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Teimuraz Vepkhvadze Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 6, ( Part - 5) June 2015, pp.01-04
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RESEARCH ARTICLE
OPEN ACCESS
On The Number of Representations of a Positive Integer By
Certain Binary Quadratic Forms Belonging To Multi-Class
Genera
TEIMURAZ VEPKHVADZE
(Department of Mathematics, Iv. Javakhishvili Tbilisi State University, Georgia,
Abstract
It is well known how to find the formulae for the number of representations of positive integers by the positive
binary quadratic forms which belong to one-class genera. In this paper we obtain the formulae for the number of
representations by certain binary forms with discriminants –80 and –128 belonging to the genera having two
classes.
Key words – positive integer, binary form, genera, number of representations.
I.
Introduction
Let r (n; f ) denote a number of representations
of a positive integer n by a positive definite
quadratic form f with a number of variables s. It is
well known that, for case s  4 , r (n; f ) can be
represented as
r (n; f )   (n; f )  (n; f ) ,
where  (n; f ) is a “singular series” and  (n; f ) is
a Fourier coefficient of cusp form. This can be
expressed in terms of the theory of modular forms
by stating that
(1)
( ; f )  E( ; f )  X ( ),
 ( ; f )  1 

 r (n; f )Q ,
n
n 1
where   H  { : Im  0} , Q  e2 i , X() is a
cusp form and
E ( , f )  1 

  (n; f )Q
n
n 1
is the Eisenstein series corresponding to f.
Siegel [1] proved that if the number of variables of a
quadratic form f is s  4 , then
(2)
E ( ; f )  F ( ; f ) ,
where F ( ; f ) denote a theta-series of a genus
containing a primitive quadratic form f (both
positive-definite and indefinite).
From formula (2) follows the well-known Siegel’s
theorem [1]: the sum of the singular series corresponding to the quadratic form f is equal to the
average number of representations of a natural
number by a genus that contains the form f.
Later Ramanathan [2] proved that for any primitive
integral quadratic form with s  3 variables (except
for zero forms with variables s  3 and zero forms
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with variables s  4 whose discriminant is a
perfect square), there is a function
E ( , z; f )  1  e
(2 m  s ) i
4
|d




1
|2

S ( f H ; q)
s
q 1 H  q 2 ( q
( H , q ) 1
m
 H ) 2 (q
s m
 H) 2
,
| q  H |
z
which he called the Eisenstein-Siegel series and
which is regular for any fixed  when Im  0 and
Re z  2  s , analytically
extendable in a
2
neighborhood of z  0 , and that
F ( ; f )  E ( , z; f ) z 0
(3)
Here m and d are respectively the inertia index and
discriminant of f, S ( fH;q) is the Gaussian sum. For
s  4 the function E ( , z; f ) z 0 coincides with the
function E( , f ) and the formula (3) with Siegel’s
formula (2).
In [3] we proved that the function E( , z; f ) is
analytically extendable in a neighborhood of z  0
also in the case where f is any nonzero integral
binary
quadratic
form
and
that
1
F ( ; f )  E ( , z; f ) z  0 further, having defined
2
the Eisenstein’s series E( , f ) by the formulas
1
E ( , z; f ) z  0 for s  2 and
2
E ( ; f )  E ( , z; f ) z 0 for s  2 , we have
E ( ; f ) 
E ( , f )  1 


1
 ( n; f )Q n for s  2 and
2 n 12
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Teimuraz Vepkhvadze Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 6, ( Part - 5) June 2015, pp.01-04
E ( , f )  1 

  (n; f )Q
n


 1/2 
 
n 1
convenient formulas are obtained for calculating of
values of the function  (n; f ) in the case where f is
any positive integral form of variables s  2 [4],
[5].
Thus, if the genus of the quadratic form f contains
one class, then according to Siegel’s theorem,
1
r (n; f )    (n; f ) for s  3 , r ( n; f )   (n; f )
2
for s  2 and in that case the problem for obtaining
“exact” formulas for r (n; f ) is solved completely.
Some papers are devoted to the case of multi-class
genera. For example Van der Blij [6], Lomadze [7],
Vepkhvadze [5] have obtained formulae for r (n; f )
for certain special binary forms, which belong to
multi-class genera. In most cases their formulae for
r (n; f ) depend upon the coefficients in the
expansion of certain products of theta functions.
In this paper we obtain formulae for the number of
representations by the binary quadratic forms
3x12  2 x1 x2  7 x22 , 3x12  2 x1 x2  7 x22 ,



