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Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8

  • Seokho Jin and Ho Park EMAIL logo
Published/Copyright: November 18, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we compute binomial convolution sums of divisor functions associated with the Dirichlet character modulo 8, which is the remaining primitive Dirichlet character modulo powers of 2 yet to be considered. To do this, we provide an explicit expression of a general modular form of even weight for Γ 0 ( 32 ) in terms of its basis, and use an identity of Huard, Ou, Spearman, and Williams. As an application, we also compute the number of representations by quadratic forms under parity conditions.

Keywords: divisor function; Dirichlet character; convolution sum; arithmetic identity; Eisenstein series
MSC 2010: 11A25; 11E25; 11F11; 11F20; 11F30

1 Introduction

For a nonnegative integer r and a positive integer n , let σ r ( n ) ≔ ∑ d ∣ n d r denote the r th power divisor function. Divisor functions are important arithmetic functions playing a fundamental role in number theory, and they appear as the coefficients of (quasi-)modular forms or the number of representations of integers by quadratic forms. In particular, by letting r k ( n ) ≔ # { ( x 1 ,…, x k ) ∈ Z k ∣ n = x 1 2 + … + x k 2 } , Jacobi showed in 1834 that r 4 ( n ) = 8 ( σ 1 ( n ) − 4 σ 1 ( n ⁄ 4 ) ) .

On the other hand, seeing the relation r 8 ( n ) = 2 r 4 ( n ) + ∑ m = 1 n − 1 r 4 ( m ) r 4 ( n − m ) , it is natural to compute the convolution sum of r 4 ( n ) , or σ 1 ( n ) . Convolution sums of divisor sums were first considered by Besge [1] with his formula

∑ m = 1 n − 1 σ 1 ( m ) σ 1 ( n − m ) = 5 12 σ 3 ( n ) + 1 12 σ 1 ( n ) − n 2 σ 1 ( n ) ,

and have been a subject of interest of mathematicians such as Liouville [2] and Ramanujan [3]. In particular, Liouville [2] introduced the notion of the binomial convolution sum of σ r ( n ) extending the aforementioned Besge’s formula and obtained the identity

∑ 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 j is the j th Bernoulli number defined by t e t − 1 = ∑ j = 0 ∞ B j t j j ! . Recently, further binomial convolution sums were considered by Kim and Bayad [4] for the modified divisor function σ r * ( n ) = ∑ d ∣ n 2 ∤ n ⁄ d d r . They obtained identities such as

∑ 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 2 σ 2 k − 1 * ( n ) , ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 ( − 1 ) m + 1 σ 2 k − 2 s − 1 * ( m ) σ 2 s + 1 * ( 2 n − m ) = n σ 2 k − 1 * ( 2 n ) .

In this case, we have simpler expressions on the right-hand sides.

On the other hand, σ r * ( n ) has another expression σ r * ( n ) = ∑ d ∣ n χ * ( n ⁄ d ) d r , where χ * is the real Dirichlet character modulo 2. For general real Dirichlet characters, it is well known (see, for example [5, Theorem 9.13]) that any primitive real Dirichlet character χ is of the form χ k ( n ) ≔ k n for a positive integer k , where k n is the Kronecker symbol. Convolution sum identities similar to those mentioned earlier but involving χ = χ − 4 were established by Cho et al. [6] about the binomial convolution sums, and recently, Aygin and Hong [7] computed single convolution sums. We also remark that there is a recent result on another kind of convolution sums of divisor functions coming from string theory by Fedosova et al. [8].

In this article, we consider the binomial convolution sums associated with χ 8 and χ − 8 . Note that there are no further primitive Dirichlet characters modulo powers of 2. The binomial convolution sums for σ k χ 1 , χ 1 , σ k χ * , χ 1 , and σ k χ − 4 , χ 1 were previously computed by Liouville [2], Kim and Bayad [9], and Cho et al. [10], respectively. For convenience, we give an explicit description of χ 8 ( n ) and χ − 8 ( n ) as follows:

χ 8 ( n ) = 1 , if  n ≡ 1 , 7 ( mod 8 ) , − 1 , if  n ≡ 3 , 5 ( mod 8 ) , 0 , if  n  is even , and χ − 8 ( n ) = 1 , if  n ≡ 1 , 3 ( mod 8 ) , − 1 , if  n ≡ 5 , 7 ( mod 8 ) , 0 , if  n  is even .

The following is the main theorem of this article. For Dirichlet characters χ and ψ , let σ k χ , ψ ( n ) ≔ ∑ d ∣ n χ ( n ⁄ d ) ψ ( d ) d k . We also let S ( l , m , k ) ≔ B 2 l + 1 , χ − 4 2 ( 2 l + 1 ) B 2 m + 1 , χ − 4 2 ( 2 m + 1 ) 4 k B 2 k , where B n , χ is a generalized Bernoulli number defined by ∑ n = 0 ∞ B n , χ n ! t n = ∑ a = 1 N χ ( a ) t a t e N t − 1 where N is the conductor of χ , and { C 2 k , i ( z ) ∣ 1 ≤ i ≤ 8 k − 9 or i = 8 k − 7 } is the basis for S 2 k ( Γ 0 ( 32 ) ) whose definition is explicitly given in Section 2.

Theorem 1.1

Let k and n be positive integers. Then, there exist real constants a 2 k , i such that the following holds:

( i ) ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) = 1 2 ( σ 2 k + 1 χ * , χ 1 ( n ) + σ 2 k χ − 4 , χ 1 ( n ) − 2 σ 2 k χ − 8 , χ 1 ( n ) ) − S ( k , 0 , k + 1 ) 2 2 k − 2 ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ n ] ∑ i a 2 k , i C 2 k , i , ( i i ) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 χ 8 , χ 1 ( m ) σ 2 s + 1 χ 8 , χ 1 ( n − m ) = 1 2 ( σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 4 , χ 1 ( n ) ) + S ( k , 0 , k + 1 ) 2 2 k − 2 ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) − [ n ] ∑ i a 2 k , i C 2 k , i , ( i i i ) ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( n − m ) = σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 8 , χ 1 ( n ) ,

where [ n ] f ( z ) for a modular form f ( z ) denotes the nth Fourier coefficient of f at the cusp i ∞ (for more details, see Section 3).

Note that Theorem 1.1 gives the asymptotics of the aforementioned binomial convolution sums, and in particular ( i i i ) gives a simple exact formula with no terms from cusp forms.

As an application, we have Corollaries 4.3 and 4.4 about (binomial) convolution sums of divisor functions associated with χ 8 and χ − 8 . For example, we have the following equality for an integer n ≥ 2 :

∑ m = 1 n − 1 σ 1 χ 8 , χ 1 ( m ) σ 1 χ 8 , χ 1 ( n − m ) = 1 4 ( σ 3 χ * , χ 1 ( n ) − σ 2 χ − 4 , χ 1 ( n ) + 20 σ 1 χ * , χ 1 ( n ) ) − [ n ] ∑ i a 2 , i C 2 , i ,

where a 2 , i are as in Theorem 1.1.

To compute the sums in ( i ) and ( i i ) of Theorem 1.1, we use the well known identity of Huard et al. ([11, Theorem 1] or Proposition 4.1 of this article). This helps reduce the level of the corresponding space of modular forms from Γ 0 ( 64 ) to Γ 0 ( 32 ) , to reduce the complexity of the computations, for instance, in finding a basis for the subspace of cusp forms. For example, in case ( i ) , we take f ( a , b , x , y ) = − 8 a − 8 b ( x − y ) 2 k and then the left-hand side of the identity in Proposition 4.1 is expressed as the following sum of a binomial convolution sum and a convolution sum of divisor functions associated with χ − 4 and χ − 8 :

2 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) − 4 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) ,

so to obtain ( i ) of Theorem 1.1, we need to compute the second sum

(1) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 k χ − 4 , χ 1 ( x ) σ 0 χ − 4 , χ 1 ( y ) .

Aygin and Hong computed the same kind of convolution sums of divisor function associate with χ − 4 with respect to x + y = n [7, Theorem 1.2]. The following theorem gives the corresponding evaluation in our case.

Theorem 1.2

Let l and m be nonnegative integers with 1 < l + m + 1 = k . Then, for each nonnegative integer n, there exist real numbers a ( l , m ) , i such that

∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = S ( l , m , k ) 2 2 k + 2 m − 1 ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ n ] ∑ i a ( l , m ) , i C 2 k , i ( z ) .

To prove this result, we note that the divisor functions in the left-hand side are the coefficients of Eisenstein series E ℓ χ − 4 , χ 1 ( z ) and E ℓ χ − 4 , χ 1 ( 2 z ) of odd weight for Γ 0 ( 32 ) associated with the Dirichlet character χ − 4 . In the following theorem, we find a general expression of a modular form f ( z ) of even weight for Γ 0 ( 32 ) in terms of its basis. We apply this theorem to E 2 l + 1 χ 1 , ψ 1 ( z ) E 2 m + 1 χ 2 , ψ 2 ( 2 z ) ∈ M 2 m + 2 l + 2 ( Γ 0 ( 32 ) ) for ( χ i , ψ i ) ∈ { ( χ 1 , χ − 4 ) , ( χ − 4 , χ 1 ) } and obtain formulas for convolution sums of divisor functions with respect to x + 2 y = n associated with the Dirichlet character χ − 4 . Let A ( 2 k ) ≔ 1 4 ( 2 2 k − 1 ) ζ ( 2 k ) for a positive integer k .

