Open Access

Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions

Journal of Inequalities and Applications20132013:225

https://doi.org/10.1186/1029-242X-2013-225

Received: 20 February 2013

Accepted: 23 April 2013

Published: 6 May 2013

Abstract

In this paper, we study the convolution sums involving restricted divisor functions, their generalizations, their relations to Bernoulli numbers, and some interesting applications.

MSC: 11B68, 11A25, 11A67, 11Y70, 33E99.

Keywords

Bernoulli numbers generating functions divisor functions convolution sums

1 Introduction

The Bernoulli polynomials B k ( x ) , which are usually defined by the exponential generating function
t e x t e t 1 = k = 0 B k ( x ) t k k ! ,
play an important role in different areas of mathematics, including number theory and the theory of finite differences. The Bernoulli polynomials satisfy the following well-known identities :
B k ( x + 1 ) B k ( x ) = k x k 1 , k 1 , d d x B k ( x ) = k B k 1 ( x ) , k 1 ,
and
j = 0 N j k = B k + 1 ( N + 1 ) B k + 1 ( 0 ) k + 1 , k 1 .

We set B k = B k ( 0 ) . It is obvious from the way the polynomials B k ( x ) are constructed that all the B k are rational numbers. It can be shown that B 2 k + 1 = 0 for k 1 , and is alternatively positive and negative for even k. The B k are called Bernoulli numbers.

Throughout the paper, we use the following arithmetical functions and q-series (sometimes defined by product expressions). For any integer N 1 , l , s N { 0 } , we define
σ l ( N ) : = d | N d l , σ s ( N ; p ) : = d | N N d 0 ( mod p ) d s , σ s ( N ) : = σ s ( N ; 2 ) = d | N N / d odd d s , σ ˜ s ( N ) : = d | N ( 1 ) d 1 d s , S i ( N ) : = k = 1 N 1 k i .
For q C with | q | < 1 , we consider the q-series:
A ( q ) : = N = 1 σ 1 ( N ) q N , B ( q ) : = N = 1 σ 3 ( N ) q N , C ( q ) : = N = 1 σ 5 ( N ) q N , N = 1 a ( N ) q N : = q N = 1 ( 1 q N ) 2 ( 1 q 2 N ) 2 ( 1 q 3 N ) 2 ( 1 q 6 N ) 2 , N = 1 b ( N ) q N : = q N = 1 ( 1 q N ) 8 ( 1 q 2 N ) 8 , N = 1 c ( N ) q N : = q N = 1 ( 1 q N ) 16 ( 1 q 2 N ) 4 , N = 1 τ ( N ) q N : = q N = 1 ( 1 q N ) 24 , N = 1 l ( N ) q N : = q 2 N = 1 ( 1 q N ) 8 ( 1 q 2 N ) 4 ( 1 q 4 N ) 8 .
The exact evaluation of the basic convolution sum
k = 1 N 1 σ 1 ( k ) σ 1 ( N k )
first appeared in a letter from Besge to Liouville in 1862. The evaluation of such sums also appear in the works of Glaisher, Lahiri, Lehmer, Ramanujan, and Skoruppa. For instance, Ramanujan [1] obtained
k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) = 1 12 ( 5 σ 3 ( N ) + ( 1 6 N ) σ ( N ) ) and k = 1 N 1 σ 1 ( k ) σ 3 ( N k ) = 1 240 [ 21 σ 5 ( N ) + ( 10 30 N ) σ 3 ( N ) σ 1 ( N ) ]
(1)
using only elementary arguments. For a , b , N N , Ramanujan showed that the sum
S a , b ( N ) : = m = 1 N 1 σ a ( m ) σ b ( N m )
can be evaluated in terms of the quantities
σ a + b + 1 ( N ) , σ a + b 1 ( N ) , , σ 3 ( N ) , σ 1 ( N )
for the nine pairs ( a , b ) N 2 satisfying
a + b = 2 , 4 , 6 , 8 , 12 , a b , a b 1 ( mod 2 ) .
For explicit evaluations of S a , b ( N ) for different pairs ( a , b ) N 2 satisfying the above conditions stated, we refer to the papers of Ramanujan [1], [2], Huard et al. [3], Lahiri [4], and Glaisher [5], respectively. Levit [6] showed that the nine arithmetic evaluations of S a , b ( N ) (with a, b both odd) are the only ones using the theory of modular forms. In [4], Lahiri has given 37 sums of the form
( m 1 , , m r ) m 1 + + n r = N m 1 a 1 m r a r σ b 1 ( m 1 ) σ b r ( m r ) ,

where a 1 , , a r N 0 : = N { 0 } , b 1 , , b r N , each of which can be expressed as a finite linear combination of σ 1 ( N ) , σ 3 ( N ) , , σ b 1 + b 2 + + b r + r 1 ( N ) with coefficients which are polynomials in N of degree at most a 1 + + a r + r 1 with rational coefficients.

In 2002, Huard et al. [3] extended Melfi’s [7] result to
k < N / 2 σ 1 ( k ) σ 3 ( N 2 k ) = 1 48 σ 5 ( N ) + 1 15 σ 5 ( N 2 ) + ( 2 3 N ) 48 σ 3 ( N ) 1 240 σ 1 ( N 2 ) , k < N / 2 σ 3 ( k ) σ 1 ( N 2 k ) = 1 240 σ 5 ( N ) + 1 12 σ 5 ( N 2 ) + ( 1 3 N ) 24 σ 3 ( N 2 ) 1 240 σ 1 ( N ) ,
(2)

where N is an arbitrary positive integer.

Glaisher [5, 8, 9] extended Besge’s formula by replacing σ 1 ( N ) in the convolution sum in (1) by other arithmetical functions; for example, he obtained
24 k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) = 6 σ 3 ( N ) 6 σ 3 ( N 2 ) 6 N σ 1 ( N ) + 6 N σ 1 ( N 2 ) = 6 { σ 3 ( N ) N σ 1 ( N ) }
(3)
and
k = 1 N 1 σ 1 ( k ) σ 3 ( N k ) = 1 16 ( σ 5 ( N ) N σ 3 ( N ) ) .
(4)
Recently, Hahn [10] showed that
16 k < N σ ˜ 1 ( k ) σ ˜ 3 ( N k ) = σ ˜ 5 ( N ) + 2 ( N 1 ) σ ˜ 3 ( N ) + σ ˜ 1 ( N ) .
It is also interesting to note that the arithmetical functions (for example, τ ( n ) and a ( n ) ), coming out as coefficients of the q-series expansion of its corresponding q-products, do appear in the explicit evaluation of certain convolution sums. For instance, Lahiri (see [11]) proved that
k = 1 N 1 k ( N k ) σ 3 ( k ) σ 3 ( N k ) = 1 540 ( N 2 σ 7 ( N ) τ ( N ) )
and from Alaca and Williams (see [12]), we observe that
( k , m ) N 2 2 k + 3 m = N σ 1 ( k ) σ 1 ( m ) = 1 120 σ 3 ( N ) + 1 30 σ 3 ( N 2 ) + 3 40 σ 3 ( N 3 ) + 3 10 σ 3 ( N 6 ) + ( 1 24 N 12 ) σ 1 ( N 2 ) + ( 1 24 N 8 ) σ 1 ( N 3 ) 1 120 a ( N ) .
(5)

In [13], Simsek (and also in [[14], (2.17)] Simsek along with Ozden and Cagul) has studied other aspects of the arithmetical function σ 1 ( n ) in connection with the classical Jacobi and Euler functions. We also refer to Kim and Lee [[15], Lemma 2.1] and [16].

