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Kim et al. Advances in Difference Equations 2013, 2013:84http://www.advancesindifferenceequations.com/content/2013/1/84

RESEARCH Open Access

Eisenstein series and their applications tosome arithmetic identities and congruencesDaeyeoul Kim1*, Aeran Kim2 and Ayyadurai Sankaranarayanan3,4

*Correspondence:[email protected] Institute for MathematicalSciences, Yuseong-daero 1689-gil,Yuseong-gu, Daejeon, 305-811,South KoreaFull list of author information isavailable at the end of the article

AbstractUtilizing the theory of elliptic curves over C to the normalized lattice �τ , itsconnection to the Weierstrass ℘-functions and to the Eisenstein series E4(τ ) andE6(τ ), we establish some arithmetic identities involving certain arithmetic functionsand convolution sums of restricted divisor functions. We also prove some congruencerelations involving certain divisor functions and restricted divisor functions.MSC: Primary 11A25; secondary 11A07; 11G99

Keywords: divisor functions; restricted divisor functions; q-series; q-products;Eisenstein series; convolution sums; congruence relations

1 IntroductionThe study of arithmetical identities and congruences is classical in number theory andsuch investigations have been carried out by several mathematicians including the leg-end Srinivasa Ramanujan. This study constitutes an important and significant part of thesubject number theory.For N ,m, r, s,d ∈ Z with d, s > and r ≥ , we define some divisor functions for our use

in the sequel. Let

σs(N) =∑d|N

ds,

and let us define the restricted divisor function

σs,r(N ;m) =∑d|N

d≡r mod m

ds.

Note that

σ (N) := σ(N) =∑d|N

d.

For a,b,n ∈N, let us define the convolution sum

Sa,b(n) :=n–∑m=

σa(m)σb(n –m).

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Ramanujan showed that the sum Sa,b(n) can be evaluated in terms of σa+b+(n),σa+b–(n),. . . ,σ(n),σ(n) for the nine pairs (a,b) ∈ N

satisfying a + b = ,, , , , a≤ b, a ≡ b ≡ (mod ). For example, explicitly, we know (see []) that

S,(n) =

,σ(n) +

(

–

n)

σ(n) –

,σ(n) ()

and (see [, p.])

S,(n) =

,σ(n) –

σ(n) +

σ(n). ()

From [], we note that for any integer n≥ , we have

∑(m,m,m)∈N

m+m+m=n

mmσ(m)σ(m)σ(m)

=

(nσ(n) +

(n – n

)σ(n) –

(n – n

)σ(n)

). ()

For an elementary proof of () and (), we refer to []. An another interesting arithmeticalidentity (which was stated by Ramanujan, see [, p.], for some analytical proofs of thisidentity, one may refer to [, p.], [, p.] and [], also []) is for n ∈N, we have

n–∑k=

σ(k + )σ(n – k) =

σ(n + ) –

σ(n + ). ()

There are some nice arithmetical identities connecting the divisor functions along withRamanujan’s τ -function. For instance, we know (see []) that

n–∑m=

m(n –m)σ(m)σ(n –m) =

nσ(n) –

τ (n) ()

and from [], we observe that for n ∈N∪ {},

t(n) =

,

(σ *(n + ) – τ (n + ) – ,τ

(n +

)), ()

where tk(n) denotes the number of representations of n as a sum of k triangular numbers,i.e. (with N =N∪ {}),

tk(n) := #

{(m, . . . ,mk) ∈ N

k

∣∣∣ n =m(m + ) + · · · +

mk(mk + )

},

σ *k (n) :=

∑d|n

nd odd

dk ,

and τ (n) is the coefficient defined by the expression:

q{( – q)

( – q

) · · ·} = ∞∑n=

τ (n)qn, q ∈C with |q| < .



For any integer k ≥ , let

rk(n) := #{(x, . . . ,xk) ∈ Z

k | n = x + · · · + xk}.