Theorem 1.3

Let k ≥ 2 and f ( z ) ∈ M 2 k ( Γ 0 ( 32 ) ) . Then, there exist constants d 2 k , i ∈ R such that

f ( z ) = A ( 2 k ) { 2 ( 2 2 k [ 0 ] 1 f + [ 0 ] 1 ⁄ 2 f ) E 2 k χ * , χ 1 ( z ) + ( [ 0 ] 1 ⁄ 4 f + [ 0 ] 3 ⁄ 4 f ) E 2 k χ * , χ 1 ( 2 z ) + ( [ 0 ] 1 ⁄ 8 f + [ 0 ] 3 ⁄ 8 f ) E 2 k χ * , χ 1 ( 4 z ) + 2 [ 0 ] 1 ⁄ 16 f ⋅ E 2 k χ * , χ 1 ( 8 z ) − 2 [ 0 ] 1 ⁄ 32 f ⋅ E 2 k χ * , χ 1 ( 16 z ) + 2 ( 2 2 k − 1 ) [ 0 ] 1 ⁄ 32 f ⋅ E 2 k χ 1 , χ 1 ( 32 z ) − 2 2 k ( [ 0 ] 1 ⁄ 4 f − [ 0 ] 3 ⁄ 4 f ) E 2 k χ − 4 , χ − 4 ( 4 z ) − 2 2 k ( [ 0 ] 1 ⁄ 8 f − [ 0 ] 3 ⁄ 8 f ) E 2 k χ − 4 , χ − 4 ( 8 z ) } + ∑ i d 2 k , i C 2 k , i ( z ) ,

where [ 0 ] r f is the constant term of of the Fourier expansion of f ( z ) at cusp r , and E 2 k χ , ψ ( z ) is the Eisenstein series associated with Dirichlet characters χ and ψ (for the definitions, see Section 2).

The organization of this article is as follows. In Section 2, we establish a basis for M 2 k ( Γ 0 ( 32 ) ) for k ≥ 2 and we prove Theorem 1.3. In Section 3, we provide convolution sums for divisor functions with the trivial character or the Dirichlet characters mod 4, and prove Theorem 1.2 as a special case of Theorem 3.4. In Section 4, we prove Theorem 1.1 and we give several binomial convolution sums for divisor functions with the Dirichlet characters mod 8. In Section 5, as an application of Theorem 1.3, we provide formulas for representations of positive integers by two types of quadratic forms with coefficients 1, 2, 4, and 8 and with certain parity conditions on the variables such that coefficients 1 and 2.

2 On the space M 2 k ( Γ 0 ( 32 ) )

In this section, we summarize some facts on the space M 2 k ( Γ 0 ( 32 ) ) that are necessary for the proof of our main theorem.

2.1 Eisenstein series

Let ℋ denote the complex upper half-plane. Let χ and ψ be Dirichlet characters mod L and mod M , respectively. For any positive integer k ≥ 3 , there is an Eisenstein series defined by

E k χ , ψ ( z ) ≔ ∑ ′ χ ( m ) ψ ( n ) ( m z + n ) k ( z ∈ ℋ ) ,

where ∑ ′ is the summation over all pairs of integers ( m , n ) except ( 0 , 0 ) . Following convention, we also denote E k ( z ) for E k χ 1 , χ 1 ( z ) . If ( χ ψ ) ( − 1 ) = ( − 1 ) k , it has the following Fourier expansion:

(2) E k χ , ψ ( z ) = 2 ( − 2 π i ) k W ( ψ ) M k ( k − 1 ) ! ∑ n = 0 ∞ σ k − 1 χ , ψ ( n ) q n ⁄ M ( q = e 2 π i z ) ,

where σ k − 1 χ , ψ ( n ) ≔ ∑ d ∣ n χ ( n ⁄ d ) ψ ( d ) d k − 1 and W ( ψ ) ≔ ∑ a = 0 M − 1 ψ ( a ) e 2 π i a ⁄ M is the Gauss sum. We record some specific values of the Gauss sums that are required in this article:

W ( χ 1 ) = 1 and W ( χ − 4 ) = 2 i .

2.2 Basis of M 2 k ( Γ 0 ( 32 ) )

The Dedekind eta function η ( z ) ≔ q 1 24 ∏ n = 1 ∞ ( 1 − q n ) is the holomorphic function that plays an important role in the theory of modular forms and has been the subject of active research. On the classification of the spaces of modular forms that are generated by eta-quotients ∏ t = 1 s η ( n i z ) t i , there is also a work of Rouse and Webb [12]. In this subsection, for each k , we find an explicit basis of cusp forms for the space S 2 k ( Γ 0 ( 32 ) ) . These basis elements are all eta quotients. We begin with the following well known necessary condition for eta-quotients.

Theorem 2.1

[13– 15] Let N be a positive integer. If f ( z ) = ∏ δ ∣ N η ( δ z ) r δ is an eta-quotient with k = 1 2 ∑ δ ∣ N r δ ∈ Z , with the additional properties that

∑ δ ∣ N δ r δ ≡ 0 ( mod 24 )

and

∑ δ ∣ N N δ r δ ≡ 0 ( mod 24 ) ,

then f ( z ) satisfies

f a z + b c z + d = χ ˜ ( d ) ( c z + d ) k f ( z ) ,

for every a b c d ∈ Γ 0 ( N ) . Here, the character χ ˜ is defined by χ ˜ ( d ) ≔ ( − 1 ) k s d , where s ≔ ∏ δ ∣ N δ r δ . Additionally, if c , d , and N be positive integers with d ∣ N and gcd ( c , d ) = 1 , then the order of vanishing of f ( z ) at the cusp c d is

N 24 gcd ( d , N ⁄ d ) d ∑ δ ∣ N gcd ( d , δ ) 2 r δ δ .

Note that the space M 2 k ( Γ 0 ( 32 ) ) has the following decomposition:

M 2 k ( Γ 0 ( 32 ) ) = E 2 k ( Γ 0 ( 32 ) ) ⊕ S 2 k ( Γ 0 ( 32 ) ) ,

where E 2 k ( Γ 0 ( 32 ) ) is the Eisenstein subspace of M 2 k ( Γ 0 ( 32 ) ) . The following lemma gives bases for the corresponding subspaces.

Lemma 2.2

Let k > 1 . The set of Eisenstein series

{ E 2 k χ 1 , χ 1 ( t z ) ∣ t = 1 , 2 , 4 , 8 , 16 , 32 } ∪ { E 2 k χ − 4 , χ − 4 ( 4 t z ) ∣ t = 1 , 2 }

is a basis for E 2 k ( Γ 0 ( 32 ) ) , and the set of eta-quotients

{ C 2 k , i ( z ) ∣ 1 ≤ i ≤ 8 k − 9 or i = 8 k − 7 } ,

where

C 2 k , i ( z ) = C i , 0 ( z ) = η 8 k − 4 i − 10 ( z ) η 7 ( 4 z ) η 4 i − 2 ( 32 z ) η 4 k − 2 i − 3 ( 2 z ) η ( 8 z ) η 2 i − 3 ( 16 z ) , if i = 4 , 8 ,…, 8 k − 12 , C i , 1 ( z ) = η 8 k − 4 i − 8 ( z ) η 2 ( 4 z ) η 2 ( 8 z ) η 4 i ( 32 z ) η 4 k − 2 i − 4 ( 2 z ) η 2 i ( 16 z ) , if i = 1 , 5 ,…, 8 k − 7 , C i , 2 ( z ) = η 8 k − 4 i − 10 ( z ) η 2 ( 4 z ) η ( 8 z ) η 4 i ( 32 z ) η 4 k − 2 i − 5 ( 2 z ) η 2 i − 2 ( 16 z ) , if i = 2 , 6 ,…, 8 k − 10 , C i , 3 ( z ) = η 8 k − 4 i − 12 ( z ) η ( 4 z ) η 2 ( 8 z ) η 4 i + 2 ( 32 z ) η 4 k − 2 i − 8 ( 2 z ) η 2 i + 1 ( 16 z ) , if i = 3 , 7 ,…, 8 k − 9 ,

is a basis for S 2 k ( Γ 0 ( 32 ) ) .

Proof

For the Eisenstein part, the result is a straightforward application of Theorem 4.5.2 of [16]. For the cusp form part, from section 6.1 in [17], for each k > 1 , we obtain

dim ( S 2 k ( Γ 0 ( 32 ) ) ) = 8 ( k − 1 ) .

On the other hand, utilizing Theorem 2.1, for any positive integer k ≥ 2 , the order of vanishing of C 2 k , i ( z ) at the cusp 1/32 can be checked as follows:

C i , 0 ( z ) : 4 , 8 , 12 ,…, 8 k − 12 , C i , 1 ( z ) : 1 , 5 , 9 ,…, 8 k − 7 , C i , 2 ( z ) : 2 , 6 , 10 ,…, 8 k − 10 , C i , 3 ( z ) : 3 , 7 , 11 ,…, 8 k − 9 .

Then, since the orders of C 2 k , i ( z ) at 1/32 are all distinct, these elements must be linearly independent. This completes the proof.□

Let A r = a b c d ∈ SL 2 ( Z ) such that − d ⁄ c = r . Then, the Fourier expansion of f ( z ) ∈ M k ( Γ 0 ( N ) ) at the cusp − d ⁄ c ∈ Q ∪ { i ∞ } is given by the Fourier series expansion of ( c z − a ) − k f ( A r − 1 z ) at the cusp i ∞ , given by

(3) ( c z − a ) − k f ( A r − 1 z ) = ∑ n ≥ 0 a r ( n ) e 2 π i ( n + κ ) z ⁄ h ,

where h is the width of Γ 0 ( N ) at the cusp r , and κ is the cusp parameter [18]. We use the notation [ n + κ ] r f ( z ) to denote a r ( n ) , and we write [ n ] , instead of [ n ] i ∞ . Moreover, the constant term of the Fourier expansion of f ( z ) at cusp r refers to the term [ 0 ] r f ( z ) . Note that if κ ≠ 0 , f is a cusp form and the constant term is 0.

Aygin and Hong [7] proved that any modular form in M 2 k ( Γ 0 ( 16 ) ) is expressed as a linear combination of the basis for M 2 k ( Γ 0 ( 16 ) ) by solving a system of linear equations with the constant terms of the Fourier expansion of the Eisenstein series at the inequivalent cusps of Γ 0 ( 16 ) . Our Theorem 1.3 establishes the same thing for Γ 0 ( 32 ) . In general, this process works for general Γ 0 ( N ) as the following proposition shows.

Proposition 2.3

Let k ≥ 4 be an even integer. For any basis { f 1 , … , f n } for the Eisenstein subspace of M k ( Γ 0 ( N ) ) or M k ( Γ 1 ( N ) ) , let a i , j ( 0 ) be the n × n matrix obtained from the following Fourier expansions of f i at the cusp r j = − d ⁄ c :

( c z − a ) − k f i ( A r j − 1 z ) = ∑ m ≥ 0 a i , j ( m ) e 2 π i ( m + κ r j ) z ⁄ h r j ,

where { r j ∣ j = 1 , … n } is a set of inequivalent cusps. Then, [ a i , j ( 0 ) ] is an invertible matrix.