Thus the study of convolution sums and their applications is classical and they play an important role in number theory. The aim of this article is to first extend and generalize Glaishers formulas (stated in (3) and (4)). We indeed study the sums (in Section 3)
k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 2 n ( N k ) ) , k = 1 N 1 σ 1 ( 2 m k ) σ 3 ( 2 n ( N k ) ) , k = 1 N 1 σ 3 ( 2 m k ) σ 1 ( 2 n ( N k ) ) and r + s + t = N σ 1 ( 2 m r ) σ 1 ( 2 n s ) σ 1 ( 2 l t ) .
Then, we study and evaluate (in Section 4) sums of the type
k = 1 N 1 σ 1 ( 3 m k ; 3 ) σ 1 ( 3 n ( N k ) ; 3 ) and k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 ) .

As applications to our study and evaluations of convolution sums, we show that A ( q ) , B ( q ) , and C ( q ) are connected by a second-order differential equation (see Theorem 3.8).

In Section 5, we also prove some interesting congruence relations involving the coefficients of modular-like functions and divisor functions (see Theorem 5.3). As a sample, we obtain that if N 1 ( mod 8 ) , then the congruence
τ ( N ) σ 11 ( N ) ( mod 1 , 415 , 168 )

holds.

In Section 6, we present a generalization of Besge’s formula by considering certain combinatorial convolution sums (see Theorem 6.3). It should be noted that Proposition 6.1, Theorem 6.3, and Remark 6.4 exhibit amply the connection between the convolution sums and the Bernoulli numbers. Finally, we record special values of a ( N ) , τ ( N ) , b ( N ) , c ( N ) , and l ( N ) (for 1 N 45 ) and some convolution formulas in the Appendix.

2 Some weighted convolution sums

Let f ˆ ( a ) g ˆ ( b ) : = 1 2 { f ( a ) g ( b ) + f ( b ) g ( a ) } . It is easily checked that
k = 1 N 1 f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 f ( k ) g ( N k ) .
Lemma 2.1 For f , g : N C and N N , we have
k = 1 N 1 k f ˆ ( k ) g ˆ ( N k ) = N 2 k = 1 N 1 f ˆ ( k ) g ˆ ( N k ) .
Proof We observe that
k = 1 N 1 k f ˆ ( k ) g ˆ ( N k ) = 1 2 k = 1 N 1 k { f ( k ) g ( N k ) + f ( N k ) g ( k ) } = 1 2 k = 1 N 1 ( N k ) { f ( N k ) g ( k ) + f ( k ) g ( N k ) } = k = 1 N 1 ( N k ) f ˆ ( k ) g ˆ ( N k ) .
Hence
2 k = 1 N 1 k f ˆ ( k ) g ˆ ( N k ) = N k = 1 N 1 f ˆ ( k ) g ˆ ( N k )
and thus
k = 1 N 1 k f ˆ ( k ) g ˆ ( N k ) = N 2 k = 1 N 1 f ˆ ( k ) g ˆ ( N k ) .

 □

Lemma 2.2 For f , g : N C and N N , we have
k = 1 N 1 k 3 f ˆ ( k ) g ˆ ( N k ) = 1 4 N 3 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) + 3 N 2 k = 1 N 1 k 2 f ˆ ( N k ) g ˆ ( k ) .
Proof We note that
k = 1 N 1 k 3 f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 ( N k ) 3 f ˆ ( N k ) g ˆ ( k ) = N 3 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) 3 N 2 k = 1 N 1 k f ˆ ( N k ) g ˆ ( k ) + 3 N k = 1 N 1 k 2 f ˆ ( N k ) g ˆ ( k ) k = 1 N 1 k 3 f ˆ ( N k ) g ˆ ( k ) .
(6)
Then, for the second term on the right-hand side of (6), we replace k by N k in Lemma 2.1 and obtain
k = 1 N 1 ( N k ) f ˆ ( N k ) g ˆ ( k ) = N 2 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) .
This shows that
k = 1 N 1 k f ˆ ( N k ) g ˆ ( k ) = N 2 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) .
So, (6) can be written as
k = 1 N 1 k 3 { f ˆ ( k ) g ˆ ( N k ) + f ˆ ( N k ) g ˆ ( k ) } = N 3 2 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) + 3 N k = 1 N 1 k 2 f ˆ ( N k ) g ˆ ( k ) .
(7)
Here the left-hand side of (7) is
k = 1 N 1 k 3 { f ˆ ( k ) g ˆ ( N k ) + f ˆ ( N k ) g ˆ ( k ) } = k = 1 N 1 k 3 [ 1 2 { f ( k ) g ( N k ) + f ( N k ) g ( k ) } + 1 2 { f ( N k ) g ( k ) + f ( k ) g ( N k ) } ] = k = 1 N 1 k 3 { f ( k ) g ( N k ) + f ( N k ) g ( k ) } = 2 k = 1 N 1 k 3 f ˆ ( k ) g ˆ ( N k ) .
Therefore we have
2 k = 1 N 1 k 3 f ˆ ( k ) g ˆ ( N k ) = 1 2 N 3 k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) + 3 N k = 1 N 1 k 2 f ˆ ( N k ) g ˆ ( k ) .

This completes the proof. □

Remark 2.3 In general, we can express
k = 1 N 1 k 2 l + 1 f ˆ ( k ) g ˆ ( N k ) for  k N { 0 }
as a combination in terms of the sum
k = 1 N 1 k 2 j f ˆ ( k ) g ˆ ( N k ) with  j = 0 , 1 , 2 , , k

with their coefficients being polynomials in N of degree at most ( 2 l + 1 ) .

Lemma 2.4 For l , N N and f , g : N C , we have
{ 1 + ( 1 ) l + 1 } k = 1 N 1 k l f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) { j = 0 l 1 ( l j ) ( 1 ) j k j N l j } .
Proof We note that (for l N ),
k = 1 N 1 k l f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 ( N k ) l f ˆ ( N k ) g ˆ ( k ) = k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) { j = 0 l ( l j ) ( 1 ) j k j N l j } = ( 1 ) l k = 1 N 1 k l f ˆ ( N k ) g ˆ ( k ) + k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) { j = 0 l 1 ( l j ) ( 1 ) j k j N l j } .
Observing the fact that
k = 1 N 1 k l f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 k l f ˆ ( N k ) g ˆ ( k )
for any l N , we obtain
{ 1 + ( 1 ) l + 1 } k = 1 N 1 k l f ˆ ( k ) g ˆ ( N k ) = k = 1 N 1 f ˆ ( N k ) g ˆ ( k ) { j = 0 l 1 ( l j ) ( 1 ) j k j N l j } .

This proves Lemma 2.4. □

Remark 2.5 By iteration process, in principle, the convolution sum k = 1 N 1 k l f ˆ ( k ) g ˆ ( N k ) can be evaluated for any odd l N .