 h 
h 
1   3   1   



2  16  h 4  2  
if   2(mod8) ,   2 ,



1 h   
 16 
for s  2 . Moreover,
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



 h 

 if   6(mod8) .
  3  1  
 h

16
16  2
Lemma
2
([4],
Theorem
2).
Let
2
2
f  ax  2bxy  cy is a positive binary quadratic
 1/2

form with determinant   ac  b2 , (a, 2b, c)  1 ,
(a, 2)  1 ,   r 2 ,   2æ 1 (2ł1 ) ,
p l ||  ,
p  || n .  (n; f ) is a “sum of a generalized singular
n  2 m ,
series” corresponding to f,
p l ||  , p  || n , u 
p2,
 p . Then
p|n
pł 2 
 2
 (n; f ) 
3x12  2 x1 x2  11x22
3x12  2 x1 x2  11x22
and
belonging to the two-class genera.

p / , p 2
p
 



 

 
|u
    1

1/2
1  
 L(1,  ) 
 p  p

p| , p  2 

where
II.
Preliminaries
Below in this paper n, , ,  are positive integers;
l, æ, ,  are non-negative integers; u is an odd
positive integer, m, h are integers, p is a prime
number;  h  is a Legendre-Jacobi symbol; r (n; f )
u
denote the number of representations of a positive
integer n by the positive binary quadratic form f
with determinant .
Lemma 1 ([4], Lemma 14). Let
L ( ; m) 
1 2
,
if
2|æ, 0    æ 3 , 2|,
2  2 2
m  a (mod8) ;
if 2|æ, 0    æ 3 , 2|,
0,
m  a (mod8) ,
or 0    æ1 , 2ł ;
ma

  1  ( 1) 2


1 (ma)  1 æ

  1  ( 1) 2
 2 2 , if 2|æ,   æ, 2|,



1  1(mod 4) ;

  mu  u1
,
u 1
 is a square-free integer. Then
L(1,  )   if   1 ;
1æ
 22 ,
4
 1/ 2

  h 
1 h 
  1,
 1/2
2
  1(mod 4) ,
4
h
  if   3(mod 4) ,
 

1 h 
2
 3/ 2 if   2 ,
2
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if
 1æ
 2 2 , if 2|æ,   æ 2 ;

if 2|æ,   æ , 1  1(mod 4) ;
1 (  1)

1
  2  (1) 4


,
1 (  1) 
1 æ1
 1æ 
1
 2 2   1  (1) 4
 (  æ  2)2 2



if 2|æ,   æ , 2|, 1  1(mod 4) ;
1 (  1)  1 ( m  a )  1 æ

1
2
  1  ( 1) 4
 2 2 , if 2|æ,



  æ1 ,
2ł , 1  1(mod 4) ;
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Teimuraz Vepkhvadze Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 6, ( Part - 5) June 2015, pp.01-04
1 (  1) 
1 æ1

1
  1  (1) 4
(  æ  1)2 2
, if 2|æ,





  æ1 , 2ł , 1  1(mod 4) .
1 2
,
if
 22
m  a (mod8) ;
0,
m  a (mod8) ,
2łæ,
if
0    æ 3 ,
2|,
2łæ, 0    æ 3 , 2|,
or 0    æ 2 , 2ł ;
1 ( m  a )  1 ( æ1)

  1  ( 1) 4
, if 2łæ,   æ1,
 2 2



2|, m   (mod8) ;
P. Kaplan and K.S. Wiliams [6] have obtained formulae for the number of representations of an even
positive integer n by the forms f 3 and f 4 .
It is obvious that
r (n; f1 )  r (n; f 2 ) .
Thus by Siegel’s theorem
1
r (n; f1 )  r (n; f 2 )   (n; f1 ) .
2
The function  (n; f1 ) may be calculated by the
Lemma 2. So we have
Theorem 1. Let n  2 5 u , (u,10)  1 . Then
r (n; f1 )  r (n; f 2 ) 
1 ( m  a )  1 ( m  a )  1 ( æ1)