Proof

Suppose that Γ is a congruence subgroup. Then, we have SL 2 ( Z ) ⋅ i ∞ = ⋃ j = 1 n Γ α j ⋅ i ∞ , where n is the number of inequivalent cusps of Γ and α j are representatives. Note that in general, there is a linear map ϕ : M k ( Γ ) → C n defined by f ↦ ( ( f ∣ k A α 1 − 1 ) ( i ∞ ) , … , ( f ∣ k A α n − 1 ) ( i ∞ ) ) , where A r ⋅ r = i ∞ and f ∣ k A r − 1 ( z ) ≔ ( c z − a ) − k f ( A r − 1 z ) for A r = a b c d ∈ SL 2 ( Z ) . Then, since the kernel is S k ( Γ ) , we have that

dim C ( E k ( Γ ) ) = dim C ( M k ( Γ ) ⁄ S k ( Γ ) ) ≤ n .

But when Γ = Γ 0 ( N ) or Γ 1 ( N ) , the equality holds as is seen in [17, Proposition 6.1 and Proposition 6.6]. This implies that in these cases, the map ϕ is an isomorphism, and hence, for any basis { f 1 , … , f n } for the Eisenstein subspace E k ( Γ ) , the vectors ϕ ( f 1 ) , … , ϕ ( f n ) must be linearly independent. This completes the proof.□

A set of inequivalent cusps of Γ 0 ( 32 ) are as follows:

R ( 32 ) = 1 , 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 3 4 , 3 8 .

The corresponding matrices such that A r − 1 ( i ∞ ) = r can be given by

A 1 ⁄ c = − 1 0 c − 1 for  c ∣ 32 , A 3 ⁄ 4 = − 1 1 − 4 3 , and A 3 ⁄ 8 = 3 − 1 − 8 3 ,

and their inverse matrices are

A 1 ⁄ c − 1 = − 1 0 − c − 1 for  c ∣ 32 , A 3 ⁄ 4 − 1 = 3 − 1 4 − 1 , and A 3 ⁄ 8 − 1 = 3 1 8 3 .

The following lemma evaluates the constant terms of the Fourier expansions of the Eisenstein series that form a basis for E 2 k ( Γ 0 ( 32 ) ) (Lemma 2.2) at the cusp in R ( 32 ) .

Lemma 2.4

Let k > 1 and r = − d c ∈ R ( 32 ) . We have

( i ) [ 0 ] r E 2 k χ 1 , χ 1 ( t z ) = gcd ( t , c ) t 2 k 2 ζ ( 2 k ) , for a n y t ∣ 32 , ( i i ) [ 0 ] r E 2 k χ − 4 , χ − 4 ( 4 t z ) = − 1 − 1 2 2 k 2 ζ ( 2 k ) , r = 1 4 t , 1 − 1 2 2 k 2 ζ ( 2 k ) , r = 3 4 t , 0 , otherwise , for a n y t = 1 , 2 .

Proof

The cases in ( i ) for t ∣ 16 and ( i i ) when t = 1 and r = 1 , 1 2 , 1 4 , 1 8 , 1 16 , 3 4 were proved by Aygin and Hong [7, Lemma 2.2], and the remaining cases can be proved in the same way. Here, we present a computation of [ 0 ] 3 ⁄ 8 E 2 k χ − 4 , χ − 4 ( 8 z ) for reader’s convenience: first we note that

E 2 k χ − 4 , χ − 4 ( 8 A 3 ⁄ 8 − 1 z ) = ∑ ′ χ − 4 ( m ) χ − 4 ( n ) ( 8 m A 3 ⁄ 8 − 1 z + n ) 2 k = ( 8 z + 3 ) 2 k ∑ ′ χ − 4 ( m ) χ − 4 ( n ) ( ( 24 m + 8 n ) z + ( 8 m + 3 n ) ) 2 k .

Then, by the Fourier expansion (3) at the cusp 3/8, we have

□ [ 0 ] 3 ⁄ 8 E 2 k χ − 4 , χ − 4 ( 8 z ) = ∑ ′ 24 m + 8 n = 0 χ − 4 ( m ) χ − 4 ( n ) ( 8 m + 3 n ) 2 k = 2 ∑ m = 1 2 ∤ m ∞ 1 m 2 k = 2 1 − 1 2 2 k ζ ( 2 k ) .

Now, we are ready to prove Theorem 1.3.

Proof of Theorem 1.3

Let k > 1 and f ( z ) ∈ M 2 k ( Γ 0 ( 32 ) ) . Then,

f ( z ) = ∑ t ∣ 32 a t E 2 k χ 1 , χ 1 ( t z ) + b 1 E 2 k χ − 4 , χ − 4 ( 4 z ) + b 2 E 2 k χ − 4 , χ − 4 ( 8 z ) + ∑ i c i C 2 k , i ( z ) ,

for some a t , b 1 , b 2 , c i ∈ C . Clearly, [ 0 ] r ∑ c i C 2 k , i ( z ) = 0 . From Lemma 2.4, we can compute the constant terms of Eisenstein series at various cusps, for instance,

[ 0 ] 1 f = ∑ t ∣ 32 a t [ 0 ] 1 E 2 k χ 1 , χ 1 ( t z ) + b 1 [ 0 ] 1 E 2 k χ − 4 , χ − 4 ( 4 z ) + b 2 [ 0 ] 1 E 2 k χ − 4 , χ − 4 ( 8 z ) = ∑ t ∣ 32 a t 1 t 2 k 2 ζ ( 2 k ) ,

and then, we obtain the following system of linear equations:

1 1 2 2 k 1 4 2 k 1 8 2 k 1 1 6 2 k 1 3 2 2 k 0 0 1 1 1 2 2 k 1 4 2 k 1 8 2 k 1 1 6 2 k 0 0 1 1 1 1 2 2 k 1 4 2 k 1 8 2 k − 1 − 1 2 2 k 0 1 1 1 1 1 2 2 k 1 4 2 k 0 − 1 − 1 2 2 k 1 1 1 1 1 1 2 2 k 0 0 1 1 1 1 1 1 0 0 1 1 1 1 2 2 k 1 4 2 k 1 8 2 k 1 − 1 2 2 k 0 1 1 1 1 1 2 2 k 1 4 2 k 0 1 − 1 2 2 k a 1 a 2 a 4 a 8 a 16 a 32 b 1 b 2 = 1 2 ζ ( 2 k ) [ 0 ] 1 f [ 0 ] 1 ⁄ 2 f [ 0 ] 1 ⁄ 4 f [ 0 ] 1 ⁄ 8 f [ 0 ] 1 ⁄ 16 f [ 0 ] 1 ⁄ 32 f [ 0 ] 3 ⁄ 4 f [ 0 ] 3 ⁄ 8 f .

Solving this, we obtain the following expression:

f ( z ) = A ( 2 k ) { ( 2 2 k + 1 [ 0 ] 1 f − 2 [ 0 ] 1 ⁄ 2 f ) E 2 k χ 1 , χ 1 ( z ) + ( − 2 2 k + 1 [ 0 ] 1 f + ( 2 2 k + 1 + 2 ) [ 0 ] 1 ⁄ 2 f − [ 0 ] 1 ⁄ 4 f − [ 0 ] 3 ⁄ 4 f ) E 2 k χ 1 , χ 1 ( 2 z ) + ( − 2 2 k + 1 [ 0 ] 1 ⁄ 2 f + ( 2 2 k + 1 ) [ 0 ] 1 ⁄ 4 f − [ 0 ] 1 ⁄ 8 f + ( 2 2 k + 1 ) [ 0 ] 3 ⁄ 4 f − [ 0 ] 3 ⁄ 8 f ) E 2 k χ 1 , χ 1 ( 4 z ) + ( − 2 2 k [ 0 ] 1 ⁄ 4 f + ( 2 2 k + 1 ) [ 0 ] 1 ⁄ 8 f − 2 [ 0 ] 1 ⁄ 16 f − 2 2 k [ 0 ] 3 ⁄ 4 f + ( 2 2 k + 1 ) [ 0 ] 3 ⁄ 8 f ) E 2 k χ 1 , χ 1 ( 8 z ) + ( − 2 2 k [ 0 ] 1 ⁄ 8 f + ( 2 2 k + 1 + 2 ) [ 0 ] 1 ⁄ 16 f − 2 [ 0 ] 1 ⁄ 32 f − 2 2 k [ 0 ] 3 ⁄ 8 f ) E 2 k χ 1 , χ 1 ( 16 z ) + ( − 2 2 k + 1 [ 0 ] 1 ⁄ 16 f + 2 2 k + 1 [ 0 ] 1 ⁄ 32 f ) E 2 k χ 1 , χ 1 ( 32 z ) + ( − 2 2 k [ 0 ] 1 ⁄ 4 f + 2 2 k [ 0 ] 3 ⁄ 4 f ) E 2 k χ − 4 , χ − 4 ( 4 z ) + ( − 2 2 k [ 0 ] 1 ⁄ 8 f + 2 2 k [ 0 ] 3 ⁄ 8 f ) E 2 k χ − 4 , χ − 4 ( 8 z ) } + ∑ i c i C 2 k , i ( z ) .

Finally, using E 2 k χ * , χ 1 ( z ) = E 2 k χ 1 , χ 1 ( z ) − E 2 k χ 1 , χ 1 ( 2 z ) and E 2 k χ * , χ 1 ( z ) = 2 2 k − 1 E 2 k χ * , χ 1 ( 2 z ) , we have

(4) E 2 k χ 1 , χ 1 ( z ) − ( 2 2 k + 1 ) E 2 k χ 1 , χ 1 ( 2 z ) + 2 2 k E 2 k χ 1 , χ 1 ( 4 z ) = − E 2 k χ * , χ 1 ( z ) .