Let
f ˆ ( a ) g ˆ ( b ) h ˆ ( c ) : = 1 6 { f ( a ) g ( b ) h ( c ) + f ( a ) g ( c ) h ( b ) + f ( b ) g ( a ) h ( c ) + f ( b ) g ( c ) h ( a ) + f ( c ) g ( a ) h ( b ) + f ( c ) g ( b ) h ( a ) } .
Lemma 2.6 For f : N C and N N . Then we have
( r , s , t ) N 3 r + s + t = N r f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) = N 3 ( r , s , t ) N 3 r + s + t = N f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) .
Proof From the cyclic transformation, r s t r , we observe that
( r , s , t ) N 3 r + s + t = N r f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) = ( r , s , t ) N 3 r + s + t = N s f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) = ( r , s , t ) N 3 r + s + t = N t f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) .
Therefore, we have
( r , s , t ) N 3 r + s + t = N N f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) = ( r , s , t ) N 3 r + s + t = N ( r + s + t ) f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) = 3 ( r , s , t ) N 3 r + s + t = N r f ˆ ( r ) f ˆ ( s ) f ˆ ( t ) .

Now Lemma 2.6 follows. □

Lemma 2.7 For f : N C and N N . Then we have (with g ( m ) = m f ( m ) for all m N )
3 ( r , s , t ) N 3 r + s + t = N r 2 f ( r ) f ( s ) f ( t ) = N 2 ( r , s , t ) N 3 r + s + t = N f ( r ) f ( s ) f ( t ) + 6 t = 1 N 2 ( r , s ) N 2 r + s = N t g ( r ) g ( s ) .
Proof As in Lemma 2.6, again by the cyclic transformation r s t r , we observe that
( r , s , t ) N 3 r + s + t = N r 2 f ( r ) f ( s ) f ( t ) = ( r , s , t ) N 3 r + s + t = N s 2 f ( r ) f ( s ) f ( t ) = ( r , s , t ) N 3 r + s + t = N t 2 f ( r ) f ( s ) f ( t )
(8)
and
( r , s , t ) N 3 r + s + t = N r s f ( r ) f ( s ) f ( t ) = ( r , s , t ) N 3 r + s + t = N s t f ( r ) f ( s ) f ( t ) = ( r , s , t ) N 3 r + s + t = N t r f ( r ) f ( s ) f ( t ) .
(9)
Therefore, we get (from (8) and (9))
( r , s , t ) N 3 r + s + t = N n 2 f ( r ) f ( s ) f ( t ) = 3 ( r , s , t ) N 3 r + s + t = N r 2 f ( r ) f ( s ) f ( t ) + 6 ( r , s , t ) N 3 r + s + t = N r s f ( r ) f ( s ) f ( t ) = 3 ( r , s , t ) N 3 r + s + t = N r 2 f ( r ) f ( s ) f ( t ) + 6 t = 1 N 2 f ( t ) ( r , s ) N 2 r + s = N t g ( r ) g ( s ) ,

where g ( m ) = m f ( m ) : N C for all m N . This proves Lemma 2.7. □

3 The convolution sum k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) and its extensions

Proposition 3.1 (a) [[17], Theorem 15.1, p.184], we have
k < N / 2 σ ( k ) σ ( N 2 k ) = 1 24 { 2 σ 3 ( N ) + ( 1 3 N ) σ ( N ) + 8 σ 3 ( N / 2 ) + ( 1 6 N ) σ ( N / 2 ) } .
(b) [[3], Theorem 3], [[17], Theorem 15.3, p.188], we have
k < N / 3 σ 1 ( k ) σ 1 ( N 3 k ) = 1 24 { σ 3 ( N ) + ( 1 2 N ) σ 1 ( N ) + 9 σ 3 ( N 3 ) + ( 1 6 N ) σ 1 ( N 3 ) } .
Proposition 3.2 Let s , N N . Then we obtain
σ s ( N ; p ) = σ s ( N ) σ s ( N / p ) .
In particular, we have
σ s ( N ) = σ s ( N ) σ s ( N / 2 ) ,

which can also be seen in [[17], p.27].

Proof We can know that
σ s ( N ; p ) : = d | N N d 0 ( mod p ) d s = d | N d s d | N N d 0 ( mod p ) d s = σ s ( N ) d | N p d s = σ s ( N ) σ s ( N / p ) .

 □

Proposition 3.3 Let N be any positive integer. For m , n N { 0 } , where 0 m n , we have
(a) k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 2 n ( N k ) ) = 2 n + m 2 ( σ 3 ( N ) N σ 1 ( N ) ) . (b) k = 1 N 1 σ 1 ( 2 m k ) σ 3 ( 2 n ( N k ) ) = 2 3 n + m 4 ( σ 5 ( N ) N σ 3 ( N ) ) . (c) k = 1 N 1 σ 3 ( 2 m k ) σ 1 ( 2 n ( N k ) ) = 2 n + 3 m 4 ( σ 5 ( N ) N σ 3 ( N ) ) . (d) k = 1 N 1 σ 1 ( 2 m k ) { σ 3 ( 2 n + 1 ( N k ) ) k σ 1 ( 2 3 n + 2 ( N k ) ) } (a) = 2 3 n + m 1 ( σ 5 ( N ) N 2 σ 1 ( N ) ) .
Proof For the sake of completeness, we just hint the proof of (b). We note that σ k ( 2 a ) = 2 a k . Since σ k is a 2 k -scalar multiplicative function (i.e., σ k ( 2 N ) = σ k ( 2 ) σ k ( N ) ), we have
k = 1 N 1 σ 1 ( 2 m k ) σ 3 ( 2 n ( N k ) ) = 2 m + 3 n k = 1 N 1 σ 1 ( k ) σ 3 ( N k ) = 2 3 n + m 4 ( σ 5 ( N ) N σ 3 ( N ) )

by (4). The proofs of (a), (c), and (d) are similar to (b). □

Remark 3.4 Using Eq. (3) and Lemma 2.1, we obtain
k = 1 N 1 k σ 1 ( k ) σ 1 ( N k ) = N 2 k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) = N 8 { σ 3 ( N ) N σ 1 ( N ) } .
(10)
Theorem 3.5 Let N (≥3) be any integer with m , n , l N { 0 } . Then we have
r + s + t = N σ 1 ( 2 m r ) σ 1 ( 2 n s ) σ 1 ( 2 l t ) = 2 m + n + l 64 ( σ 5 ( N ) 3 N σ 3 ( N ) + 2 N 2 σ 1 ( N ) ) .
Proof Let r 1 l 1 = p m 1 N and l 1 0 ( mod p ) for some r 1 , l 1 N and prime p. Since p does not divide l 1 , we can write r 1 = p m 1 d for some d N . Therefore, we have
σ s ( p m 1 N ; p ) = r 1 | p m 1 N p m 1 N r 1 0 ( mod p ) r 1 s = d | N N d 0 ( mod p ) ( p m 1 d ) s = p m 1 s d | N N d 0 ( mod p ) d s = p m 1 s σ s ( N ; p ) .
(11)
We note that
r + s + t = N σ 1 ( 2 m r ) σ 1 ( 2 n s ) σ 1 ( 2 l t ) = 2 m + n + l t = 1 N 2 σ 1 ( t ) r + s = N t σ 1 ( r ) σ 1 ( s ) = 2 m + n + l t = 1 N 2 σ 1 ( t ) { 1 4 ( σ 3 ( N t ) ( N t ) σ 1 ( N t ) ) } ( by (3) ) = 2 m + n + l 4 { t = 1 N 2 σ 1 ( t ) σ 3 ( N t ) t = 1 N 2 ( N t ) σ 1 ( t ) σ 1 ( N t ) } = 2 m + n + l 4 { t = 1 N 2 σ 1 ( t ) σ 3 ( N t ) k = 1 N 2 k σ 1 ( N k ) σ 1 ( k ) }

and hence, we use Eq. (1) and Eq. (10) to obtain the result. □

Corollary 3.6 Let
f ( p ) { k = 1 p 1 σ 1 ( 2 m k ) σ 1 ( 2 n ( p k ) ) , k = 1 p 1 σ 1 ( 2 m k ) σ 3 ( 2 n ( p k ) ) , r + s + t = p σ 1 ( 2 m r ) σ 1 ( 2 n s ) σ 1 ( 2 l t ) } ,
where p = 2 q + 1 is an odd prime. Then
f ( p ) = a S 4 ( q + 1 ) + b S 3 ( q + 1 ) + c S 2 ( q + 1 ) + d S 1 ( q + 1 ) ,
where the coefficients a, b, c, d are listed in Table 1.
Table 1

