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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 277–283 DOI: 10.18514/MMN.2017.1955
REPRESENTATION NUMBERS BY SUMS OF QUADRATICFORMS X2
1 CX1X2CX22 IN SIXTEEN VARIABLES
BULENT KOKLUCE AND MUBERRA GUREL
Received 11 April, 2016
Abstract. Let R.a1; :::;a8In/ denote the number of representations of an integer n by the forma1.x
21 C x1x2C x
22/C a2.x
23 C x3x4C x
24/C a3.x
25 C x5x6C x
26/C a4.x
27 C x7x8C x
28/C
a5.x29Cx9x10Cx
210/Ca6.x
211Cx11x12Cx
212/Ca7.x
213Cx13x14Cx
214/
Ca8.x215 C x15x16 C x
216/. In this article, we derive formulae for R.1;2;2;2;2;2;2;2In/,
R.1;1;1;2;2;2;2;2In/, R.1;1;1;1;1;2;2;2In/ andR.1;1;1;1;1;1;1;2In/. These formulae aregiven in terms of the function �7.n/ and the numbers �8;2.n/ and �8;6.n/.
2010 Mathematics Subject Classification: 11A25; 11E25
Keywords: quadratic forms, sum of divisors functions, convolution sums, representation num-bers
1. INTRODUCTION
Let N; Z and Q denote the set of positive integers, integers and rational numbersrespectively and let N0 DN[f0g: For a1; :::;a8 2N and n 2N0 we letR.a1; :::;a8In/ denote the representation number of n by the form a1.x
21Cx1x2C
x22/Ca2.x
23Cx3x4Cx
24/Ca3.x
25Cx5x6Cx
26/Ca4.x
27Cx7x8Cx
28/Ca5.x
29C
x9x10Cx210/Ca6.x
211Cx11x12Cx
212/Ca7.x
213Cx13x14Cx
214/
Ca8.x215Cx15x16Cx
216/, that is
R.a1; :::;a8In/ (1.1)
WD card
(.x1; :::;x16/ 2Z16 W nD
8XkD1
ak.x22k�1
Cx2k�1x2kCx22k/
).
If l of a1; :::;a8 are equal, say
ai D aiC1 D :::D aiCl�1 D a (1.2)
for convenience we indicate this in R.a1; :::;a8In/ by writing al forai ;aiC1; :::;aiCl�1. For k 2N and n 2Q the sum of divisor function is defined by
c 2017 Miskolc University Press
278 BULENT KOKLUCE AND MUBERRA GUREL
�k.n/D
8<:
Xd2Nd jn
dk; if n 2N;
0; if n 2Q;n …N:
(1.3)
Ramakrishnan and Sahu [8] have recently proved formulae for R.18In/ andR.14;24In/ for all n 2 N. In the present paper, motivated from the work of Ra-makrishnan and Sahu we derive formulae for R.11;27In/, R.13;25In/, R.15;23In/
and R.17;21In/ by using the method of Alaca, Alaca and Williams (see for example[1]). These formulae are given in terms of the function �7.n/ and the numbers �8;2.n/
and �8;6.n/ which are introduced in [8]. Kokluce [4] has derived formulae for thequadratic forms in twelve and sixteen variables which are sums of quadratic formswith discriminant �23 by using the method developed by Lomadze [7].
2. PRELIMINARIES AND STATEMENT OF THE THEOREM
For a;b;r; s;n 2N with a � b we define the convolution sum Wr;s
a;b.n/ by
Wr;s
a;b.n/ WD
X.l;m/2N2
alCbmDn
�r.l/�s.m/: (2.1)
In recent years, some of the convolution sums have been evaluated explicitly. Inthis study, we require the evaluations for W 3;3
1;1 .n/, W3;3
1;2 .n/, W3;3
1;3 .n/, W3;3
2;3 .n/,W
3;31;6 .n/: The convolution sum
W3;3
1;1 .n/D1
120�7.n/�
1
120�3.n/; (2.2)
was evaluated explicitly by Ramanujan [9]. Formulae for the sum W3;3
1;2 .n/ wasfirstly given by Cheng and Williams [2]. Formulae for many other convolution sumsof this type have also been given by the authors in that study. Ramakrishnan andSahu [8] have recently evaluated the following four formulae containing formulaefor W 3;3
1;2 .n/ as well.