Hence, we can simplify the aforementioned form as follows:

□ f ( z ) = A ( 2 k ) { 2 ( 2 2 k [ 0 ] 1 f + [ 0 ] 1 ⁄ 2 f ) E 2 k χ * , χ 1 ( z ) + ( [ 0 ] 1 ⁄ 4 f + [ 0 ] 3 ⁄ 4 f ) E 2 k χ * , χ 1 ( 2 z ) + ( [ 0 ] 1 ⁄ 8 f + [ 0 ] 3 ⁄ 8 f ) E 2 k χ * , χ 1 ( 4 z ) + 2 [ 0 ] 1 ⁄ 16 f ⋅ E 2 k χ * , χ 1 ( 8 z ) − 2 [ 0 ] 1 ⁄ 32 f ⋅ E 2 k χ * , χ 1 ( 16 z ) + 2 ( 2 2 k − 1 ) [ 0 ] 1 ⁄ 32 f ⋅ E 2 k χ 1 , χ 1 ( 32 z ) − 2 2 k ( [ 0 ] 1 ⁄ 4 f − [ 0 ] 3 ⁄ 4 f ) E 2 k χ − 4 , χ − 4 ( 4 z ) − 2 2 k ( [ 0 ] 1 ⁄ 8 f − [ 0 ] 3 ⁄ 8 f ) E 2 k χ − 4 , χ − 4 ( 8 z ) } + ∑ i c i C 2 k , i ( z ) .

3 Convolution sums for σ k χ , ψ when χ , ψ ∈ { χ 1 , χ − 4 }

In this section, we compute the convolution sums for σ k χ , ψ when χ , ψ ∈ { χ 1 , χ − 4 } to obtain formulas for convolution sums of divisor functions with respect to x + 2 y = n associated with the Dirichlet character χ − 4 . The result ( i ) of Theorem 3.4 gives the evaluation for the sum (1). Aygin and Hong computed the same kind of convolution sums of divisor function associate with χ − 4 with respect to x + y = n ([7, Theorem 1.2]). Our result in fact reproves some of the results of Aygin and Hong (Corollary 3.6).

We begin with the following lemma, which computes the Fourier expansion of the Dedekind eta function at the cusps in R ( 32 ) .

Lemma 3.1

Let m ∣ 32 be a positive integer.

(i) If c ∣ 32 , then

η ( m A 1 ⁄ c − 1 z ) = e π i 12 ( m ⁄ c + 2 ) c m ( − c z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e c n 2 π i 12 m ( c z + m ⁄ c + 1 ) , if c ∣ m , i e − c π i 12 m ( − c z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e m n 2 π i 12 , if m ∣ c a n d m ≠ c .

(ii)

η ( m A 3 ⁄ 4 − 1 z ) = e π i 12 ( 3 m ⁄ 4 − 4 ) 4 m ( 4 z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e n 2 π i 3 m ( 4 z + m ⁄ 4 − 1 ) , if 4 ∣ m , − i e π i 12 ( 4 ⁄ m + m ) ( 4 z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e m n 2 π i 12 , if m ∣ 4 a n d m ≠ 4 .

(iii)

η ( m A 3 ⁄ 8 − 1 z ) = e π i 12 ( 3 m ⁄ 8 − 2 ) 8 m ( 8 z + 3 ) 1 2 ∑ n = 1 ∞ 12 n e 2 n 2 π i 3 m ( 8 z − m ⁄ 8 + 3 ) , if 8 ∣ m , 8 ⁄ m 3 e π i 4 ( − 72 ⁄ m + m + 2 ) ( 8 z + 3 ) 1 2 ∑ n = 1 ∞ 12 n e m n 2 π i 12 , if m ∣ 8 a n d m ≠ 8 .

Proof

These can be proved as in the proof of [18, Proposition 2.1].□

We give two more lemmas for the constant terms of Eisenstein series at the cusps in R ( 32 ) that appear in the basis for M k ( Γ 0 ( 32 ) ) . Lemma 3.2 gives the constant terms of the Eisenstein series of small weight, and Lemma 3.3 deals with the Eisenstein series E k χ 1 , χ − 4 and E k χ − 4 , χ 1 for positive odd integers k > 1 .

Lemma 3.2

Let r = − d c ∈ R ( 32 ) and t = 1 , 2 . Then, we have

( i ) [ 0 ] r E 1 χ 1 , χ − 4 ( 4 t z ) = − i gcd ( c , t ) t L ( 1 , χ − 4 ) , if c < 2 t , 0 , if c = 2 t , 2 L ( 1 , χ − 4 ) , otherwise , ( i i ) [ 0 ] r E 1 χ − 4 , χ 1 ( t z ) = − 2 gcd ( c , t ) t L ( 1 , χ − 4 ) , if c < 2 t , 0 , if c = 2 t , − 4 i L ( 1 , χ − 4 ) , otherwise , ( i i i ) [ 0 ] r E 2 χ − 4 , χ − 4 ( 4 t z ) = − 1 − 1 2 2 2 ζ ( 2 ) , if r = 1 ⁄ 4 t , 1 − 1 2 2 2 ζ ( 2 ) , if r = 3 ⁄ 4 t , 0 , otherwise ,

where L ( s , χ ) ≔ ∑ n = 1 ∞ χ ( n ) n s .

Proof

From [7, Proof of Lemma 4.1], the following Eisenstein series can be expressed in terms of eta-quotients:

E 1 χ 1 , χ − 4 ( 4 z ) = π 2 η 10 ( 2 z ) η 4 ( z ) η 4 ( 4 z ) , E 1 χ − 4 , χ − 4 ( 4 z ) = − i π 2 η 4 ( 2 z ) η 4 ( 8 z ) η 4 ( 4 z ) , and E 1 χ − 4 , χ 1 ( z ) = − 2 i ⋅ E 1 χ 1 , χ − 4 ( 4 z ) = − i π η 10 ( 2 z ) η 4 ( z ) η 4 ( 4 z ) .

We will only compute [ 0 ] r E 1 χ 1 , χ − 4 ( 4 z ) here since the other cases can be proved similarly. Let ord r ( f ( z ) ) be the order of f ( z ) at the cusp r = − d c . By [18, Proposition 2.1], we have

ord r ( η ( m z ) ) = 1 24 m ( gcd ( c , m ) ) 2 .

Let T r ( z ) ≔ η 10 ( 2 A r − 1 z ) η 4 ( A r − 1 z ) η 4 ( 4 A r − 1 z ) . Then, the order of T r ( z ) at the cusp r is 1 24 ( 5 gcd ( c , 2 ) 2 − 4 − gcd ( c , 4 ) 2 ) ≥ 0 and vanishes precisely when c ≠ 2 (Note that c is a power of 2.). Therefore, we have

[ 0 ] 1 ⁄ 2 E 1 χ 1 , χ − 4 ( 4 z ) = 0 .

Now, we consider the case when c ≠ 2 . Consider the case when r = 1 . Using ( i ) of Lemma 3.1, we have

T 1 ( z ) = e 4 π i 12 1 2 ( − z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e n 2 π i 24 ( z + 3 ) 10 i e − π i 12 ( − z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e n 2 π i 12 4 e 6 π i 12 1 4 ( − z − 1 ) 1 2 ∑ n = 1 ∞ 12 n e n 2 π i 48 ( z + 5 ) 4 = − 1 2 ( z + 1 ) i + ∑ n = 1 ∞ a n q n ⁄ 32 .

Then, ( z + 1 ) − 1 T 1 ( z ) = − 1 2 i + ∑ n = 1 ∞ a n q n ⁄ 32 . Hence, we obtain [ 0 ] 1 E 1 χ 1 , χ − 4 ( 4 z ) = − i π 4 . The other cases can be proved in the same way.□

Lemma 3.3

Let r = − d c ∈ R ( 32 ) and t = 1 , 2 . For l ∈ N , we have

( i ) [ 0 ] r E 2 l + 1 χ 1 , χ − 4 ( 4 t z ) = 2 L ( 2 l + 1 , χ − 4 ) , if c > 2 t , 0 , otherwise , ( i i ) [ 0 ] r E 2 l + 1 χ − 4 , χ 1 ( t z ) = − 2 gcd ( c , t ) t 2 l + 1 L ( 2 l + 1 , χ − 4 ) , if c < 2 t , 0 , otherwise .

Proof

This lemma can also be proved using the same methods as in the proof of Lemma 2.4.□

Now, we are ready to compute convolution sums of divisor functions of the form σ l χ , ψ ( x ) for the case of odd l , providing a proof of Theorem 3.4.

Theorem 3.4

Let l and m be nonnegative integers with 1 < l + m + 1 = k , and let l ′ and m ′ be positive integers with l ′ + m + 1 = k and l + m ′ + 1 = k . Then, for each nonnegative integer, n there exist real numbers a ( l , m ) , i , b ( l ′ , m ) , i , c ( l , m ′ ) , i , d ( l ′ , m ) , i , e ( 0 , m ) , i and f ( 0 , m ) , i such that

( i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = S ( l , m , k ) 2 2 k + 2 m − 1 ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ n ] ∑ i a ( l , m ) , i C 2 k , i ( z ) , ( i i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l ′ χ 1 , χ − 4 ( x ) σ 2 m χ 1 , χ − 4 ( y ) = S ( l ′ , m , k ) 2 2 k − 1 ( σ 2 k − 1 ( n ⁄ 4 ) − 2 2 k σ 2 k − 1 ( n ⁄ 8 ) ) + [ n ] ∑ i b ( l ′ , m ) , i C 2 k , i ( z ) , ( i i i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m ′ χ 1 , χ − 4 ( y ) = [ n ] ∑ i c ( l , m ′ ) , i C 2 k , i ( z ) , ( i v ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l ′ χ 1 , χ − 4 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = [ n ] ∑ i d ( l ′ , m ) , i C 2 k , i ( z ) , ( v ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 0 χ − 4 , χ 1 ( x ) σ 2 m χ 1 , χ − 4 ( y ) = 4 S ( 0 , k − 1 , k ) 2 2 k − 1 ( σ 2 k − 1 ( n ⁄ 4 ) − 2 2 k σ 2 k − 1 ( n ⁄ 8 ) ) + ∑ i e ( 0 , m ) , i C 2 k , i ( z ) , ( v i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 0 χ 1 , χ − 4 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = 2 − 4 k + 3 S ( 0 , k − 1 , k ) 2 2 k − 1 σ 2 k − 1 χ * , χ 1 ( n ) + ∑ i f ( 0 , m ) , i C 2 k , i ( z ) .