Coefficients for p = 2 q + 1

f ( p ) : =

a

b

c

d

k = 1 p 1 σ 1 ( 2 m k ) σ 1 ( 2 n ( p k ) )

0

0

3 2 m + n + 1

2 m + n + 1

k = 1 p 1 σ 1 ( 2 m k ) σ 3 ( 2 n ( p k ) )

5 2 m + 3 n + 1

2 m + 3 n + 2

5 2 m + 3 n

2 m + 3 n

r + s + t = p σ 1 ( 2 m r ) σ 1 ( 2 n s ) σ 1 ( 2 l t )

5 2 m + n + l 1

3 2 m + n + l

2 m + n + l + 1

2 m + n + l 1

Remark 3.7 If p = 2 q + 1 is any odd prime, then
k = 1 2 q σ 1 ( k ) σ 1 ( 2 q + 1 k ) k = 1 2 q σ 1 ( k ) σ 3 ( 2 q + 1 k ) r + s + t = 2 q + 1 σ 1 ( r ) σ 1 ( s ) σ 1 ( t ) 0 ( mod S 1 ( q + 1 ) )
from Table 2.
Table 2

Convolution formulas for prime p = 2 q + 1

Convolution sum

f ( 2 q + 1 ) : =

k = 1 p 1 σ 1 ( k ) σ 1 ( p k )

1 3 q ( 10 q 2 + 9 q 1 )

k = 1 p 1 σ 1 ( k ) σ 1 ( p k )

2 q 2 ( q + 1 )

k = 1 p 1 σ 1 ( k ) σ 3 ( p k )

1 15 q ( 42 q 4 + 75 q 3 + 50 q 2 + 15 q 2 )

k = 1 p 1 σ 1 ( k ) σ 3 ( p k )

q 2 ( q + 1 ) ( 2 q 2 + 2 q + 1 )

r + s + t = p σ 1 ( r ) σ 1 ( s ) σ 1 ( t )

1 12 q 2 ( 14 q 3 + 5 q 2 8 q + 1 )

r + s + t = p σ 1 ( r ) σ 1 ( s ) σ 1 ( t )

1 2 q 4 ( q + 1 )

Now, we compare the values of the convolution sums in Table 3 [[17], p.148]. For almost all N N , we find that A ( N ) > B ( N ) and C ( N ) > D ( N ) (Figure 1). As an application to Theorem 3.5, we have the following.
Figure 1

A ( N ) , B ( N ) , C ( N ) , and D ( N ) .

Table 3

Convolution formulas

Convolution sum

 

A ( N ) : = k = 1 N 1 σ 1 ( k ) σ 1 ( N k )

1 12 ( 5 σ 3 ( N ) + ( 1 6 N ) σ 1 ( N ) )

B ( N ) : = k = 1 N 1 σ 1 ( k ) σ 1 ( N k )

1 4 ( σ 3 ( N ) N σ 1 ( N ) )

C ( N ) : = r + s + t = N σ 1 ( r ) σ 1 ( s ) σ 1 ( t )

7 192 σ 5 ( N ) + ( 5 96 5 32 N ) σ 3 ( N ) + ( 1 192 1 16 N + 1 8 N 2 ) σ 1 ( N )

D ( N ) : = r + s + t = N σ 1 ( r ) σ 1 ( s ) σ 1 ( t )

1 64 ( σ 5 ( N ) 3 N σ 3 ( N ) + 2 N 2 σ 1 ( N ) )

Theorem 3.8 The q-series A ( q ) , B ( q ) , C ( q ) are connected by the differential equation,
64 A 3 ( q ) = C ( q ) 3 q d B ( q ) d q + 2 q 2 d 2 A ( q ) d q 2 + 2 q d A ( q ) d q ,
(12)

where q d B ( q ) d q = N = 1 N σ 3 ( N ) q N and q 2 d 2 A ( q ) d q 2 = N = 1 N ( N 1 ) σ 1 ( N ) q N .

Proof We note that
64 A 3 ( q ) = 64 ( l = 1 σ 1 ( l ) q l ) ( m = 1 σ 1 ( m ) q m ) ( n = 1 σ 1 ( n ) q n ) = N = 3 ( 64 l + m + n = N σ 1 ( l ) σ 1 ( m ) σ 1 ( n ) ) q N = N = 3 { σ 5 ( N ) 3 N σ 3 ( N ) + 2 N 2 σ 1 ( N ) } q N
(13)

from Theorem 3.5. We note that σ 5 ( N ) 3 N σ 3 ( N ) + 2 N 2 σ 1 ( N ) is zero for N = 1 and N = 2 . Thus the differential Eq. (12) follows. □

Remark 3.9 We also note that from Eq. (1) and Eq. (3), we can determine the equations
16 A ( q ) B ( q ) = C ( q ) q d B ( q ) d q
(14)
and
4 A 2 ( q ) = B ( q ) q d A ( q ) d q .
(15)
From (14) and (15), Eq. (12) can also be deduced. Using [[18], (28)] and (15), we also get
( τ 2 , τ ) = 32 π 4 ( 4 A 2 ( q ) + q d A ( q ) d q ) ,
where
( z ; Λ τ ) = 1 z 2 + ω Λ τ , ω 0 { 1 ( z ω ) 2 1 ω 2 }

and Λ τ = Z + τ Z ( τ H the complex upper-half plane) is a lattice and z C .

4 The convolution sum k = 1 N 1 σ 1 ( k ; 3 ) σ 1 ( N k ; 3 ) and its extensions

Theorem 4.1 Let N be a positive integer. And let m , n N { 0 } . Then we have
k = 1 N 1 σ 1 ( 3 m k ; 3 ) σ 1 ( 3 n ( N k ) ; 3 ) = 3 n + m 1 { σ 3 ( N ; 3 ) N σ 1 ( N ; 3 ) } .
Proof From Proposition 3.2 and Eq. (11), we can deduce that
k = 1 N 1 σ 1 ( 3 m k ; 3 ) σ 1 ( 3 n ( N k ) ; 3 ) = 3 n + m k = 1 N 1 { σ 1 ( k ) σ 1 ( k 3 ) } { σ 1 ( N k ) σ 1 ( N k 3 ) } = 3 n + m { k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) k = 1 N 1 σ 1 ( k ) σ 1 ( N k 3 ) k = 1 N 1 σ 1 ( k 3 ) σ 1 ( N k ) + k = 1 N 1 σ 1 ( k 3 ) σ 1 ( N k 3 ) } .
Now, we note that
k = 1 N 1 σ 1 ( k ) σ 1 ( N k 3 ) = t < N / 3 σ 1 ( N 3 t ) σ 1 ( t ) , k = 1 N 1 σ 1 ( k 3 ) σ 1 ( N k ) = t < N / 3 σ 1 ( t ) σ 1 ( N 3 t ) , k = 1 N 1 σ 1 ( k 3 ) σ 1 ( N k 3 ) = t < N / 3 σ 1 ( t ) σ 1 ( N 3 t ) .