W3;3
1;2 .n/D�1
240�3.n/�
1
240�3.
n
2/C
1
2040�7.n/C
2
255�7.
n
2/C
1
272�8;2.n/;
(2.3)
W3;3
1;3 .n/D�1
240�3.n/�
1
240�3.
n
3/C
1
9840�7.n/C
81
9840�7.
n
3/C
1
246�8;3.n/;
(2.4)
REPRESENTATION NUMBERS OF SUMS OF QUADRATIC FORMS 279
W3;3
2;3 .n/
D�1
240�3.
n
2/�
1
240�3.
n
3/C
1
167280�7.n/C
1
10455�7.
n
2/C
27
55760�7.
n
3/
C27
3485�7.
n
6/C
7
8160�8;2.n/C
189
2720�8;2.
n
3/C
1
820�8;3.n/C
4
205�8;3.
n
2/
�1
480�8;6.n/; (2.5)
and
W3;3
1;6 .n/
D�1
240�3.n/�
1
240�3.
n
6/C
1
167280�7.n/C
1
10455�7.
n
2/C
27
55760�7.
n
3/
C27
3485�7.
n
6/C
7
8160�8;2.n/C
189
2720�8;2.
n
3/C
1
820�8;3.n/C
4
205�8;3.
n
2/
C1
480�8;6.n/; (2.6)
where �8;2.n/, �8;3.n/, �8;6.n/ are n-th Fourier coefficients of the newforms�8;2.n/, �8;3.n/, �8;6.n/ given in [8]. In a recent publication Kokluce and Eser[5] have completed the evaluations of the convolution sums W 1;3
1;12.n/, W1;3
3;4 .n/,W
1;34;3 .n/, and W 1;3
12;1.n/:
We now state our main result.
Theorem 1. Let n 2N then,
(i)
R.11;27In/D
6
85�7.n/�
516
85�7.
n
2/�
486
85�7.
n
3/C
41796
85�7.
n
6/C
9
17�8;2.n/
�729
17�8;2.
n
3/C
27
5�8;6.n/; (2.7)
(ii)
R.13;25In/D
18
85�7.n/�
528
85�7.
n
2/�
1458
85�7.
n
3/C
42768
85�7.
n
6/C
27
17�8;2.n/
�2187
17�8;2.
n
3/C
81
5�8;6.n/; (2.8)
(iii)
R.15;23In/D
66
85�7.n/�
576
85�7.
n
2/�
5346
85�7.
n
3/C
46656
85�7.
n
6/�
54
17�8;2.n/
280 BULENT KOKLUCE AND MUBERRA GUREL
C4374
17�8;2.
n
3/C
162
5�8;6.n/; (2.9)
(iv)
R.17;21In/D
258
85�7.n/�
768
85�7.
n
2/�
20898
85�7.
n
3/C
62208
85�7.
n
6/�
72
17�8;2.n/
C5832
17�8;2.
n
3/C
216
5�8;6.n/: (2.10)
3. PROOF OF THEOREM 1
To prove the theorem we need the representation number formulae for the follow-ing octonary quadratic forms,
f1 WD x21Cx1x2Cx
22Cx
23Cx3x4Cx
24Cx
25Cx5x6Cx
26Cx
27Cx7x8Cx
28 ; (3.1)
f2 WD x21Cx1x2Cx
22Cx
23Cx3x4Cx
24Cx
25Cx5x6Cx
26C2.x
27Cx7x8Cx
28/
(3.2)and
f3 WD (3.3)
x21Cx1x2Cx
22C2.x
23Cx3x4Cx
24/C2.x
25Cx5x6Cx
26/C2.x
27Cx7x8Cx
28/.
For l 2 N0 we set ri .l/ Dcard˚.x1; :::;x8/ 2Z8 W l D fi .x1; :::;x8/
. Obviously
ri .0/D 1 for i 2 f1;2;3g. The representation number formula
r1.l/D 24�3.l/C216�3.l
3/; l 2N; (3.4)
is given by Lomadze [6]. Kokluce [3] has obtained formulae for the number ofrepresentation of a positive integer l by the forms f2 and f3: He has proved that,
r2.l/D 18�3.l/�48�3.l
2/�162�3.
l
3/C432�3.
l
6/; l 2N; (3.5)
r3.l/D 6�3.l/�36�3.l
2/�54�3.
l
3/C324�3.
l
6/; l 2N: (3.6)
Proof. (i) We just prove part (i) in detail as the rest can be proved similarly.It is clear that
R.11;27In/D
Xl;m2N0
2lCmDn
r1.l/r3.m/
REPRESENTATION NUMBERS OF SUMS OF QUADRATIC FORMS 281
D r1.0/r3.n/C r1.n
2/r3.0/C
Xl;m2N
2lCmDn
r1.l/r3.m/.
Thus using (3.4) and (3.6) we have
R.11;27In/� .6�3.n/�36�3.
n
2/�54�3.
n
3/C324�3.
n
6/C24�3.
n
2/C216�3.
n
6//
D
Xl;m2N
2lCmDn
.24�3.l/C216�3.l
3//.6�3.m/�36�3.
m
2/�54�3.
m
3/C324�3.
m
6//
D 144X
l;m2N2lCmDn
�3.l/�3.m/�864X
l;m2N2lCmDn
�3.l/�3.m
2/
�1296X
l;m2N2lCmDn
�3.l/�3.m
3/C7776
Xl;m2N
2lCmDn
�3.l/�3.m
6/
C1296X
l;m2N2lCmDn
�3.l
3/�3.m/�7776
Xl;m2N
2lCmDn
�3.l
3/�3.
m
2/
�11664X
l;m2N2lCmDn
�3.l
3/�3.
m
3/C69984
Xl;m2N
2lCmDn
�3.l
3/�3.
m
6/
D 144W3;3
1;2 .n/�864W3;3
1;1 .n
2/�1296W
3;32;3 .n/C7776W
3;31;3 .
n
2/C1296W
3;31;6 .n/
�7776W3;3
1;3 .n
2/�11664W
3;31;2 .
n
3/C69984W
3;31;1 .
n
6/:
Appealing to (2.2)-(2.6) and adding 6�3.n/� 36�3.n2/� 54�3.
n3/C 324�3.
n6/C
24�3.n2/C216�3.
n6/ to both sides of the equation we obtain the asserted formula.