Proof

From section 4.6 and section 4.8 in [16], we see that

E 1 χ 1 , χ − 4 ( 4 z ) , E 1 χ 1 , χ − 4 ( 8 z ) , E 1 χ − 4 , χ 1 ( z ) , E 1 χ − 4 , χ 1 ( 2 z ) ∈ M 1 ( Γ 0 ( 32 ) ) .

Let l and m be nonnegative integers with l + m > 0 , and put k = l + m + 1 . We will give a proof for ( i ) only since the cases ( i i ) – ( v i ) can be proved in a similar way to ( i ) . First, we note that

E 2 l + 1 χ − 4 , χ 1 ( z ) E 2 m + 1 χ − 4 , χ 1 ( 2 z ) ∈ M 2 k ( Γ 0 ( 32 ) ) .

By Theorem 1.3, ( i i ) of Lemma 3.2 and ( i i ) of Lemma 3.3, we have

E 2 l + 1 χ − 4 , χ 1 ( z ) E 2 m + 1 χ − 4 , χ 1 ( 2 z ) = 2 2 k − 2 m + 2 A ( 2 k ) L ( 2 l + 1 , χ − 4 ) L ( 2 m + 1 , χ − 4 ) E 2 k χ * , χ 1 ( z ) + ∑ n = 0 ∞ a ( l , m ) , i C 2 k , i ( z ) ,

for some complex numbers a ( l , m ) , i .

Note that when k ≥ 2 , from (2), the aforementioned Eisenstein series have the Fourier expansions

E 2 l + 1 χ − 4 , χ 1 ( z ) = i 2 4 l + 3 ( 2 l + 1 ) L ( 2 l + 1 , χ − 4 ) B 2 l + 1 , χ − 4 ∑ n = 0 ∞ σ 2 l χ − 4 , χ 1 ( n ) q n , E 2 k χ * , χ 1 ( z ) = − 8 k B 2 k ζ ( 2 k ) ∑ n = 0 ∞ σ 2 k − 1 χ * , χ 1 ( n ) q n .

Then, we have

− 2 4 l + 4 m + 6 ( 2 l + 1 ) ( 2 m + 1 ) L ( 2 l + 1 , χ − 4 ) L ( 2 m + 1 , χ − 4 ) B 2 l + 1 , χ − 4 B 2 m + 1 , χ − 4 ∑ n = 0 ∞ σ 2 l χ − 4 , χ 1 ( n ) q n ∑ n = 0 ∞ σ 2 m χ − 4 , χ 1 ( n ) q 2 n = 2 2 k − 2 m + 2 4 ζ ( 2 k ) ( 2 2 k − 1 ) L ( 2 l + 1 , χ − 4 ) L ( 2 m + 1 , χ − 4 ) − 8 k ζ ( 2 k ) B 2 k ∑ n = 0 ∞ σ 2 k − 1 χ * , χ 1 ( n ) q n + ∑ n = 0 ∞ a ( l , m ) , i C 2 k , i ( z ) ,

from which we obtain

∑ n = 0 ∞ ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) q n = S ( l , m , k ) 2 2 k + 2 m − 1 ( 2 2 k − 1 ) ∑ n = 0 ∞ σ 2 k − 1 χ * , χ 1 ( n ) q n + ∑ i a ′ ( l , m ) , i C 2 k , i ( z ) .

Therefore, we have

□ ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = S ( l , m , k ) 2 2 k + 2 m − 1 ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ n ] ∑ i a ′ ( l , m ) , i C 2 k , i ( z ) .

We also consider the case of even l and give the corresponding result in the following theorem for the convolution sums of σ 2 l χ , ψ ( x ) and σ 2 m χ , ψ ( y ) for nonnegative integers x and y with a positive integer x + 2 y where l and m are nonnegative integers such that m + l > 0 .

Theorem 3.5

Let l , m , and m ′ be positive integers such that l , m ′ > 1 , l + m = k , and l + m ′ = k . Then, for each nonnegative integer n, there exist complex numbers a ( l , m ) , i , b ( l , m ′ ) , i , and c ( l , m ) , i such that

( i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = R ( l , m , k ) 2 2 m ( 2 2 m − 1 ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ⁄ 2 ) + [ n ] ∑ i a ( l , m ) , i C 2 k , i ( z ) , ( i i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l − 1 χ − 4 , χ − 4 ( x ) σ 2 m ′ − 1 ( y ) = R ( l , m ′ , k ) 2 2 l ( 2 2 l − 1 ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b ( l , m ′ ) , i C 2 k , i ( z ) , ( i i i ) ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 l − 1 χ − 4 , χ − 4 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = [ n ] ∑ i c ( l , m ) , i C 2 k , i ( z ) ,

where R ( l , m , k ) = − B 2 l 4 l B 2 m 4 m 4 k B 2 k .

Proof

This theorem can be proved as in the proof of Theorem 3.4.□

As a corollary of Theorems 3.4 and 3.5, we can reprove some parts of Theorem 1.2 in Aygin-Hong’s study [7].

Corollary 3.6

[7, Theorem 1.2, (1.2), (1.3), and (1.6)] Let l and m be nonnegative integers and l ′ > 1 such that 1 < l + m + 1 = k and l ′ + m = k . Then, for each nonnegative integer n, there exist complex numbers a ( l , m ) , i , b ( l , m ) , i , and c ( l ′ , m ) , i such that

( i ) ∑ ( x , y ) ∈ N 0 2 x + y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = 4 S ( l , m , k ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ 2 n ] ∑ i a ( l , m ) , i C 2 k , i , ( i i ) ∑ ( x , y ) ∈ N 0 2 x + y = n σ 2 l χ 1 , χ − 4 ( x ) σ 2 m χ 1 , χ − 4 ( y ) = S ( l , m , k ) 2 2 k − 1 ( σ 2 k − 1 ( n ⁄ 2 ) − 2 2 k σ 2 k − 1 ( n ⁄ 4 ) ) + [ 2 n ] ∑ i b ( l , m ) , i C 2 k , i , ( i i i ) ∑ ( x , y ) ∈ N 0 2 x + y = n σ 2 l ′ − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = R ( l ′ , m , k ) 2 2 m ( 2 2 m − 1 ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ 2 n ] ∑ i c ( l ′ , m ) , i C 2 k , i ( z ) .

Proof

Let χ ∈ { χ − 4 , χ 8 , χ − 8 } . For nonnegative integers l and x , we have

(5) σ 2 l χ , χ 1 ( 2 x ) = 2 2 l σ 2 l χ , χ 1 ( x ) .

Suppose that n is a positive integer. If x + 2 y = 2 n for nonnegative integers x and y , then x is even. Replace x by 2 x , then from Theorem 3.4 and (5), we obtain

∑ ( x , y ) ∈ N 0 2 x + y = n σ 2 l χ − 4 , χ 1 ( x ) σ 2 m χ − 4 , χ 1 ( y ) = 4 S ( l , m , k ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ * , χ 1 ( n ) + [ 2 n ] ∑ i a ( l , m ) , i C 2 k , i , j ,

and ( i ) is proved. ( i i ) is shown in a similar way. For ( i i i ) , we replace n by 2 n in ( i ) of Theorem 3.5 and use σ k ( 2 n ) = ( 2 k + 1 ) σ k ( n ) − 2 k σ ( n ⁄ 2 ) for positive integers k and n , then ∑ ( x , y ) ∈ N 0 2 x + 2 y = 2 n σ 2 l ′ − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) is

∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n σ 2 l ′ − 1 ( 2 x ′ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n x ′ ≡ 1 ( 2 ) ( 2 2 l ′ − 1 + 1 ) σ 2 l ′ − 1 ( x ′ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) + ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n x ′ ≡ 0 ( 2 ) ( ( 2 2 l ′ − 1 + 1 ) σ 2 l ′ − 1 ( x ′ ) − 2 2 l ′ − 1 σ 2 l ′ − 1 ( x ′ ⁄ 2 ) ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) .

From this equality and ( i ) and ( i i ) of Theorem 3.5, we have

( 2 2 l ′ − 1 + 1 ) ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n σ 2 l ′ − 1 ( x ′ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = ( 2 2 l ′ − 1 + 1 ) ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n x ′ ≡ 0 ( 2 ) σ 2 l ′ − 1 ( x ′ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) + ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n x ′ ≡ 0 ( 2 ) σ 2 l ′ − 1 ( x ′ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = ∑ ( x , y ) ∈ N 0 2 x + 2 y = 2 n σ 2 l ′ − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) + 2 2 l ′ − 1 ∑ ( x ′ , y ) ∈ N 0 2 x ′ + y = n x ′ ≡ 0 ( 2 ) σ 2 l ′ − 1 ( x ′ ⁄ 2 ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = ∑ ( x , y ) ∈ N 0 2 x + 2 y = 2 n σ 2 l ′ − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) + 2 2 l ′ − 1 ∑ ( x ″ , y ) ∈ N 0 2 2 x ″ + y = n σ 2 l ′ − 1 ( x ″ ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = ( 2 2 l ′ − 1 + 1 ) R ( l ′ , m , k ) 2 2 m ( 2 2 m − 1 ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b ¯ ( l ′ , m ) , i C 2 k , i ( z ) for some b ¯ ( l ′ , m ) , i ∈ C .

Hence, we obtain

□ ∑ ( x , y ) ∈ N 0 2 x + y = n σ 2 l ′ − 1 ( x ) σ 2 m − 1 χ − 4 , χ − 4 ( y ) = R ( l ′ , m , k ) 2 2 m ( 2 2 m − 1 ) 2 2 k ( 2 2 k − 1 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b ¯ ( l ′ , m ) , i C 2 k , i ( z ) .

4 Proof of Theorem 1.1

In this section, we prove Theorem 1.1, which is our main theorem. First let us recall the identity of Huard et al. [11].