Therefore, the theorem follows from Eq. (1) and Proposition 3.1(b). □

Theorem 4.2 Let N be a positive integer and let m , n N { 0 } . Then we have
k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 ) = 2 m 3 n 120 [ 36 { σ 3 ( N ) σ 3 ( N 3 ) } 20 N σ 1 ( N ) 15 N σ 1 ( N ; 3 ) a ( N ) ] .
Proof We note that
k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 ) = 2 m 3 n k = 1 N 1 { σ 1 ( k ) σ 1 ( k 2 ) } { σ 1 ( N k ) σ 1 ( N k 3 ) } = 2 m 3 n { k = 1 N 1 σ 1 ( k ) σ 1 ( N k ) k < N / 3 σ 1 ( N 3 k ) σ 1 ( k ) k < N / 2 σ 1 ( k ) σ 1 ( N 2 k ) + 2 k + 3 m = N σ 1 ( k ) σ 1 ( m ) } .

Now, the theorem follows from (1), Proposition 3.1(a), (b), and Eq. (5). □

Remark 4.3 We can compare the two sums
k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 ) , k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 3 n ( N k ) )
as follows (see Table 4). Here we find the formula for the sum k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 3 n ( N k ) ) in a similar way as in Theorem 4.2 (see [[16], Theorem 2.5]).
Table 4

k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 ) and k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 3 n ( N k ) )

 

Convolution formula

k = 1 N 1 σ 1 ( 2 m k ; 2 ) σ 1 ( 3 n ( N k ) ; 3 )

2 m 3 n 120 [ 36 { σ 3 ( N ) σ 3 ( N 3 ) } 20 N σ 1 ( N ) 15 N σ 1 ( N ; 3 ) a ( N ) ]

k = 1 N 1 σ 1 ( 2 m k ) σ 1 ( 3 n ( N k ) )

1 240 [ ( 2 m + 3 3 ) ( 3 n + 3 7 ) σ 3 ( N ) 8 ( 2 m 1 ) ( 3 n + 3 7 ) σ 3 ( N 2 ) 27 ( 2 m + 3 3 ) ( 3 n 1 ) σ 3 ( N 3 ) + 216 ( 2 m 1 ) ( 3 n 1 ) σ 3 ( N 6 ) 5 { 3 2 m + 2 3 n + 1 + 6 ( 7 2 m 3 n 2 3 n 2 m ) N } σ 1 ( N ) + 20 ( 2 m 1 ) ( 2 3 n + 1 N 1 ) σ 1 ( N 2 ) + 15 ( 3 n 1 ) ( 3 2 m + 1 N 1 ) σ 1 ( N 3 ) 6 ( 2 m 1 ) ( 3 n 1 ) a ( N ) ]

5 Congruence relations of coefficients of certain modular-like functions

Lemma 5.1 Let n 2 . Then we have Table 5.
Table 5

Some convolution formulas

Convolution sum

Convolution formula

k = 1 N 1 σ 1 ( k ) σ 5 ( N k )

1 408 { 12 σ 7 ( N ) 17 N σ 5 ( N ) + 5 b ( N ) }

k = 1 N 1 σ 3 ( k ) σ 3 ( N k )

1 136 { σ 7 ( N ) b ( N ) }

k = 1 N 1 σ 1 ( k ) σ 7 ( N k )

1 992 { 17 σ 9 ( N ) 31 N σ 7 ( N ) + 448 l ( N ) + 14 c ( N ) }

k = 1 N 1 σ 3 ( k ) σ 5 ( N k )

1 496 { σ 9 ( N ) 32 l ( N ) c ( N ) }

k = 1 N 1 σ 5 ( k ) σ 5 ( N k )

1 2 , 764 { σ 11 ( N ) τ ( N ) + 692 τ ( N 2 ) }

k = 1 N 1 σ 1 ( k ) σ 9 ( N k )

1 27 , 640 { 310 σ 11 ( N ) 691 N σ 9 ( N ) + 381 τ ( N ) + 109 , 488 τ ( N 2 ) }

k = 1 N 1 σ 3 ( k ) σ 7 ( N k )

1 22 , 112 { 17 σ 11 ( N ) 17 τ ( N ) 13 , 112 τ ( N 2 ) }

Proof We use Table 15 in the Appendix, and the proof of Lemma 5.1 is now similar to the proof of Theorem 4.1. □

Remark 5.2 It is easy to observe that
k = 1 N 1 σ 5 ( k ) σ 5 ( N k ) = 8 17 k = 1 N 1 σ 3 ( k ) σ 7 ( N k )

when N is odd.

As an application to the explicit evaluation of convolution sums, using Table 5 we prove the following theorem.

Theorem 5.3 If N is odd, then

(a) b ( N ) N σ 5 ( N ) ( mod 48 ) . In particular N 1 ( mod 8 ) , we have b ( N ) N σ 5 ( N ) ( mod 192 ) .

(b) b ( N ) σ 7 ( N ) ( mod 2 , 176 ) . In particular N 1 ( mod 8 ) , we have b ( N ) σ 7 ( N ) ( mod 17 , 408 ) .

(c) τ ( N ) σ 11 ( N ) ( mod 176 , 896 ) . In particular N 1 ( mod 8 ) , we have τ ( N ) σ 11 ( N ) ( mod 1 , 415 , 168 ) .

Proof Since the proofs of (a), (b), and (c) are similar, we only prove (c). When N is odd, from Table 5, we see that
5 , 528 k = 1 N 1 2 σ 5 ( k ) σ 5 ( N k ) = σ 11 ( N ) τ ( N ) .
(16)
Since N is odd, k N k ( mod 2 ) , and hence either k is even or N k is even. Then, by (11), σ 5 ( k ) = 2 5 σ 5 ( k 2 ) or σ 5 ( N k ) = 2 5 σ 5 ( N k 2 ) . So, in general,
τ ( N ) σ 11 ( N ) ( mod 176 , 896 ) .
In particular, for N 1 ( mod 8 ) , we have Table 6 and
k = 1 N 1 2 σ 5 ( k ) σ 5 ( N k ) = k 0 , 1 ( mod 4 ) N 1 2 σ 5 ( k ) σ 5 ( N k ) + k 2 , 7 ( mod 8 ) N 1 2 σ 5 ( k ) σ 5 ( N k ) + k 3 , 6 ( mod 8 ) N 1 2 σ 5 ( k ) σ 5 ( N k ) .
(17)
If n 7 ( mod 8 ) , then there exists a prime p 7 ( mod 8 ) satisfying p 2 i 1 | n and p 2 i n . Thus
σ 5 ( n ) = σ 5 ( n p 2 i 1 ) σ 5 ( p 2 i 1 ) = σ 5 ( n p 2 i 1 ) ( 1 + p 5 + p 5 2 + + p 5 ( 2 i 1 ) ) 0 ( mod 8 ) .
(18)
Therefore, we obtain
σ 5 ( 8 k + 2 ) σ 5 ( 8 k + 7 ) 2 5 σ 5 ( 4 k + 1 ) σ 5 ( 8 k + 7 ) 0 ( mod 2 8 ) .
Arguing in a similar manner as in (18), we get
σ 5 ( 8 k + 3 ) 0 ( mod 4 )
and
σ 5 ( 4 k + 3 ) 0 ( mod 4 ) .
Therefore, we obtain
σ 5 ( 8 k + 3 ) σ 5 ( 8 k + 6 ) 2 5 σ 5 ( 8 k + 3 ) σ 5 ( 4 k + 3 ) 0 ( mod 2 9 ) .
(19)
Using σ 5 ( 4 l ) = 2 10 σ 5 ( l ) , (18), and Table 6, we get
k = 1 k 0 , 1 ( mod 4 ) N 1 2 σ 5 ( k ) σ 5 ( N k ) 0 ( mod 2 10 )
(20)
and
k = 1 k 2 , 3 ( mod 4 ) N 1 2 σ 5 ( k ) σ 5 ( N k ) 0 ( mod 2 8 ) .
(21)
Therefore, by (16), (20), and (21), we have
τ ( N ) σ 11 ( N ) ( mod 1 , 415 , 168 ) .
 □
Table 6

k and N k by mod 8

k (mod8)

