv) in the
following theorem. G. Ligozat [26] determined the order of vanishing of an eta function at the cusps of €0 .N /,
which is condition (v) or (v0 ) in Theorem 2.2.
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
449
The following theorem is proved in L. J. P. Kilford’s book [27, p. 99] and G. Köhler’s book [28, p. 37]; we will
apply that theorem to determine eta quotients, f .z/, which belong to Mk .€0 .N //, and especially those eta quotients
which are in Sk .€0 .N //.
Theorem 2.2 (M. Newman and G. Ligozat ). Let N 2 N, D.N / be the set of all positive divisors of N , ı 2 D.N /
Q
and rı 2 Z. Let furthermore f .z/ D
rı .ız/ be an -quotient. If the following five conditions are satisfied
ı2D.N /
(i)
P
ı rı 0 .mod 24/,
ı2D.N /
(ii)
P
ı2D.N /
(iii)
N
ı
rı 0 .mod 24/,
ı rı is a square in Q,
Q
ı2D.N /
(iv) 0 <
P
rı 0 .mod 4/
ı2D.N /
(v) for each d 2 D.N / it holds that
then f .z/ 2 Mk .€0 .N //, where k
P
gcd .ı;d /2
ı
rı 0,
ı2D.N /
P
D 21
rı .
ı2D.N /
Moreover, the -quotient f .z/ belongs to Sk .€0 .N // if (v) is replaced by
P gcd .ı;d /2
rı > 0.
(v’) for each d 2 D.N / it holds that
ı
ı2D.N /
2.2 Convolution sums W.˛;ˇ/ .n/
Recall that given ˛; ˇ 2 N such that ˛ ˇ, the convolution sum is defined by Equation 2.
As observed by A. Alaca et al. [11], we can assume that gcd.˛; ˇ/ D 1. Let q 2 C be such that jqj < 1. Then
the Eisenstein series L.q/ and M.q/ are defined as follows:
L.q/ D E2 .q/ D 1
24
M.q/ D E4 .q/ D 1 C 240
1
X
.n/q n ;
(5)
3 .n/q n :
(6)
nD1
1
X
nD1
The following two relevant results are essential for the sequel of this work and are a generalization of the extraction
of the convolution sum using Eisenstein forms of weight 4 for all pairs .˛; ˇ/ 2 N2 . Their proofs are given by
E. Ntienjem [21].
Lemma 2.3. Let ˛; ˇ 2 N. Then
.˛ L.q ˛ /
ˇ L.q ˇ //2 2 M4 .€0 .˛ˇ//:
Theorem 2.4. Let ˛; ˇ 2 N be such that ˛ and ˇ are relatively prime and ˛ < ˇ. Then
.˛ L.q ˛ /
ˇ L.q ˇ //2 D.˛
ˇ/2 C
1 X
nD1
n
n
240 ˛ 2 3 . / C 240 ˇ 2 3 . /
˛
ˇ
n
6n/ . / C 48 ˇ .˛
˛
1152 ˛ˇ W.˛;ˇ/ .n/ q n :
C 48 ˛ .ˇ
n
6n/ . /
ˇ
(7)
450
E. Ntienjem
3 Evaluation of the convolution sums W.˛;ˇ/ .n/, where
˛ˇ D 22 ; 44; 52
In this section, we give explicit formulae for the convolution sums W.1;22/ .n/, W.2;11/ .n/, W.1;44/ .n/, W.4;11/ .n/,
W.1;52/ .n/ and W.4;13/ .n/.
3.1 Bases for E4 .€0 .˛ˇ// and S4 .€0 .˛ˇ// with ˛ˇ D 44; 52
We observe the following inclusion relations
M4 .€0 .11// M4 .€0 .22// M4 .€0 .44//
(8)
M4 .€0 .13// M4 .€0 .26// M4 .€0 .52//:
(9)
Therefore, it suffices to correspondingly determine the basis of the spaces M4 .€0 .44// and M4 .€0 .52//, respectively.