An another identity worth mentioning (see []) is

r(n) =

σ ** (n) +

((–)n–τ (n) – τ

(n

)), ()

where σ **k (n) :=

∑d|n(–)ddk and τ ( n ) = if n

is not an integer.From (), () and (), it is immediate (and interesting) to note that, for n ∈N,

nσ(n) +(n – n

)σ(n) –

(n – n

)σ(n) ≡ (mod ), ()

σ(n + ) – σ(n + )≡ (mod ) ()

and

nσ(n) – τ (n)≡ (mod ). ()

The proofs of all these identities and congruences heavily depend upon the theory ofmod-ular functions and the properties of Eisenstein series. Later some of these identities havebeen proved using only elementary techniques.Define the integers a(n) (for n ∈N) by

∞∑n=

a(n)qn = q∞∏n=

( – qn

), q ∈C with |q| < . ()

Also define the integers b(n) (for n ∈ N) by

∞∑n=

b(n)qn = q∞∏n=

( – qn

)( – qn), q ∈C with |q| < . ()

Note that a(n) ≡ if n ≡ (mod ) and b(n) ≡ if n ≡ (mod ). It has been proved byKenneth Williams (see [, ]) that (for n ∈ N),

r(n) = σ(n) – σ

(n

)+ a(n) ()

and

A(n) =

σ(n) +

σ

(n

)+

σ

(n

)+σ

(n

)

+(

–

n)

σ(n) +(

–n)

σ

(n

)–

b(n),

where r(n) is as mentioned before and for s,n ∈N,

As(n) :=∑k∈Nk< n

s

σ(k)σ(n – sk).



It should be mentioned that Bernoulli identities associated with the Weierstrass℘-function have been studied by Chang and Srivastava in []. Families of Weierstrasstype functions and their applications have been investigated by Chang, Srivastava andWuin []. It is also interesting to note that the families of Weierstrass type functions, Webertype functions and their applications have been studied by Aygunes and Simsek in [].A few more related references are [, ] and [].Though there are plenty of identities and congruences involving various arithmetic

functions available in the literature, practically nothing seriously known involving re-stricted divisor functions.For any integerM ≥ withM = pe · · ·perr , we define ordpj M := ej.Throughout the paper, q = eπ iτ where τ ∈ h = {x + yi | y > } unless otherwise specified

hereafter. The aim of this article is to prove some arithmetical identities involving certainarithmetic functions and convolution sums of restricted divisor functions. We also estab-lish some congruence relations similar to (), () and (). More precisely, we prove thefollowing theorems.

Theorem .(i) For any integerM ≥ , we have

σ(M) ≡ σ(M) + σ,(M; ) (mod ).

(ii) Moreover, ifM is odd or ordpM is odd for an odd prime p, then

σ(M) ≡ σ(M) + σ,(M; ) (mod ).

Theorem . For any integer M ≥ , we have

–σ(M) + σ,(M; ) ≡ σ(M) – σ(M) (mod ).

Theorem . Let N ∈N. Then, we have

∑n+l+m–=Nn,l,m≥

σ,(n – ; )σ,(l – ; )σ,(m – ; )

=

[σ(N – ) – a(N – )

],

where

∞∑N=

a(N)qN = q∞∏N=

( – qN

).Corollary . For N ∈N, we have

(N + )a(N – ) ≡ (N + )σ(N – ) (mod ).

In particular, if N ≡ , (mod ), then we have

a(N – )≡ σ(N – ) (mod ).



Theorem . Let N ∈N. Then we have

N–∑M=

M–∑m=

σ,(N – – M; )σ,(M –m; )σ,(m; )

=

,[σ(N – ) – σ(N – ) + σ(N – ) + a(N – )

].

Remark The main idea in proving Theorems . and . is to obtain q-series expansionsfor E(τ ) and E(τ ) with coefficients being restricted divisor functions and their convolu-tion sums. Then we have to compare these expressions with the already known q-seriesexpressions of E(τ ) and E(τ ).

The paper is organized as follows. In Section , we express g(τ ) and g(τ ) in terms ofq-products. In Section , g(τ ) and g(τ ) are transformed into expressions involving S andS where S and S are q-series expressionswith coefficients as restricted divisor functions.Then we obtain q-series expressions for E(τ ) and E(τ ) with coefficients being restricteddivisor functions and their convolution sums. Thenwe compare these expressions of E(τ )and E(τ ) with already known expressions. Section concludes the proofs of Theorems .and .. In proving arithmetical identities involving the coefficient a(n) (where a(n) is de-fined as in ()), we need to study the quantities S and SS and in turn the convolutionsums of restricted divisor functions along with a(n) come out very naturally.