Proposition 4.1

[11, Theorem 1] Let f : Z 4 → C be a function such that

f ( a , b , x , y ) − f ( x , y , a , b ) = f ( − a , − b , x , y ) − f ( x , y , − a , − b ) ,

for all a , b , x , y ∈ Z . Then, for all n ∈ N , we have the identity

(6) ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n ( f ( a , b , x , − y ) − f ( a , − b , x , y ) + f ( a , a − b , x + y , y ) − f ( a , a + b , y − x , y ) + f ( b − a , b , x , x + y ) − f ( a + b , b , x , x − y ) ) = ∑ d ∣ n ∑ x ∈ N x < d ( f ( 0 , n ⁄ d , x , d ) + f ( n ⁄ d , 0 , d , x ) + f ( n ⁄ d , n ⁄ d , d − x , − x ) − f ( x , x − d , n ⁄ d , n ⁄ d ) − f ( x , d , 0 , n ⁄ d ) − f ( d , x , n ⁄ d , 0 ) ) .

We also need the following properties of the Kronecker symbol obtainable via modulo 8 consideration.

Lemma 4.2

For integers a and b, we have

( i ) − 8 a − b − 8 a − − 8 a + b − 8 a = − 2 − 4 a − 4 b ⁄ 2 , ( i i ) 8 a − b 8 a − 8 a + b 8 a = 2 − 4 a − 4 b ⁄ 2 .

We now prove Theorem 1.1 using Proposition 4.1 and Theorem 3.4.

Proof of Theorem 1.1

We take f ( a , b , x , y ) = − 8 a − 8 b ( x − y ) 2 k and apply Proposition 4.1. Since − − 8 a = − 8 − a for a ∈ Z , it follows that f ( a , b , x , y ) − f ( x , y , a , b ) = f ( − a , − b , x , y ) − f ( x , y , − a , − b ) . Then, the left-hand side of (6) is

∑ ( a , b , x , y ) ∈ N 4 a x + b y = n − 8 a − 8 b ( x + y ) 2 k − − 8 a − 8 − b ( x − y ) 2 k + − 8 a − 8 a − b x 2 k − − 8 a − 8 a + b x 2 k + − 8 b − a − 8 b y 2 k − − 8 a + b − 8 b y 2 k = U 1 + 2 U 2 ,

where

U 1 ≔ ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n − 8 a − 8 b ( x + y ) 2 k + − 8 a − 8 b ( x − y ) 2 k , U 2 ≔ ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n − 8 a − 8 a − b x 2 k − − 8 a − 8 a + b x 2 k .

First, U 1 can be computed as follows:

U 1 = ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n − 8 a − 8 b ∑ s = 0 2 k 2 k s ( x 2 k − s y s + x 2 k − s ( − y ) s ) = 2 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 ∑ x ∣ m − 8 m ⁄ x x 2 k − 2 s ∑ y ∣ n − m − 8 ( n − m ) ⁄ y y 2 s = 2 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) .

Using ( i ) of Lemma 4.2, U 2 is computed as follows, too:

U 2 = − 2 ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n − 4 a − 4 b ⁄ 2 x 2 k = − 2 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) .

Combining these, the left-hand side of (6) is equal to

2 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) − 2 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) .

On the other hand, the right-hand side of (6) is V 1 − 2 V 2 , where

V 1 ≔ ∑ d ∣ n ∑ x = 1 d − 1 − 8 n ⁄ d − 8 n ⁄ d d 2 k = ∑ d ∣ n 2 ∤ n ⁄ d d 2 k ( d − 1 ) = σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ * , χ 1 ( n ) , V 2 ≔ ∑ d ∣ n ∑ x = 1 d − 1 − 8 x − 8 d ( n ⁄ d ) 2 k ,

and V 2 can be computed as follows:

V 2 = ∑ d ∣ n − 8 d ( n ⁄ d ) 2 k ∑ x = 1 d − 8 x − ∑ d ∣ n − 8 d 2 ( n ⁄ d ) 2 k = ∑ d ∣ n d ≡ 1 ( 8 ) ( n ⁄ d ) 2 k + 2 ∑ d ∣ n d ≡ 3 ( 8 ) ( n ⁄ d ) 2 k − ∑ d ∣ n d ≡ 5 ( 8 ) ( n ⁄ d ) 2 k − ∑ d ∣ n 2 ∤ d ( n ⁄ d ) 2 k = ∑ d ∣ n n ⁄ d ≡ 1 ( 8 ) d 2 k + 2 ∑ d ∣ n n ⁄ d ≡ 3 ( 8 ) d 2 k − ∑ d ∣ n n ⁄ d ≡ 5 ( 8 ) d 2 k − σ 2 k χ * , χ 1 ( n ) .

Hence, the whole right-hand side of (6) is

σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ * , χ 1 ( n ) − 2 ∑ d ∣ n n ⁄ d ≡ 1 ( 8 ) d 2 k + 2 ∑ d ∣ n n ⁄ d ≡ 3 ( 8 ) d 2 k − ∑ d ∣ n n ⁄ d ≡ 5 ( 8 ) d 2 k − σ 2 k χ * , χ 1 ( n ) = σ 2 k + 1 χ * , χ 1 ( n ) + σ 2 k χ * , χ 1 ( n ) − 2 ∑ d ∣ n n ⁄ d ≡ 1 ( 8 ) d 2 k + 2 ∑ d ∣ n n ⁄ d ≡ 3 ( 8 ) d 2 k − ∑ d ∣ n n ⁄ d ≡ 5 ( 8 ) d 2 k = σ 2 k + 1 χ * , χ 1 ( n ) + σ 2 k χ − 4 , χ 1 ( n ) − 2 σ 2 k χ − 8 , χ 1 ( n ) ,

and therefore, we obtain the following equality:

(7) ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) = 1 2 ( σ 2 k + 1 χ * , χ 1 ( n ) + σ 2 k χ − 4 , χ 1 ( n ) − 2 σ 2 k χ − 8 , χ 1 ( n ) ) + 2 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) .

Since σ k χ − 4 , χ 1 ( 0 ) = 0 , the convolution sum ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) can be expressed as

∑ m = 0 n σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) = ∑ ( x , y ) ∈ N 0 2 x + 2 y = n σ 2 k χ − 4 , χ 1 ( x ) σ 0 χ − 4 , χ 1 ( y ) ,

and finally, by ( i ) of Theorem 3.4, we obtain the following expression for the left-hand side of (6), completing the proof of ( i ) :

∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s χ − 8 , χ 1 ( m ) σ 2 s χ − 8 , χ 1 ( n − m ) = 1 2 ( σ 2 k + 1 χ * , χ 1 ( n ) + σ 2 k χ − 4 , χ 1 ( n ) − 2 σ 2 k χ − 8 , χ 1 ( n ) ) − S ( k , 0 , k + 1 ) 2 2 k ( 2 2 k + 2 − 1 ) σ 2 k + 1 χ * , χ 1 ( n ) + 2 [ n ] ∑ i a 2 k , i C 2 k , i .

The proof of ( i i ) is similar to the proof of ( i ) ; hence, we give only a sketch of it. In this case, we take f ( a , b , x , y ) = 8 a 8 b ( x − y ) 2 k for Theorem 4.1. Then, by the similar reasoning, the left-hand side of (6) is shown to be equal to

2 ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 χ 8 , χ 1 ( m ) σ 2 s + 1 χ 8 , χ 1 ( n − m ) + 2 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) .

The right-hand side of (6) is similarly computed to be equal to σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 4 , χ 1 ( n ) . Thus, we obtain

(8) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 χ 8 , χ 1 ( m ) σ 2 s + 1 χ 8 , χ 1 ( n − m ) = 1 2 ( σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 4 , χ 1 ( n ) ) − 2 ∑ m = 1 n − 1 σ 2 k χ − 4 , χ 1 ( m ) σ 0 χ − 4 , χ 1 ( ( n − m ) ⁄ 2 ) ,

and by ( i ) of Theorem 3.4, we obtain the desired expression.

Finally, ( i i i ) is obtained by adding identities (7) and (8), which completes the proof.□

Corollary 4.3

For positive integers k and n, we have

( i ) ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( 2 m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − 2 m ) = 2 2 k ( σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 8 , χ 1 ( n ) ) , ( i i ) ∑ s = 0 2 k 2 k s ∑ m = 1 n σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( 2 m − 1 ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − 2 m + 1 ) = 2 2 k σ 2 k + 1 χ * , χ 1 ( n ) , ( i i i ) ∑ s = 0 2 k 2 k s ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) = − 2 2 k σ 2 k χ − 8 , χ 1 ( n ) .

Proof

Let k and n be positive integers. By ( i i i ) of Theorem 1.1 and (5), we obtain

∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( 2 m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − 2 m ) = 2 2 k ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( n − m ) = 2 2 k ( σ 2 k + 1 χ * , χ 1 ( n ) − σ 2 k χ − 8 , χ 1 ( n ) ) .

Hence, ( i ) is true.

Now, we prove ( i i i ) . We first divide the sum ∑ s = 0 2 k 2 k s ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) into two sums according to the parity of m , then the sum is computed as follows:

= ∑ s = 0 2 k 2 k s ∑ m = 1 2 ∣ m 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) − ∑ m = 1 2 ∤ m 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) = ∑ s = 0 2 k 2 k s 2 ∑ m = 1 2 ∣ m 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) − ∑ m = 1 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) = ∑ s = 0 2 k 2 k s 2 ∑ m = 1 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( 2 m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − 2 m ) − ∑ m = 1 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) .

From ( i i i ) of Theorem 1.1, ( i ) of Corollary 4.3 and (5), the last expression is computed as − 2 2 k σ 2 k χ − 8 , χ 1 ( n ) , and hence, we have

∑ s = 0 2 k 2 k s ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) = − 2 2 k σ 2 k χ − 8 , χ 1 ( n ) .

Finally, we compute ∑ s = 0 2 k 2 k s ∑ m = 1 n σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( 2 m − 1 ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − 2 m + 1 ) and prove ( i i ) using

□ = ∑ s = 0 2 k 2 k s ∑ m = 1 2 ∣ m 2 n − 1 σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) − ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − s χ ( − 1 ) s + 1 8 , χ 1 ( m ) σ s χ ( − 1 ) s + 1 8 , χ 1 ( 2 n − m ) = 2 2 k σ 2 k + 1 χ * , χ 1 ( n ) .

Taking k = 1 in ( i )   Theorem 1.1 and taking k = 1 , 2 in ( i i ) of Theorem 1.1, we also obtain the values of the convolution sums of divisor functions associated with χ 8 and χ − 8 . Here, S ( 1 , 0 , 2 ) = 15 and S ( 2 , 0 , 3 ) = 315 2 .