0

1

2

3

4

5

6

7

N − k (mod8)

1

0

7

6

5

4

3

2

Remark 5.4 The same result (c) of Theorem 5.3 has been obtained by Lahiri by a different approach considering the Eisenstein series and so on (see [[4], (11.1)-(11.40)], [[19], p.28]).

Corollary 5.5 Suppose that N = 176 , 896 a + b is a prime number with ( 176 , 896 , b ) = 1 . Then τ ( N ) 0 for b 1 .

Proof Using Mathematica 8.0, we first find b with 1 b < 176 , 896 satisfying ( 176 , 896 , b ) = 1 . Then if N = 176 , 896 a + b is a prime number, then we find that by Theorem 5.3 (c), using Mathematica 8.0,
τ ( N ) σ 11 ( N ) b 11 + 1 0 ( mod 176 , 896 )

except for b = 1 . Thus the corollary follows. □

Example 5.6 Some values of b ( N ) , N σ 5 ( N ) , σ 7 ( N ) , τ ( N ) , and σ 11 ( N ) are listed in Table 7.
Table 7

Some values of b ( N ) , N σ 5 ( N ) , σ 7 ( N ) , τ ( N ) , and σ 11 ( N )

N

1

3

5

7

9

b(N)

1

12

−210

1,016

−2,043

N σ 5 ( N )

1

732

15,630

117,656

533,637

σ 7 ( N )

1

2,188

78,126

823,544

4,785,157

τ(N)

1

252

4,830

−16,744

−113,643

σ 11 ( N )

1

177,148

48,828,126

1,977,326,744

31,381,236,757

Corollary 5.7 Let N N . Then we have Table 8.
Table 8

Some convolution formulas

Convolution sum

Convolution formula

k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 408 { 768 σ 7 ( N ) + 5 b ( 2 N ) 320 b ( N ) }

k = 1 N σ 3 ( 2 k 1 ) σ 3 ( 2 N ( 2 k 1 ) )

1 136 { 64 σ 7 ( N ) b ( 2 N ) + 64 b ( N ) }

k = 1 N σ 1 ( 2 k 1 ) σ 7 ( 2 N ( 2 k 1 ) )

1 496 { 2 , 176 σ 9 ( N ) + 224 d ( 2 N ) 57 , 344 l ( N ) + 7 c ( 2 N ) 1 , 792 c ( N ) }

k = 1 N σ 3 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 496 { 256 σ 9 ( N ) 32 d ( 2 N ) + 8 , 192 l ( N ) c ( 2 N ) + 256 c ( N ) }

k = 1 N σ 5 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 2 , 764 { 1 , 024 σ 11 ( N ) τ ( 2 N ) + 1 , 716 τ ( N ) 708 , 608 τ ( N 2 ) }

k = 1 N σ 1 ( 2 k 1 ) σ 9 ( 2 N ( 2 k 1 ) )

1 27 , 640 { 317 , 440 σ 11 ( N ) + 381 τ ( 2 N ) 280 , 656 τ ( N ) 112 , 115 , 712 τ ( N 2 ) }

k = 1 N σ 3 ( 2 k 1 ) σ 7 ( 2 N ( 2 k 1 ) )

1 22 , 112 { 17 , 408 σ 11 ( N ) 17 τ ( 2 N ) + 4 , 296 τ ( N ) + 13 , 426 , 688 τ ( N 2 ) }

In particular, if N is odd, then we have Table 9.
Table 9

Some convolution formulas for odd N

Convolution sum

Convolution formula

k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 17 { 32 σ 7 ( N ) 15 b ( N ) }

k = 1 N σ 3 ( 2 k 1 ) σ 3 ( 2 N ( 2 k 1 ) )

1 17 { 8 σ 7 ( N ) + 9 b ( N ) }

k = 1 N σ 1 ( 2 k 1 ) σ 7 ( 2 N ( 2 k 1 ) )

1 496 { 2 , 176 σ 9 ( N ) + 224 d ( 2 N ) 57 , 344 l ( N ) + 7 c ( 2 N ) 1 , 792 c ( N ) }

k = 1 N σ 3 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 496 { 256 σ 9 ( N ) 32 d ( 2 N ) + 8 , 192 l ( N ) c ( 2 N ) + 256 c ( N ) }

k = 1 N σ 5 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) )

1 691 { 256 σ 11 ( N ) + 435 τ ( N ) }

k = 1 N σ 1 ( 2 k 1 ) σ 9 ( 2 N ( 2 k 1 ) )

1 691 { 7 , 936 σ 11 ( N ) 7 , 245 τ ( N ) }

k = 1 N σ 3 ( 2 k 1 ) σ 7 ( 2 N ( 2 k 1 ) )

1 691 { 544 σ 11 ( N ) + 147 τ ( N ) }

Proof Since the proofs for the convolution sums are similar, we only prove k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) . We write k = 1 2 N 1 σ 1 ( k ) σ 5 ( 2 N k ) as follows:
k = 1 2 N 1 σ 1 ( k ) σ 5 ( 2 N k ) = k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) + k = 1 N 1 σ 1 ( 2 k ) σ 5 ( 2 N 2 k ) = k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) + 2 6 k = 1 N 1 σ 1 ( k ) σ 5 ( N k )
(22)
by (11). Also since 2 k 1 and 2 N ( 2 k 1 ) are odd, we find that
k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) = k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) .

So, we obtain the formula for k = 1 N σ 1 ( 2 k 1 ) σ 5 ( 2 N ( 2 k 1 ) ) by (22) and Table 5. In particular, if N is odd, then b ( 2 N ) = b ( 2 ) b ( N ) , because b ( N ) is multiplicative. Therefore the proof is complete. □

By eliminating b ( N ) , c ( N ) , and l ( N ) in Table 5, we can obtain the following example.

Example 5.8 Let N N . Then for k = 2 , 3 , 4 , we have
s = 0 k 1 ( 2 k 2 s + 1 ) ( m = 1 N 1 σ 2 k 2 s 1 ( m ) σ 2 s + 1 ( N m ) ) = 1 2 { σ 2 k + 1 ( N ) N σ 2 k 1 ( N ) } .

For k = 2 , we refer to Eq. (1).