We use the dimension formulae for the space of Eisenstein forms and the space of cusp forms in T. Miyake’s
book [29, Thrm 2.5.2, p. 60] or W. A. Stein’s book [23, Prop. 6.1, p. 91] to deduce that dim.E4 .€0 .44/// D
dim.E4 .€0 .52/// D 6, dim.S4 .€0 .44// D 15 and dim.S4 .€0 .52// D 18.
Let D.44/ D f1; 2; 4; 11; 22; 44g and D.52/ D f1; 2; 4; 13; 26; 52g be the sets of all positive divisors of 44 and
52, respectively.
Theorem 3.1.
(a) The sets BE;44 D f M.q t / j t 2 D.44/ g and BE;52 D f M.q t / j t 2 D.52/ g are bases of E4 .€0 .44// and
E4 .€0 .52//, respectively.
(b) Let 1 i 15 and 1 j 18 be positive integers.
Let ı1 2 D.44/ and .r.i; ı1 //i;ı1 be the Table 3 of the powers of .ı1 z/.
Let ı2 2 D.52/ and .r.j; ı2 //j;ı2 be the Table 4 of the powers of .ı2 z/.
Q
Q
Let furthermore Ai .q/ D
r.i;ı1 / .ı1 z/ and Bj .q/ D
r.j;ı2 / .ı2 z/ be selected elements of
ı1 2D.44/
ı2 2D.52/
S4 .€0 .44// and S4 .€0 .52//, respectively.
Then the sets BS;44 D f Ai .q/ j 1 i 15 g and BS;52 D f Bj .q/ j 1 j 18 g are bases of S4 .€0 .44//
and S4 .€0 .52//, repectively.
(c) The sets BM;44 D BE;44 [ BS;44 and BM;52 D BE;52 [ BS;52 constitute bases of M4 .€0 .44// and
M4 .€0 .52//, respectively.
For 1 i 15 and 1 j 18 let in the sequel Ai .q/ be expressed in the form
1
P
ai .n/q n and Bj .q/ be
nD1
expressed in the form
1
P
bj .n/q n .
nD1
Proof. We give the proof for the case ˛ˇ D 44. The case ˛ˇ D 52 is proved similarly.
(a) By Theorem 5.8 in Section 5.3 of W. A. Stein [23, p. 86] M.q t / is in M4 .€0 .t // for each t which is an element
of D.44/. Since E4 .€0 .44// has a finite dimension, it suffices to show that M.q t / with t 2 D.44/ are linearly
independent. Suppose that x t 2 C with t 2 D.44/. We prove this by induction on the elements of the set D.44/
which is assumed to be ascendantly ordered.
The case t D 1 2 D.44/ is obvious since comparing the coefficients of q t on both sides of the equation
x t M.q t / D 0 clearly gives x t D 0.
Suppose now that the cardinality of the set D.44/ is greater than 1 and that M.q t / are linearly independent for
all tj44 and t t1 for a given t1 with 1 < t1 < 44. Let C be the proper non-empty subset of D.44/ which contains
all positive divisors of 44 less than or equal to t1 . Note that all positive divisors of t1 constitute a subset of C . Let
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
451
us consider the non-empty subset C [ ft 0 g of D.44/, wherein t 0 is the next ascendant element of D.44/ which is
greater than t1 the greatest element of the set C . Then
X
X
0
x t M.q t / D
x t M.q t / C x t 0 M.q t / D 0:
t 2C [ft 0 g
t2C
By the induction hypothesis it holds that x t D 0 for all t 2 C . So, we obtain from the above equation that x t 0 D 0
0
when we compare the coefficient of q t on both sides of the equation.
Hence, the solution is x t D 0 for all t such that t is a positive divisor of 44. Therefore, the set BE;44 is linearly
independent. Hence, the set BE;44 is a basis of E4 .€0 .44//.
(b) The Ai .q/ with 1 i 15 are obtained from an exhaustive search using Theorem 2.2 .i / .v 0 /. Hence, each
Ai .q/ is an element of the space S4 .€0 .44//.