1
2
 2   1  (1) 4
, if 2łæ,
 2 2


  æ1, 2|, m   (mod 4) ;
1 ( m   )  1 ( æ1)

1
  1  (1) 4
, if 2łæ,   æ,
 2 2



2ł ,
m  1a (mod 4) ;
1 ( m  a )  1 ( m  a )  1 ( æ1)

1
2
  1  (1) 4
, if 2łæ,
 2 2



  æ,
2ł , m  1a (mod 4) .
  p   na   1 
  p 2 , if l    1 , 2 | ;

  p 
if l    1 , 2ł ;
0,
 p  1  
   p l   1  
    p l      l 
 l /2
 1  
1 1
p



    p   2 
p
p
 



 




,
if l   , 2 | l , 2 |  ;
   p l   1     p l      l  1 l / 2
 1  
p ,
   1  
 

2
  p  p   p 
if l   , 2 | l , 2 ł  ;
  p l    1  p (   l ) na   1 (l 1)


,
 1  
 
 p 2
p
p








if l   , 2 ł l .
III.
Basic results
The set of forms with discriminant –80 splits
into to genera of forms and each genus consists two
reduced classes of forms which are respectively
f1  3x12  2 x1 x2  7 x22 , f 2  3x12  2 x1 x2  7 x22
and
f3  x12  20 x22 , f 4  4 x12  5 x22 .
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1   u 
1 

2   5  
 5 




 
|u
for   0 , u  3(mod 4) ,

  1  ( 1)

 u 
 5 
 
 5 




 
|u
for 2|,   2 , u  3(mod 4) ,
or
for   2 , 2ł ,
u  1(mod 4) ,
 0 otherwise.
The set of forms with discriminant –128 splits into
two genera of forms and each genus consists two
reduced classes of forms which are respectively
f5  3x12  2 x1 x2  11x22 , f6  3x12  2 x1 x2  11x22
and
2
2
f7  x1  32 x2 , f8  4 x12  4 x1 x2  9 x22 .
P. Kaplan and K.S. Williams [8] have obtained formulae for the number of representations of an even
positive integer n by the forms f 6 and f 7 .
It is obvious that
r (n; f5 )  r (n; f6 ) .
Thus by Siegel’s theorem
1
r (n; f 5 )  r (n; f 6 )   (n; f 5 ) .
2
The function  (n; f5 ) can be calculated by the
Lemma 2.
So we have
Theorem 2. Let n  2 u , (u, 2)  1 . Then
r (n; f5 )  r (n; f 6 ) 
 2 
 for   0 ,



 
|u
u  3(mod8) ,
2
 2 




 
|u
and
for   2 , u  3(mod8) ,
for   4 , u  1(mod8) ,
or u  3(mod8) ,
 0 otherwise.
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Teimuraz Vepkhvadze Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 5, Issue 6, ( Part - 5) June 2015, pp.01-04
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References
[1] C.L. Siegel, Uber die analytische theorie der
quadratischen formen. Annals of mathematics,
36, 1935, 527-606.
[2] K.G. Ramanathan, On the analytic theory of
quadratic forms, Acta Arithmetica, 21, 1972,
423-436.
[3] T.V. Vepkhvadze, On a formula of Siegel,
Acta Arithmetica, 40, 1981/82, 125-142 (in
Russian).
[4] T.V.Vepkhvadze, The representation of
numbers by positive Gaussian binary quadratic
forms. Trudy Razmadze Tbilissi Mathematical
Institute 40, 1971, 21-58 (in Russian).
[5] T. Vepkhvadze, Modular properties of thetafunctions and representation of numbers by
positive
quadratic
forms.
Georgian
Mathematical Journal, 4, 1997, 385-400.
[6] F. Van der Blij, Binary quadratic forms of
discriminant – 23, Indagationes Mathematicae,
14, 1952, 489-503.
[7] G.A. Lomadze, On the representation of
numbers by binary diagonal quadratic forms,
Mathematical sbornik, 68, 119, 1965, 283-312
(in Russian).
[8] P. Kaplan and K.S. Williams, On the number
of representations of a positive integer by a
binary quadratic form. Acta Arithmetica, 114,
1, 2004, 87-98.
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