Corollary 4.4

Let n be a positive integer. Then, we have

( i ) ∑ m = 1 n − 1 σ 2 χ − 8 , χ 1 ( m ) σ 0 χ − 8 , χ 1 ( n − m ) = 1 4 ( σ 3 χ * , χ 1 ( n ) + σ 2 χ − 4 , χ 1 ( n ) − 2 σ 2 χ − 8 , χ 1 ( n ) − 20 σ 1 χ * , χ 1 ( n ) ) + [ n ] ∑ i a 2 , i C 2 , i , ( i i ) ∑ m = 1 n − 1 σ 1 χ 8 , χ 1 ( m ) σ 1 χ 8 , χ 1 ( n − m ) = 1 4 ( σ 3 χ * , χ 1 ( n ) − σ 2 χ − 4 , χ 1 ( n ) + 20 σ 1 χ * , χ 1 ( n ) ) − [ n ] ∑ i a 2 , i C 2 , i , ( i i i ) ∑ m = 1 n − 1 σ 1 χ 8 , χ 1 ( m ) σ 3 χ 8 , χ 1 ( n − m ) = 1 16 σ 5 χ * , χ 1 ( n ) − σ 4 χ − 4 , χ 1 ( n ) + 21 4 σ 3 χ * , χ 1 ( n ) − [ n ] ∑ i a 4 , i C 4 , i ,

where a 2 , i and a 4 , i are the constants of Theorem 1.1.

5 Representations by quadratic forms

In this section, we compute the number of representations of a nonnegative integer by certain quadratic forms to prove Theorems 5.1 and 5.2. Let α , β , γ and δ be nonnegative integers, and we denote the vector ( α , β , γ , δ ) by v . We consider the following number of representations by quadratic forms:

N ( v ; n ) ≔ # ( x 1 ,…, x α + β + γ + δ ) ∈ Z α + β + γ + δ ∣ n = ∑ i = 1 α x i 2 + 2 ∑ i = 1 β x α + i 2 + 4 ∑ i = 1 γ x α + β + i 2 + 8 ∑ i = 1 δ x α + β + γ + i 2 , U ( v ; n ) ≔ # ( x 1 ,…, x α + β + γ + δ ) ∈ Z α + β + γ + δ ∣ n = ∑ i = 1 α + β x i 2 + 2 ∑ i = 1 γ + δ x α + β + i 2 , where x 1 ,…, x α , x α + β + 1 ,…, x α + β + γ are even and x α + 1 ,…, x α + β , x α + β + γ + 1 ,…, x α + β + γ + δ are odd } .

In fact, N ( α , 0 , 0 , 0 ; n ) is equal to r α ( n ) of Jacobi, and a formula for the number U ( 0 , 4 , 0 , 0 ; 4 n ) was obtained by Hurwitz [19] by the use of the arithmetic of quaternions.

Previously, Lemire [20] and Aygin-Hong [7] presented formulas for N ( α , β , γ , 0 ; n ) under certain conditions on α , β , and γ . Note that N ( 0 , β , γ , δ ; n ) = N ( α , β , γ , 0 ; n ⁄ 2 ) when α = 0 , which reduces to the previous results. We obtain the following further result for N ( v ; n ) when both α and δ are nonzero.

Theorem 5.1

Let k > 1 be a positive integer, and α , β , γ , and δ be nonnegative integers such that α + γ and β + δ are even and α + β + γ + δ = 4 k . Then, N ( v ; n ) is given as follows:

(i) When α , β , γ , δ ≠ 0 ,

N ( v ; n ) = 2 B ( β , γ , δ ) ( − 1 ) k σ 2 k − 1 χ * , χ 1 ( n ) − 2 β + 2 γ + 3 δ 2 σ 2 k − 1 n 16 + 2 2 k + β + 2 γ + 3 δ 2 σ 2 k − 1 n 32 + [ n ] ∑ i a v , i C 2 k , i ( z ) , for some a v , i ∈ R .

(ii) When β = 0 and α , γ , δ ≠ 0 ,

N ( v ; n ) = ( − 1 ) k + 1 B ( 0 , γ , δ ) σ 2 k − 1 χ * , χ 1 ( n ) + 2 γ + δ 2 cos ( α π ⁄ 4 ) σ 2 k − 1 χ * , χ 1 n 2 − ( − 1 ) k 2 2 γ + 3 δ 2 σ 2 k − 1 n 16 + ( − 1 ) k 2 2 k + 2 γ + 3 δ 2 σ 2 k − 1 n 32 + 2 − 2 k + γ + δ 2 sin ( α π ⁄ 4 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b v , i C 2 k , i ( z ) , for s o m e b v , i ∈ R .

(iii) When α , β , δ ≠ 0 and γ = 0 ,

N ( v ; n ) = 2 − 3 δ 2 B ( β , 0 , δ ) σ 2 k − 1 χ * , χ 1 ( n ) + 2 β + 2 δ 2 cos ( δ π ⁄ 4 ) σ 2 k − 1 χ * , χ 1 ( n ⁄ 4 ) − 2 β + 3 δ 2 σ 2 k − 1 ( n ⁄ 16 ) + 2 2 k + β + 3 δ 2 σ 2 k − 1 ( n ⁄ 32 ) − 2 − 2 k + β + 2 δ 2 sin ( δ π ⁄ 4 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ⁄ 2 ) + [ n ] ∑ i c v , i C 2 k , i ( z ) , for s o m e c v , i ∈ R .

(iv) When α , δ ≠ 0 and β , γ = 0 ,

N ( v ; n ) = ( − 1 ) k B ( 0 , 0 , δ ) σ 2 k − 1 χ * , χ 1 ( n ) + 2 δ 2 cos ( α π ⁄ 4 ) σ 2 k − 1 χ * , χ 1 ( n ⁄ 2 ) + ( − 1 ) k 2 − δ cos ( δ π ⁄ 4 ) σ 2 k − 1 χ * , χ 1 ( n ⁄ 4 ) − ( − 1 ) k 2 3 δ 2 σ 2 k − 1 ( n ⁄ 16 ) + ( − 1 ) k 2 2 k + 3 δ 2 σ 2 k − 1 ( n ⁄ 32 ) + 2 − 2 k − δ 2 sin ( α π ⁄ 4 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ) − 2 − 2 k − δ sin ( δ π ⁄ 4 ) σ 2 k − 1 χ − 4 , χ − 4 ( n ⁄ 2 ) + [ n ] ∑ i d v , i C 2 k , i ( z ) , for s o m e d v , i ∈ R ,

where B ( β , γ , δ ) = − 2 − β + 2 γ + 3 δ 2 + 1 2 2 k − 1 2 k B 2 k .

Proof

From [21, Corollary 1.3.4.], we have the following eta-quotient expression:

(9) ∑ n = − ∞ ∞ q n 2 = ∏ n = 1 ∞ ( 1 + q 2 n + 1 ) 2 ( 1 − q 2 n ) = η 5 ( 2 z ) η 2 ( z ) η 2 ( 4 z ) .

From this equality, we can express the generating function of N ( α , β , γ , δ ; n ) as the eta-quotient

f ( α , β , γ , δ ; z ) ≔ η − 2 α ( z ) η 5 α − 2 β ( 2 z ) η − 2 α + 5 β − 2 γ ( 4 z ) η − 2 β + 5 γ − 2 δ ( 8 z ) η − 2 γ + 5 δ ( 16 z ) η − 2 δ ( 32 z ) .

Suppose that α + γ and β + δ are even, and α + β + γ + δ = 4 k for some integer k > 1 . By Theorem 2.1, f ( α , β , γ , δ ; z ) is shown to be an element of M 2 k ( Γ 0 ( 32 ) ) .

Now, we assume that α , β , γ , and δ are positive integers. N ( α , β , γ , 0 ; n ) , N ( α , 0 , γ , 0 ; n ) , and N ( α , β , 0 , 0 ; n ) were already computed in [7, Theorem 6.1]. Here, we consider N ( α , β , γ , δ ; n ) , N ( α , 0 , γ , δ ; n ) , N ( α , β , 0 , δ ; n ) , and N ( α , 0 , 0 , δ ; n ) .

Now, we let v = ( α , β , γ , δ ) and write f ( v ; z ) for f ( α , β , γ , δ ; z ) . By [18, Proposition 2.1], the order of f ( v ; z ) at the cusp r ∈ R ( 32 ) is

0 , if  r = 1 , 1 ⁄ 32 , α 4 , if  r = 1 ⁄ 2 , β 2 , if  r = 1 ⁄ 4 , 3 ⁄ 4 , γ , if  r = 1 ⁄ 8 , 3 ⁄ 8 , 2 δ , if  r = 1 ⁄ 16 .

Now, we prove ( i ) . Suppose α , β , γ , δ ≠ 0 . Then, we obtain

[ 0 ] r f ( v ; z ) = ( − 1 ) k 2 − 2 k − b + 2 c + 3 d 2 , if r = 1 , 1 , if r = 1 ⁄ 32 , 0 , otherwise .

By Theorem 1.3, we obtain

f ( v ; z ) = A ( 2 k ) 2 − β + 2 γ + 3 δ 2 + 1 ( − 1 ) k E 2 k χ * , χ 1 ( z ) − 2 E 2 k χ * , χ 1 ( 16 z ) + 2 ( 2 2 k − 1 ) E 2 k χ 1 , χ 1 ( 32 z ) + ∑ i a v , i C 2 k , i ( z ) = − 2 2 2 k − 1 2 k B 2 k ∑ n = 0 ∞ 2 − β + 2 γ + 3 δ 2 + 1 ( − 1 ) k σ 2 k − 1 χ * , χ 1 ( n ) q n − 2 σ 2 k − 1 χ * , χ 1 ( n ) q 16 n + 2 ( 2 2 k − 1 ) σ 2 k − 1 ( n ) q 32 n + ∑ i a v , i C 2 k , i ( z ) , for some a v , i ∈ C .

Hence, we have

N ( v ; n ) = − 2 − β + 2 γ + 3 δ 2 + 2 2 2 k − 1 2 k B 2 k ( − 1 ) k σ 2 k − 1 χ * , χ 1 ( n ) − 2 β + 2 γ + 3 δ 2 σ 2 k − 1 n 16 + 2 2 k + β + 2 γ + 3 δ 2 σ 2 k − 1 n 32 + [ n ] ∑ i a v , i C 2 k , i ( z ) .