Since the dimension of S4 .€0 .44// is 15, it suffices to show that the set f Ai .q/ j 1 i 15g is linearly
15
P
independent. Suppose that xi 2 C and
xi Ai .q/ D 0. Then
iD1
15
X
xi Ai .q/ D
1 X
15
X
.
xi ai .n/ /q n D 0
nD1 i D1
iD1
which gives the following homogeneous system of linear equations
15
X
ai .n/ xi D 0;
1 n 15:
(10)
i D1
A simple computation using software for symbolic scientific computation shows that the determinant of the matrix
of this homogeneous system of linear equations is non-zero. So, xi D 0 for all 1 i 15. Hence, the set
f Ai .q/ j 1 i 15 g is linearly independent and therefore a basis of S4 .€0 .44//.
(c) Since M4 .€0 .44// D E4 .€0 .44// ˚ S4 .€0 .44//, the result follows from (a) and (b).
According to Equation 8 the basis elements Ai .q/, where 1 i 5, are contained in S4 .€0 .22//. The basis element
A2 .q/ is the only element of the space S4 .€0 .11// that we are able to generate with the help of Theorem 2.2. Even
though the basis element A14 .q/ looks like an element of S4 .€0 .22//, it cannot be generated at level 22 using
Theorem 2.2.
To evaluate the convolution sums W.1;22/ .n/ and W.2;11/ .n/, we determine two additional basis elements of
S4 .€0 .22// which are
A06 .q/ D
1
X
.2z/3 .11z/5 .22z/
D
a60 .n/q n ;
.z/
nD1
9
A07 .q/ D
7
.2z/ .11z/
D
5 .z/3 .22z/
1
X
a70 .n/q n :
nD1
Due to Equation 9 the basis elements Bj .q/, where 1 j 7 and j D 15; 17, belong to S4 .€0 .26//. We are
unable to generate any elements of the space S4 .€0 .13// using Theorem 2.2. We note that B2j .q/ D Bj .q 2 /,
where 4 j 7, B16 .q/ D B15 .q 2 / and B18 .q/ D B17 .q 2 /. Therefore, one can easily replicate the evaluation of
the convolution sums W.1;26/ .n/ and W.2;13/ .n/ shown by E. Ntienjem [21].
3.2 Evaluation of W.˛;ˇ/ .n/ where ˛ˇ D 22 ; 44; 52
Lemma 3.2. We have
2
.L.q /
22 L.q
22
2
// D 441 C
1 X
nD1
3312
12672
n
3 .n/ C
3 . /
61
61
2
452
E. Ntienjem
n
6557760
n
12096
45792
110880
3 . / C
3 . / C
a1 .n/ C
a2 .n/
61
11
61
22
61
61
19872
73728
50688
C
a3 .n/ C
a4 .n/
a5 .n/ C 22176 a60 .n/ C 864 a70 .n/ q n ;
61
61
61
.2 L.q 2 /
11 L.q 11 //2 D 81 C
1 X
15840
nD1
61
3 .n/ C
(11)
37440
n
3 . /
61
2
1626768
n
494208
n
36864
357408
3 . /
3 . / C
a1 .n/ C
a2 .n/
61
11
61
22
61
61
1539072
834048
1160352
0
0
a3 .n/ C
a4 .n/ C
a5 .n/ 22176 a6 .n/ 864 a7 .n/ q n ;
C
61
61
61
C
44 L.q 44 //2 D 1849 C
.L.q/
1 X
124464
nD1
61
3 .n/
577662336
n
3 . /
40565
2
n
174240
n
62064288
n
2525690112
n
68986368
3 . /
3 . / C
3 . / C
3 . /
C
5795
4
61
11
5795
22
5795
44
1440
82927872
887345568
1676429568
C
a1 .n/
a2 .n/
a3 .n/
a4 .n/
61
5795
5795
5795
2804007168
3753380736
13356288
4226609664
a5 .n/ C
a6 .n/
a7 .n/ C
a8 .n/
5795
5795
19
5795
633600
527332608
7679232
a9 .n/
a10 .n/ C
a11 .n/
19
1159
19
15231744
131079168
317952
12595968
a12 .n/
a13 .n/ C
a14 .n/
a15 .n/ q n ;
95
95
19
95
4
.4 L.q /
11 L.q
11
2
// D 49 C
1 X