Now we present some convolution formulas in Table 10.
Table 10

Some convolution formulas

Convolution sum

Convolution formula

k = 1 N σ 1 ( 2 k ) σ 5 ( 2 N 2 k )

1 8 , 568 [ 4 { 85 σ 7 ( 2 N ) 4 , 032 σ 7 ( N ) } 357 ( 4 N 1 ) σ 5 ( 2 N ) + 17 σ 1 ( 2 N ) 105 { b ( 2 N ) 64 b ( N ) } ]

k = 1 N σ 3 ( 2 k ) σ 3 ( 2 N 2 k )

1 2 , 040 { 17 σ 7 ( 2 N ) 960 σ 7 ( N ) 17 σ 3 ( 2 N ) + 15 b ( 2 N ) 960 b ( N ) }

k = 1 N σ 1 ( 2 k ) σ 7 ( 2 N 2 k )

1 14 , 880 { 341 σ 9 ( 2 N ) 65 , 280 σ 9 ( N ) 620 ( 3 N 1 ) σ 7 ( 2 N ) 31 σ 1 ( 2 N ) 6 , 720 l ( 2 N ) + 1 , 720 , 320 l ( N ) 210 c ( 2 N ) + 53 , 760 c ( N ) }

k = 1 N σ 3 ( 2 k ) σ 5 ( 2 N 2 k )

1 156 , 240 { 341 σ 9 ( 2 N ) 80 , 640 σ 9 ( N ) 651 σ 5 ( 2 N ) + 310 σ 3 ( 2 N ) + 10 , 080 l ( 2 N ) 25 , 80 , 480 l ( N ) + 315 c ( 2 N ) 80 , 640 c ( N ) }

k = 1 N σ 5 ( 2 k ) σ 5 ( 2 N 2 k )

1 174 , 132 { 65 σ 11 ( 2 N ) 64 , 512 σ 11 ( N ) + 691 σ 5 ( 2 N ) 693 τ ( 2 N ) 108 , 108 τ ( N ) + 44 , 642 , 304 τ ( N 2 ) }

k = 1 N σ 1 ( 2 k ) σ 9 ( 2 N 2 k )

1 912 , 120 [ 3 { 4 , 550 σ 11 ( 2 N ) 3 , 491 , 840 σ 11 ( N ) } 7 , 601 ( 12 N 5 ) σ 9 ( 2 N ) + 3 , 455 σ 1 ( 2 N ) + 3 { 7 , 359 τ ( 2 N ) + 3 , 087 , 216 τ ( N ) + 1 , 233 , 272 , 832 τ ( N 2 ) } ]

k = 1 N σ 3 ( 2 k ) σ 7 ( 2 N 2 k )

1 331 , 680 [ 3 { 91 σ 11 ( 2 N ) 87 , 040 σ 11 ( N ) } 1 , 382 σ 7 ( 2 N ) 691 σ 3 ( 2 N ) + 3 { 685 τ ( 2 N ) 21 , 480 τ ( N ) 67 , 133 , 440 τ ( N 2 ) } ]

6 Certain combinatorial convolution sum

The four basic theta functions are defined below following the notation of Whittaker and Watson [[20], p.464]. Let τ C be such that Im ( τ ) > 0 . Set q = e π i τ so that | q | < 1 . For z C , we define (as in [17])
θ 1 ( z , q ) : = 2 N = 1 ( 1 ) N 1 q ( 2 N 1 ) 2 4 sin ( 2 N 1 ) z , θ 2 ( z , q ) : = 2 N = 1 q ( 2 N 1 ) 2 4 cos ( 2 N 1 ) z , θ 3 ( z , q ) : = 1 + 2 N = 1 q N 2 cos 2 N z , θ 4 ( z , q ) : = 1 + 2 N = 1 ( 1 ) N q N 2 cos 2 N z .
Jacobi (see [21, 22]) proved that
θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 1 ( z , q ) θ 4 ( z , q ) = 4 m = 0 q 2 m + 1 2 1 q 2 m + 1 sin ( 2 m + 1 ) z
(23)
and
θ 2 2 ( 0 , q ) θ 3 2 ( 0 , q ) θ 1 2 ( z , q ) θ 4 2 ( z , q ) = 8 m = 0 m q m 1 q 2 m ( 1 cos 2 m z ) .
(24)
From (23) and (24), we deduce that
m = 1 m q m 1 q 2 m ( 1 cos 2 m z ) = 2 ( m = 0 q 2 m + 1 2 1 q 2 m + 1 sin ( 2 m + 1 ) z ) 2 .
(25)
Equating coefficients of q N ( N N ) on the left- and right-hand sides of (25) (see [[17], p.16]), then we obtain the arithmetical equality involving the trigonometric functions
m N m | N N m odd m ( 1 cos 2 m z ) = ( a , b , x , y ) N 4 a x + b y = 2 N a , b , x , y odd ( cos ( a b ) z cos ( a + b ) z ) .
(26)
If we expand each cosine in powers of z using
1 cos z = k = 1 ( 1 ) k 1 z 2 k 2 k !
and equate coefficients of z 2 k 2 k ! ( k N ), then we obtain
m N m | N N m odd 2 2 k m 2 k + 1 = ( a , b , x , y ) N 4 a x + b y = 2 N a , b , x , y odd ( ( a + b ) 2 k ( a b ) 2 k ) , k , N N .
(27)

A generalized Besge formula due to Liouville is as follows.

Proposition 6.1 (See [[17], Theorem 12.3])

Let k N and N N , where k , N 2 . Then we have
s = 0 k 1 ( 2 k 2 s + 1 ) ( m = 1 N 1 σ 2 k 2 s 1 ( m ) σ 2 s + 1 ( N m ) ) = 2 k + 3 4 k + 2 σ 2 k + 1 ( N ) + ( k 6 N ) σ 2 k 1 ( N ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 j σ 2 k + 1 2 j ( N ) ,

where B 2 j is the 2jth Bernoulli number.

Let σ s , o o ( N ) : = d | N d odd n d odd d s . Now we prove the following lemma.

Lemma 6.2 We have
s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 2 N 1 σ 2 k 2 s 1 , o o ( m ) σ 2 s + 1 , o o ( 2 N m ) = 1 4 σ 2 k + 1 ( 2 N ) .
Proof First we consider
( a , b , x , y ) N 4 , a x + b y = 2 N , a b x y 1 mod 2 ( ( a + b ) 2 k ( a b ) 2 k ) = ( a , b , x , y ) N 4 , a x + b y = 2 N , a , b , x , y odd ( r = 0 2 k ( 2 k r ) a 2 k r b r r = 0 2 k ( 2 k r ) ( 1 ) r a 2 k r b r ) = 2 ( a , b , x , y ) N 4 , a x + b y = 2 N , a , b , x , y odd r = 0 , r odd 2 k ( 2 k r ) a 2 k r b r = 2 s = 0 k 1 ( 2 k 2 s + 1 ) ( a , b , x , y ) N 4 , a x + b y = 2 N , a , b , x , y odd a 2 k 2 s 1 b 2 s + 1 = 2 s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 2 N 1 ( a | m , a odd , m a odd a 2 k 2 s 1 ) ( b | 2 N m , b odd , 2 N m b odd b 2 s + 1 ) = 2 s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 2 N 1 σ 2 k 2 s 1 , o o ( m ) σ 2 s + 1 , o o ( 2 N m ) .
Now the right-hand side of (27) is
d N , d | N , d N 2 2 2 k d 2 k + 1 = 2 2 k { d | N d 2 k + 1 d | N 2 d 2 k + 1 } = 2 2 k { σ 2 k + 1 ( N ) σ 2 k + 1 ( N 2 ) } = 2 2 k σ 2 k + 1 ( N ) = 1 2 σ 2 k + 1 ( 2 N )
by (11). Therefore we obtain
s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 N 1 σ 2 k 2 s 1 , o o ( m ) σ 2 s + 1 , o o ( N m ) = 1 4 σ 2 k + 1 ( N ) .