Cases ( i i ) , ( i i i ) , and ( i v ) can be proved in a similar way to ( i ) . For reader’s convenience, we record [ 0 ] r f ( v ; z ) for each case:

( i i ) If α , γ , δ ≠ 0 and β = 0 , then

[ 0 ] r f ( v ; z ) = ( − 1 ) k 2 − ( 2 − u ) k − 2 γ + 3 δ 2 e α u 8 π i d , if  r = d ⁄ 2 u   for  u = 0 , 2 , 1 , if r = 1 ⁄ 32 , 0 , otherwise .

( i i i ) If α , β , δ ≠ 0 and γ = 0 , then

[ 0 ] r f ( v ; z ) = ( − 1 ) k 2 − 2 k − β + 3 δ 2 , if r = 1 , 2 − δ 2 e − δ 4 π i d , if r = d ⁄ 8 , 1 , if r = 1 ⁄ 32 , 0 , otherwise .

( i v ) If α , δ ≠ 0 and β , γ = 0 , then

□ [ 0 ] r f ( v ; z ) = ( − 1 ) k 2 − 2 k − 3 δ 2 , if  r = 1 , ( − 1 ) k 2 − 4 δ c e α 4 π i d , if  r = d ⁄ c  for  c = 4 , 8 , 1 , if r = 1 ⁄ 32 , 0 , otherwise .

For U ( v ; n ) , after the work of the Hurwitz and Deutsch [22], Cho and Park [23], and Cho [24] obtained the number of representations of integers by quadratic forms x 2 + y 2 + z 2 + w 2 , and x 2 + y 2 + 2 z 2 + 2 w 2 with certain parity conditions on the variables x , y , z and w . Here, we obtain the following result for general U ( v ; n ) .

Theorem 5.2

Let k > 1 be a positive integer, and α , β , γ and δ be nonnegative integers such that α + β and γ + δ are even and α + β + γ + δ = 4 k . Then, U ( v ; n ) is given as follows:

  1. When α , β , γ , δ ≠ 0 .

    (1) If β is even, then

    U ( v ; n ) = C ( γ , δ ) σ 2 k − 1 χ * , χ 1 ( n ) + ( − 1 ) β ⁄ 2 + δ 2 2 k − 1 σ 2 k − 1 χ * , χ 1 n 2 + [ n ] ∑ i a v , i C 2 k , i ( z ) , for s o m e a v , i ∈ R .

    (2) If β is odd, then

    U ( v ; n ) = C ( γ , δ ) ( − 1 ) β + 2 δ + 1 2 2 σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b v , i C 2 k , i ( z ) , for s o m e b v , i ∈ R .

  2. When α = 0 and β , γ , δ ≠ 0 :

    (1) If γ ≡ δ ( mod 4 ) , then

    U ( v ; n ) = C ( γ , δ ) ( 1 + ( − 1 ) β ⁄ 2 + δ ) σ 2 k − 1 χ * , χ 1 ( n ) + ( − 1 ) β + 2 δ 4 2 4 k − 1 σ 2 k − 1 χ * , χ 1 n 4 + [ n ] ∑ i c v , i C 2 k , i ( z ) , for s o m e c v , i ∈ R .

    (2) If γ ≡ δ + 2 ( mod 4 ) , then

    U ( v ; n ) = C ( γ , δ ) ( 1 + ( − 1 ) β ⁄ 2 + δ ) σ 2 k − 1 χ * , χ 1 ( n ) + ( − 1 ) β + 2 δ + 2 4 2 4 k − 1 σ 2 k − 1 χ − 4 , χ − 4 n 4 + [ n ] ∑ i d v , i C 2 k , i ( z ) , for s o m e d v , i ∈ R .

  3. When β , γ = 0 and α , δ ≠ 0 :

    U ( v ; n ) = 2 C ( 0 , δ ) σ 2 k − 1 χ * , χ 1 ( n ) + ( − 1 ) k 2 α + 3 δ − 4 2 σ 2 k − 1 χ * , χ 1 n 8 + [ n ] ∑ i e v , i C 2 k , i ( z ) , for s o m e e v , i ∈ R .

  4. When α , β , γ = 0 and δ ≠ 0

    U ( v ; n ) = C ( 0 , δ ) σ 2 k − 1 χ * , χ 1 n 4 + ( − 1 ) k 2 2 k − 1 σ 2 k − 1 χ * , χ 1 n 8 + [ n ] ∑ i f v , i C 2 k , i ( z ) , for s o m e f v , i ∈ R ,

    where C ( γ , δ ) = ( − 1 ) k + 1 2 − 4 k − γ + δ 2 + 2 2 2 k − 1 ⋅ 2 k B 2 k .

Proof

Using [21, Corollary 1.3.4.], we obtain

(10) ∑ n = − ∞ ∞ q ( 2 n + 1 ) 2 = 2 η 2 ( 16 z ) η ( 8 z ) .

Then, by identities (9) and (10), the generating function of U ( v ; n ) , where v = ( α , β , γ , δ ) , is given by the eta-quotient

g ( v ; z ) = 2 β + δ η − 2 α ( 4 z ) η 5 α − β − 2 γ ( 8 z ) η − 2 α + 2 β + 5 γ − δ ( 16 z ) η − 2 γ + 2 δ ( 32 z ) .

Suppose that α + β and γ + δ are even and α + β + γ + δ = 4 k ( k > 1 ) . By Theorem 2.1, we obtain

g ( v ; z ) ∈ M 2 k ( Γ 0 ( 32 ) ) .

By [18, Proposition 2.1], the order of g ( v ; z ) at the cusp r ∈ R ( 32 ) is

0 , if  r = 1 , 1 ⁄ 2 , 1 ⁄ 4 , 3 ⁄ 4 , α , if  r = 1 ⁄ 8 , 3 ⁄ 8 , β + 2 γ , if  r = 1 ⁄ 16 , β + 2 δ , if  r = 1 ⁄ 32 .

Now, we prove ( i ) . Suppose α , β , γ , δ ≠ 0 . Then,

[ 0 ] r g ( v ; z ) = ( − 1 ) k 2 − 6 k − γ + δ 2 , if r = 1 , ( − 1 ) k + β 2 − 4 k − γ + δ 2 , if r = 1 ⁄ 2 , ( − 1 ) k + δ 2 − 2 k − γ + δ 2 i β , if r = 1 ⁄ 4 , ( − 1 ) k + δ 2 − 2 k − γ + δ 2 ( − i ) β , if r = 3 ⁄ 4 , 0 , otherwise .

Using a method similar to the proof of Theorem 5.2, we have

g ( v ; z ) = ( − 1 ) k + 1 2 − 4 k − γ + δ 2 + 2 2 2 k − 1 ⋅ 2 k B 2 k 2 ( 1 + ( − 1 ) β ) ∑ n = 0 ∞ σ 2 k − 1 χ * , χ 1 ( n ) q n + ( − 1 ) δ 2 2 k ( i β + ( − i ) β ) ∑ n = 0 ∞ σ 2 k − 1 χ * , χ 1 ( n ) q 2 n − i ( − 1 ) δ ( i β − ( − i ) β ) ∑ n = 0 ∞ σ 2 k − 1 χ − 4 , χ − 4 ( n ) q n + ∑ i a v , i C 2 k , i ( z ) .

Thus, if β is even, then

U ( v ; n ) = ( − 1 ) k + 1 2 − 4 k − γ + δ 2 + 2 2 2 k − 1 ⋅ 2 k B 2 k ( 4 σ 2 k − 1 χ * , χ 1 ( n ) + ( − 1 ) β ⁄ 2 + δ 2 2 k + 1 σ 2 k − 1 χ * , χ 1 ( n ⁄ 2 ) ) + [ n ] ∑ i a v , i C 2 k , i ( z ) ,

and if β is odd, then

U ( v ; n ) = ( − 1 ) k + 1 2 − 4 k − γ + δ 2 + 2 2 2 k − 1 ⋅ 2 k B 2 k ( − 1 ) β + 2 δ + 1 2 2 σ 2 k − 1 χ − 4 , χ − 4 ( n ) + [ n ] ∑ i b v , i C 2 k , i ( z ) .

Cases ( i i ) , ( i i i ) , and ( i v ) can be proved in a similar way to ( i ) . For reader’s convenience, we record [ 0 ] r g ( v ; z ) for each case:

( i i ) If β , γ , δ ≠ 0 and α = 0 , then

[ 0 ] r g ( v ; z ) = ( − 1 ) k 2 − ( 6 − 2 u ) k − γ + δ 2 , if  r = 1 ⁄ 2 u for u = 0 , 1 , ( − 1 ) k + δ 2 − 2 k − γ + δ 2 i β , if  r = 1 ⁄ 4 , 3 ⁄ 4 , 2 − β + δ 2 e d ( − γ + δ ) π i ⁄ 4 , if  r = d ⁄ 8 , 0 , otherwise .

( i i i ) If α , δ ≠ 0 and β , γ = 0 , then

[ 0 ] r g ( v ; z ) = ( − 1 ) k 2 − ( 6 − 2 u ) k − δ 2 , if  r = d ⁄ 2 u  for  u = 0 , 1 , 2 , 1 , if  r = 1 ⁄ 16 , 0 , otherwise .

( i v ) If δ ≠ 0 and α , β , γ = 0 , then

□ [ 0 ] r g ( v ; z ) = ( − 1 ) k 2 − ( 8 − 2 u ) k , if  r = d ⁄ 2 u  for  u = 0 , 1 , 2 , 3 , 1 , if  r = 1 ⁄ 16 , 0 , if  r = 1 ⁄ 32 .

Acknowledgements

The authors would like to express sincere thanks to the anonymous referees for suggesting many useful comments.

  1. Funding information: S. Jin was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00253814). H. Park was supported by the National Research Foundation of Korea (NRF-2019R1C1C1010211).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-12-20
Revised: 2024-09-19
Accepted: 2024-10-15
Published Online: 2024-11-18

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0089
Keywords for this article
divisor function; Dirichlet character; convolution sum; arithmetic identity; Eisenstein series
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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