nD1
52 L.q 52 //2 D 2601 C
1 X
6109008
nD1
1243
3 .n/
456504084816
n
3 . /
6064597
2
254592
n
7361952
n
4829528827344
n
3 . /
3 . /
3 . /
41
4
1243
13
6064597
26
434738304
n
3066144
498157179048
C
3 . /
b1 .n/ C
b2 .n/
41
52
1243
6064597
927327070704
442577500560
8530413669648
C
b3 .n/
b4 .n/
b5 .n/
6064597
6064597
6064597
10161699732288
10388366352
1040832
b6 .n/
b7 .n/ C
b8 .n/
6064597
1243
41
329100929664
C 7488 b9 .n/ C
b10 .n/ C 27456 b11 .n/
147917
C
(13)
80121888
n
110880
3 .n/ C
3 . /
61
5795
2
48338688
n
1817904
n
98480448
n
27320832
n
3 . / C
3 . /
3 . /
3 . /
5795
4
61
11
5795
22
5795
44
110880
174857472
1169427168
2114189568
C
a1 .n/ C
a2 .n/ C
a3 .n/ C
a4 .n/
61
5795
5795
5795
3511080576
13318272
3641762304
3025513728
a5 .n/
a6 .n/ C
a7 .n/
a8 .n/
C
5795
5795
19
5795
633600
663913728
7679232
15231744
C
a9 .n/ C
a10 .n/
a11 .n/ C
a12 .n/
19
1159
19
95
131079168
317952
12595968
C
a13 .n/
a14 .n/ C
a15 .n/ q n ;
95
19
95
.L.q/
(12)
(14)
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
15249288510144
47009664
b12 .n/ C 17472 b13 .n/ C
b14 .n/
6064597
41
25166713896
4167031826880
126425023920
b15 .n/ C
b16 .n/
b17 .n/
551327
6064597
6064597
868608
C
b18 .n/ q n ;
41
.4 L.q 4 /
13 L.q 13 //2 D 81 C
1 X
3066144
1243
nD1
453
(15)
240061230672
n
3 . /
6064597
2
3 .n/
139392
n
45798672
n
53922031824
n
C
3 . / C
3 . /
3 . /
41
4
1243
13
6064597
26
20290176
n
3066144
212735819880
C
3 . /
b1 .n/ C
b2 .n/
41
52
1243
6064597
400561037808
5152459820400
251848851024
b3 .n/
b4 .n/
b5 .n/
C
6064597
6064597
6064597
5408748312192
5489355312
150336
b6 .n/
b7 .n/ C
b8 .n/
6064597
1243
41
151016538432
7488 b9 .n/ C
b10 .n/ 27456 b11 .n/
147917
8224832431680
544896
b12 .n/ 17472 b13 .n/
b14 .n/
6064597
41
11115614088
2056953609600
64745693328
b15 .n/ C
b16 .n/
b17 .n/
551327
6064597
6064597
2304
b18 .n/ q n :
41
(16)
Proof. We just prove the case .4 L.q 4 / 11 L.q 11 //2 . The other cases are proved similarly.
It follows from Lemma 2.3 that .4 L.q 4 / 11 L.q 11 //2 2 M4 .€0 .44//. Hence, by Theorem 3.1 (c), there exist
Xı ; Yj 2 C; 1 j 15 and ı 2 D.44/, such that
.4 L.q 4 /
11 L.q 11 //2 D
X
Xı M.q ı / C
D
Yj Aj .q/
j D1
ı2D.44/
X
15
X
Xı C
ı2D.44/
1 X
nD1
240
X
ı2D.44/
mS
X
n
3 . / Xı C
aj .n/ Yj q n :
ı
(17)
j D1
We equate the right hand side of Equation 17 with that of Equation 7 when setting .˛; ˇ/ D .4; 11/ to obtain
1 X
nD1
240
15
1 X
X
n
n
n
3 . / Xı C
aj .n/ Yj q n D
3840 3 . / C 29040 3 . /
ı
4
11
nD1
ı2D.44/
j D1
n
n
C192 .11 6 n/ . / C 528 .4 6 n/ . / 50688 W.4;11/ .n/ q n :
4
11
X
We now take the coefficients of q n for which n is in
f 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 20; 22; 44 g:
This results in a system of linear equations whose unique solution determines the values of the unknown Xı for all
ı 2 D.44/ and the values of the unkown Yj for all 1 j 15. Hence, we obtain the stated result.