 □

As a consequence of Proposition 6.1 and Lemma 6.2, we have the following.

Theorem 6.3
s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 N 1 σ 2 k 2 s 1 ( 2 m ) σ 2 s + 1 ( 2 N 2 m ) = 2 k + 3 4 k + 2 σ 2 k + 1 ( 2 N ) 1 4 σ 2 k + 1 ( 2 N ) + ( k 6 2 N ) σ 2 k 1 ( 2 N ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 j σ 2 k + 1 2 j ( 2 N ) ,

where B 2 j is the 2jth Bernoulli number.

Proof From Proposition 6.1, we get
s = 0 k 1 ( 2 k 2 s + 1 ) ( m = 1 2 N 1 σ 2 k 2 s 1 ( m ) σ 2 s + 1 ( 2 N m ) ) = 2 k + 3 4 k + 2 σ 2 k + 1 ( 2 N ) + ( k 6 2 N ) σ 2 k 1 ( 2 N ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 j σ 2 k + 1 2 j ( 2 N )
(28)
by replacing N 2 N . From Lemma 6.2 we know that
s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 2 N 1 σ 2 k 2 s 1 , o o ( m ) σ 2 s + 1 , o o ( 2 N m ) = 1 4 σ 2 k + 1 ( 2 N ) = s = 0 k 1 ( 2 k 2 s + 1 ) m = 1 N σ 2 k 2 s 1 ( 2 m 1 ) σ 2 s + 1 ( 2 N 2 m + 1 ) .
(29)

By subtracting (29) from (28), the theorem follows. □

Remark 6.4 It is well known that
B 0 = 1 , B 1 = 1 2 , B 2 = 1 6 , B 2 j + 1 = 0 ( j 1 ) , .
In Proposition 6.1, taking N = 1 , we obtain
0 = 2 k + 3 4 k + 2 + ( k 6 1 ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 j = ( 1 2 k + 1 + 1 2 ) + ( k 6 1 ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 j = 1 2 k + 1 ( 2 k + 1 0 ) B 0 + 1 2 k + 1 ( 2 k + 1 1 ) B 1 + 1 2 k + 1 ( 2 k + 1 2 ) B 2 + 1 2 k + 1 j = 3 2 k ( 2 k + 1 j ) B j
and this implies the well-known identity involving the Bernoulli numbers B j
1 2 k + 1 j = 0 2 k ( 2 k + 1 j ) B j = 0 .
(30)
In Theorem 6.3, taking N = 1 , we obtain
0 = ( 1 2 k + 1 + 1 2 ) ( 1 + 2 2 k + 1 ) + ( k 6 1 1 ) ( 1 + 2 2 k 1 ) 1 4 2 2 k + 1 + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 k + 1 2 j ( 1 + 2 2 k + 1 2 j ) .
Then we have
0 = ( 1 2 k + 1 + 1 2 ) + ( k 6 1 ) + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 k + 1 2 j + 1 2 k + 1 2 2 k + 1 + 2 2 k 2 2 k 1 + k 6 2 2 k 1 + 1 2 k + 1 j = 2 k ( 2 k + 1 2 j ) B 2 k + 1 2 j 2 2 k + 1 2 j 1 2 2 k 1 2 2 k 1 .
(31)
Using (30) and (31), we get
0 = 1 2 k + 1 ( 2 k + 1 0 ) 2 2 k + 1 1 2 k + 1 ( 2 k + 1 1 ) 2 2 k + 1 1 ( 1 2 ) + 1 2 k + 1 ( 2 k + 1 2 ) 2 2 k + 1 2 + 1 2 k + 1 j = 3 2 k ( 2 k + 1 j ) ( 1 ) j B 2 k + 1 2 j 2 2 k + 1 2 j ,
and hence again we obtain a more general relation between Bernoulli numbers and the Faulhaber sum
1 2 k + 1 j = 0 2 k ( 2 k + 1 j ) ( 1 ) j B 2 k + 1 j 2 2 k + 1 j = 1 + 2 2 k = S 2 k ( 3 ) .
(32)

See, for example, [23].

Appendix

The values of a ( N ) , b ( N ) , c ( N ) , l ( N ) , and τ ( N ) (for ( 1 N 45 )) are given in Tables 11, 12, 13, 14, and 15 respectively. We first write the corresponding product expression into a q-series sum of a finite number of terms (i.e., q-series sum of N number of terms) with an error which tends to zero as N . Then we use Mathematica 8.0 to compare the corresponding coefficients up to N 45 .
Table 11

a ( N ) for N ( 1 N 45 )

N

a ( N )

1

1

2

−2

3

−3

4

4

5

6

6

6

7

−16

8

−8

9

9

10

−12

11

12

12

−12

13

38

14

32

15

−18

16

16

17

−126

18

−18

19

20

20

24

21

48

22

−24

23

168

24

24

25

−89

26

−76

27

−27

28

−64

29

30

30

36

31

−88

32

−32

33

−36

34

252

35

−96

36

36

37

254

38

−40

39

−114

40

−48

41

42

42

−96

43

−52

44

48

45

54

Table 12

b ( N ) for N ( 1 N 45 )

N

b ( N )

1

1

2

−8

3

12

4

64

5

−210

6

−96

7

1,016

8

−512

9

−2,043

10

1,680

11

1,092

12

768

13

1,382

14

−8,128

15

−2,520

16

4,096

17

14,706

18

16,344

19

−39,940

20

−13,440

21

12,192

22

−8,736

23

68,712

24

−6,144

25

−34,025

26

−11,056

27

−50,760

28

65,024

29

−102,570

30

20,160

31

227,552

32

−32,768

33

13,104

34

−117,648

35

−213,360

36

−130,752

37

160,526

38

319,520

39

16,584

40

107,520

41

10,842

42

−97,536

43

−630,748

44

69,888

45

429,030

Table 13

c ( N ) for N ( 1 N 45 )

N

c ( N )

1

1

2

−16

3

100

4

−256

5

−154

6

2,496

7

−4,536

8

−4,096

9

23,085

10

−13,920

11

−38,996

12

39,936

13

37,806

14

15,232

15

−146,472

16

−65,536

17

311,442

18

−74,448

19

128,244

20

−222,720

21

−912,352

22

898,368

23

767,192

24

638,976

25

−1,405,121

26

−2,849,504

27

2,699,496

28

243,712

29

1,659,454

30

2,171,520

31

−5,370,592

32

−1,048,576

33

−1,704,144

34

3,962,592

35

2,533,552

36

−1,191,168

37

8,258,694

38

−5,046,080

39

−14,176,264

40

−3,563,520

41

12,739,178

42

−2,376,192

43

11,055,308

44

14,373,888

45

−12,992,274

Table 14

l ( N ) for N ( 1 N 45 )

N

l ( N )

1

0

2

1

3

−8

4

16

5

32

6

−156

7

112

8

256

9

−576

10

870

11

−536

12

−2,496

13

4,384

14

−952

15

336

16

4,096

17

−17,472

18

4,653

19

5,848

20

13,920

21

33,152

22

−56,148

23

−17,584

24

−39,936

25

6,528

26

178,094

27

−11,088

28

−15,232

29

−171,872

30

−135,720

31

126,912

32

65,536

33

326,976

34

−247,662

35

−105,056

36

74,448

37

−123,488

38

315,380

39

−425,200

40

222,720

41

−350,848

42

148,512

43

706,728

44

−898,368

45

532,512

Table 15

τ ( N ) for N ( 1 N 45 )