(ii) It is clear that
R.13;25In/D
Xl;m2N0
2lCmDn
r1.l/r2.m/:
Using equations (3.4), (3.5) and (2.1) we have
R.13;25In/� .18�3.n/�48�3.
n
2/�162�3.
n
3/C432�3.
n
6/C24�3.
n
2/
C216�3.n
6//
D 432W3;3
1;2 .n/�1152W3;3
1;1 .n
2/�3888W
3;32;3 .n/C10368W
3;31;3 .
n
2/C3888W
3;31;6 .n/
282 BULENT KOKLUCE AND MUBERRA GUREL
�10368W3;3
1;3 .n
2/�34992W
3;31;2 .
n
3/C93312W
3;31;1 .
n
6/:
Appealing to (2.2)-(2.6) we obtain the desired result .(iii) It is clear that
R.15;23In/D
Xl;m2N0lCmDn
r1.l/r3.m/.
Using equations (3.4), (3.6) and (2.1) we have
R.15;23In/� .6�3.n/�36�3.
n
2/�54�3.
n
3/C324�3.
n
6/C24�3.n/
C216�3.n
3//
D 144W3;3
1;1 .n/�864W3;3
1;2 .n/�1296W3;3
1;3 .n/C7776W3;3
1;6 .n/C1296W3;3
1;3 .n/
�7776W3;3
2;3 .n/�11664W3;3
1;1 .n
3/C69984W
3;31;2 .
n
3/:
Appealing to (2.2)-(2.6) we obtain the desired result.(iv) Clearly
R.17;21In/D
Xl;m2N0lCmDn
r1.l/r2.m/.
Using equations (3.4), (3.5) and (2.1) we have
R.17;21In/� .18�3.n/�48�3.
n
2/�162�3.
n
3/C432�3.
n
6/C24�3.n/
C216�3.n
3//
D 432W3;3
1;1 .n/�1152W3;3
1;2 .n/�3888W3;3
1;3 .n/C10368W3;3
1;6 .n/C3888W3;3
1;3 .n/
�10368W3;3
2;3 .n/�34992W3;3
1;1 .n
3/C93312W
3;31;2 .
n
3/:
Appealing to (2.2)-(2.6) we obtain the desired result. �
REFERENCES
[1] A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sumsX
lC12mDn
�.l/�.m/
andX
3lC4mDn
�.l/�.m/,” Adv. Theoretical Appl. Math., vol. 1, no. 1, pp. 27–48, 2006.
[2] N. Cheng and K. S. Williams, “Evaluation of some convolutions sums involving the sum of divisorfunctions,” Yokohama Math. J., vol. 52, pp. 39–57, 2005.
[3] B. Kokluce, “Representation numbers of two octonary quadratic forms,” Int. J. Number Theory,vol. 9, no. 7, pp. 1641–1648, 2013, doi: 10.1142/S1793042113500486.
REPRESENTATION NUMBERS OF SUMS OF QUADRATIC FORMS 283
[4] B. Kokluce, “Cusp forms in S6.�0.23//, S8.�0.23// and the number of representations of numbersby some quadratic forms in 12 and 16 variables,” Ramanujan J., vol. 34, pp. 187–208, 2014, doi:10.1007/s11139-013-9480-4.
[5] B. Kokluce and H. Eser, “Evaluation of some convolutions sums and representation of integersby certain quadratic forms in twelve variables,” Int. J. Number Theory, Accepted, 2016, doi:10.1142/S1793042117500415.
[6] G. Lomadze, “Representation of numbers by sums of the quadratic forms x21 Cx1x2Cx
22 ,” Acta
Arith., vol. 54, pp. 9–36, 1989.[7] G. Lomadze, “On the number of representations of integers by a direct sum of binary quad-
ratic forms with discriminant �23,” Georgian Math. J., vol. 4, no. 6, pp. 523–532, 1997, doi:10.1023/A:1022103424996.
[8] B. Ramakrishnan and B. Sahu, “On the number of representations of an integer by certain quad-ratic forms in sixteen variables,” Int. J. Number Theory, vol. 10, no. 8, pp. 1929–1937, 2014, doi:10.1142/S1793042114500638.
[9] S. Ramanujan, “On certain arithmetical functions,” Trans. Cambridge Philos. Soc., vol. 22, pp.159–184, 1916.
Authors’ addresses
Bulent KokluceFatih University, Elementary Mathematics Education, Buyukcekmece, 34500E-mail address: [email protected]
Muberra GurelIstanbul, TurkeyE-mail address: [email protected]