Our main result of this section is as follows.
Theorem 3.3. Let n be a positive integer. Then
W.1;22/ .n/ D
17
3 .n/
1464
1
n
35
n
125
n
3 . / C
3 . / C
3 . /
122
2
488
11
366
22
454
E. Ntienjem
1
1
1
n
21
159
1
n/ .n/ C .
n/ . /
a1 .n/
a2 .n/
24 88
24 4
22
2684
5368
69
32
2
7 0
3 0
a3 .n/
a4 .n/ C
a5 .n/
a6 .n/
a .n/;
5368
671
61
8
88 7
C.
W.2;11/ .n/ D
W.1;44/ .n/ D
W.4;11/ .n/ D
5
n
5
3 .n/ C
3 . / C
488
366
2
1
1
n
1
C.
n/ . / C .
24 44
2
24
4029
668
a3 .n/
a4 .n/
5368
671
137
n
39
n
3 . / C
3 . /
1464
11
122
22
1
n
16
1241
n/ . /
a1 .n/
a2 .n/
8
11
671
5368
3 0
362
7
a5 .n/ C a60 .n/ C
a .n/;
671
8
88 7
13
501443
n
1361
n
55
n
3 .n/ C
3 . /
3 . / C
3 . /
366
1784860
2
5795
4
976
11
19591
n
9878
n
1
1
3 . / C
3 . / C .
n/ .n/
92720
22
17385
44
24 176
1
n
5
35993
1
n/ . /
a1 .n/ C
a2 .n/
C.
24 4
44
10736
127490
3081061
66147
1217017
3258143
C
a3 .n/ C
a4 .n/ C
a5 .n/
a6 .n/
1019920
11590
127490
254980
917233
25
20807
527
a7 .n/
a8 .n/ C
a9 .n/ C
a10 .n/
C
38
63745
38
2318
303
601
2586
69
a11 .n/ C
a12 .n/ C
a13 .n/
a14 .n/
38
190
95
209
497
C
a15 .n/;
190
35
25291
n
4178
n
11
n
3 .n/
3 . / C
3 . /
3 . /
976
92720
2
17385
4
732
11
n
539
n
1
1
n
15543
3 . / C
3 . / C .
n/ . /
C
46360
22
5795
44
24 44
4
1
1
n
35
75893
C.
n/ . /
a1 .n/
a2 .n/
24 16
11
976
127490
4060511
917617
1313157
3047813
a3 .n/
a4 .n/
a5 .n/ C
a6 .n/
1019920
127490
127490
254980
790313
25
288157
1051
a7 .n/ C
a8 .n/
a9 .n/
a10 .n/
76
63745
38
25498
303
601
2586
69
C
a11 .n/
a12 .n/
a13 .n/ C
a14 .n/
38
190
95
209
497
a15 .n/;
190
W.1;52/ .n/ D
97
731577059
n
17
n
5899
n
3 .n/ C
3 . /
3 . / C
3 . /
1243
582201312
2
164
4
59664
13
7739629531
n
81757
n
1
1
C
3 . /
3 . / C .
n/ .n/
582201312
26
492
52
24 208
1
1
n
31939
6918849709
C.
n/ . / C
b1 .n/
b2 .n/
24 4
52
775632
5045744704
19319313973
236419605
4556844909
b3 .n/ C
b4 .n/ C
b5 .n/
7568617056
194067104
194067104
1357064601
5549341
139
C
b6 .n/ C
b7 .n/
b8 .n/
48516776
39776
328
1
65925667
11
2036496863
b9 .n/
b10 .n/
b11 .n/ C
b12 .n/
8
1775004
24
48516776
7
3139
349537693
b13 .n/
b14 .n/ C
b15 .n/
24
164
458704064
(18)
(19)
(20)
(21)
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
556494635
67534735
b16 .n/ C
b17 .n/
48516776
194067104
W.4;13/ .n/ D
29
b18 .n/;
82
31939
5001275639
n
47
n
24049
n
3 .n/ C
3 . / C
3 . / C
3 . /
775632
7568617056
2
6396
4
387816
13
1123375663
n
17613
n
1
1
n
C
3 . /
3 . / C .
n/ . /
7568617056
26
2132
52
24 52
4
1
1
n
31939
2954664165
C.
n/ . / C
b1 .n/
b2 .n/
24 16
13
775632
5045744704
8345021621
35780970975
5246851063
b3 .n/ C
b4 .n/ C
b5 .n/
7568617056
7568617056
2522872352
4695094021
38120523
261
C
b6 .n/ C
b7 .n/
b8 .n/
315359044
517088
4264
786544471
11
42837668915
1
b10 .n/ C
b11 .n/ C
b12 .n/
C b9 .n/
8
46150104
24
1892154264
473
154383529
7
b13 .n/ C
b14 .n/ C
b15 .n/
C
24
2132
458704064
5356650025
1348868611
1
b16 .n/ C
b17 .n/ C
b18 .n/:
946077132
7568617056
1066
455
(22)
(23)
Proof. We prove the case W.4;13/ .n/ as the other cases are proved similarly.
We equate the right hand side of Equation 16 with that of Equation 7 when setting .˛; ˇ/ D .4; 13/, namely
1 X
nD1
n
n
3840 3 . / C 40560 3 . / C 192 .13
4
13
C 624 .4
n
/
13
59904 W.4;13/ .n/ q n D
3 .n/
240061230672
n
3 . /
6064597
2
6 n/ .
1 X
3066144
nD1
n
6 n/ . /
4
1243
n
45798672
n
53922031824
n
20290176
n
139392
3 . / C
3 . /
3 . / C
3 . /
C
41
4
1243
13
6064597
26
41
52
3066144
212735819880
251848851024
b1 .n/ C
b2 .n/ C
b3 .n/
1243
6064597
6064597
400561037808
5152459820400
5408748312192
b4 .n/
b5 .n/
b6 .n/
6064597
6064597
6064597
5489355312
150336
151016538432
b7 .n/ C
b8 .n/ 7488 b9 .n/ C
b10 .n/
1243
41
147917
544896
8224832431680
b12 .n/ 17472 b13 .n/
b14 .n/
27456 b11 .n/
6064597
41
11115614088
2056953609600
64745693328
b15 .n/ C
b16 .n/
b17 .n/
551327
6064597
6064597
2304
b18 .n/ q n :
41
We then solve for W.4;13/ .n/ to obtain the stated result.
4 Number of representations of a positive integer n by the
qctonary quadratic form using W.˛;ˇ/ .n/ when ˛ˇ D 44; 52
Let n 2 N0 and the number of representations of n by the quaternary quadratic form x12 C x22 C x32 C x42 be denoted
by r4 .n/. That means,
r4 .n/ D card.f.x1 ; x2 ; x3 ; x4 / 2 Z4 j m D x12 C x22 C x32 C x42 g/:
456
E. Ntienjem
We set r4 .0/ D 1. For all n 2 N, the following Jacobi’s identity is proved in K. S. Williams’ book [30, Thrm 9.5, p.
83]
n
r4 .n/ D 8 .n/ 32 . /:
(24)
4
Let furthermore the number of representations of n by the octonary quadratic form
a .x12 C x22 C x32 C x42 / C b .x52 C x62 C x72 C x82 /
be denoted by N.a;b/ .n/. That means,
N.a;b/ .n/ D card.f.x1 ; x2 ; x3 ; x4 ; x5 ; x6 ; x7 ; x8 / 2 Z8 j n D a .x12 C x22 C x32 C x42 / C b .x52 C x62 C x72 C x82 /g/:
We infer the following result:
Theorem 4.1. Let n 2 N and .a; b/ D .1; 11/; .1; 13/. Then
n
n
n
N.1;11/ .n/ D8 .n/ 32 . / C 8 . / 32 . /
4
11
44
n
C 64 W.1;11/ .n/ C 1024 W.1;11/ . / 256 W.4;11/ .n/ C W.1;44/ .n/ ;
4
n
n
n
N.1;13/ .n/ D8 .n/ 32 . / C 8 . / 32 . /
4
13
52
n
C 64 W.1;13/ .n/ C 1024 W.1;13/ . / 256 W.4;13/ .n/ C W.1;52/ .n/ :
4
Proof. We only prove N.1;11/ .n/ since that for N.1;13/ .n/ is done similarly.
It holds that
X
n
N.1;11/ .n/ D
r4 .l/r4 .m/ D r4 .n/r4 .0/ C r4 .0/r4 . / C
11
2
.l;m/2N0
lC11 mDn
X
r4 .l/r4 .m/:
.l;m/2N2
lC11 mDn
We make use of Equation 24 to derive
N.1;11/ .n/ D 8 .n/
n
n
32 . / C 8 . /
4
11
32 .
n
/C
52
X
.8 .l/
.l;m/2N2
lC11 mDn
l
32 . //.8 .m/
4
32 .
m
//:
4
We observe that
l
32 . //.8 .m/
4
The evaluation of
.8 .l/
32 .
m
// D 64 .l/ .m/
4
W.1;11/ .n/ D
l
256 . / .m/
4
X
256 .l/ .
m
l
m
/ C 1024 . / . /:
4
4
4
.l/ .m/
.l;m/2N2
lC11 mDn
is shown by E. Royer [10, Thrm 1.3]. We map l to 4l to infer
X
l
W.4;11/ .n/ D
. / .m/ D
4
2
.l;m/2N
lC11 mDn
X
.l/ .m/:
.l;m/2N2
4 lC11 mDn
The evaluation of W.4;11/ .n/ is given in Equation 21. We next map m to 4m to conclude
X
X
m
W.1;44/ .n/ D
.l/ . / D
.l/ .m/:
4
2
2
.l;m/2N
lC11 mDn
.l;m/2N
lC44 mDn
The evaluation of W.1;44/ .n/ is provided by Equation 20. We simultaneously map l to 4l and m to 4m to deduce
X
X
l
m
n
. / . / D
.l/ .m/ D W.1;11/ . /:
4
4
4
2
2
.l;m/2N
lC11 mDn
.l;m/2N
lC11 mD n
4
Again, E. Royer [10, Thrm 1.3] has proved the evaluation of W.1;11/ .n/.
We then put these evaluations together to obtain the stated result for N.1;11/ .n/.
Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
457
Tables
Table 3. Exponents of -functions being basis elements of S4 .€0 .44//
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
4
11
22
44
6
4
2
0
-2
0
0
0
3
0
1
2
0
-3
0
-2
0
2
4
6
2
-3
0
0
-2
-3
0
2
9
0
0
0
0
0
0
2
5
4
1
6
4
0
0
0
2
6
4
2
0
-2
0
0
0
-1
0
-3
2
0
1
0
-2
0
2
4
6
2
5
0
0
-2
5
-4
-2
1
-4
0
0
0
0
0
2
1
4
5
6
4
8
8
0
10
Table 4. Exponents of -functions being basis elements of S4 .€0 .52//
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
2
4
13
26
52
1
3
1
3
1
3
1
0
2
0
2
0
2
0
-1
0
7
0
5
3
3
1
1
-1
-1
3
1
1
-1
3
1
1
5
-1
-3
7
0
0
0
0
0
0
0
1
1
1
1
-1
-1
-1
0
5
0
-3
3
1
3
1
3
1
3
0
-2
0
-2
0
-2
0
5
0
-3
0
-1
1
1
3
3
5
5
1
3
3
5
1
3
3
-1
5
7
-3
0
0
0
0
0
0
0
3
3
3
3
5
5
5
0
-1
0
7
Acknowledgement: I am indebtedly thankful to the anonynuous referee for fruitful comments and suggestions on
a draft of this paper.
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