2 q-product expressions for the Eisenstein series g2(τ ) and g3(τ )Let �τ = Z + τZ (τ ∈ H the complex upper half plane) be a lattice and z ∈ C. TheWeierstrass ℘ function relative to �τ is defined by the series

℘(z;�τ ) =z

+∑ω∈�τ

ω �=

{

(z –ω)–

ω

},

and the Eisenstein series of weight k for �τ with k > is the series

Gk(�τ ) =∑ω∈�τ

ω �=

ω–k .

We shall use the notations ℘(z) and Gk instead of ℘(z;�τ ) and Gk(�τ ), respectively,when the lattice �τ has been fixed. Then the Laurent series for ℘(z) about z = is givenby

℘(z) = z– +∞∑k=

(k + )Gk+zk .

As is customary, by setting

g(τ ) = g(�τ ) = G and g(τ ) = g(�τ ) = G,

the algebraic relation between ℘(z) and ℘ ′(z) becomes

℘ ′(z) = ℘(z) – g(τ )℘(z) – g(τ ).



We use the following q-product expressions:

P =∞∏n=

( – qn

), P =

∞∏n=

( – qn–

),

P =∞∏n=

( + qn

), P =

∞∏n=

( + qn–

).

Theorem . We have

g(τ ) =π

P(P – qP

P + qP

)

and

g(τ ) =π

P

(–P

+ PP

+ P

P – P

).

To prove Theorem ., we need the following lemma.

Lemma . Let e = ℘( τ ), e = ℘( ) and e = ℘( τ+

).() e – e = πP

P .

() e – e = πPP

.() e – e = πqP

P .

Proof See []. �

Proof of Theorem . From [, p.], we observe that e, e and e are the roots of theequation

℘(z) – g(τ )℘(z) – g(τ ) = .

Therefore, we have

e + e + e = ,

ee + ee + ee = –g(τ )

and

eee =g(τ )

.

By the above equations and Lemma ., we deduce that

e + e = e – (–e – e) = e – e = –πPP

,



and the following three identities:

e =[(e + e) + (e – e)

]=(–πP

P – πqP

P)

= –π

P(P + qP

), ()

e = e + πqPP

= –

π

P(P – qP

)

()

and

e = e + πPP

= –

π

P(–P

+ qP). ()

Using (), () and (), we obtain the identities for g(τ ) and g(τ ), namely

g(τ ) = –(ee + ee + ee)

= –π

P[(P + qP

)(–P

+ qP)

+(P – qP

)(–P

+ qP)

+(P + qP

)(P – qP

)]

=π

P(P – qP

P + qP

)

()

and

g(τ ) = eee

= –π

P

(P + qP

)(–P

+ qP)(P – qP

)

=π

P

(–P

+ PP

+ P

P – P

). ()

This proves the theorem. �

3 Eisenstein series and divisor functionsWe use the q-series and q-products notions

S :=∑N odd

σ,(N ; )qN ,

S :=∑

N≥ even

σ,(N ; )qN ,

(a;q)∞ := (a)∞ :=∏n≥

( – aqn

)

in the following. Some identities of the basic hypergeometric series type are quoted by Fine(see []). Some of these identities (in a similar form) can also be found in [] and []. It



should be mentioned that some generalizations and basic q-extensions of Bernoulli, Eulerand Genocchi polynomials have been studied recently by Srivastava (see []). We alsorefer to [] in which zeta and q-zeta function, associated series and integrals have beeninvestigated by Srivastava andChoi.Wemention below two identities (see [, p., p.])for our further use. These are

(q;q)∞(q)∞(q;q)∞

= + ∞∑N=

qN( + (–)N

) ∑ω|N

ω odd

ω ()

and

q(q;q)∞(q;q)∞

=∑N odd

σ (N)qN . ()

Using (), () and the facts,

∞∏n=

( – qn–

)=

∞∏n=

( – qn)( – qn)

,∞∏n=

( + qn–

)=

∞∏n=

( – qn)( – qn)

( – qn)( – qn)

and

∞∏n=

( + qn

)=

∞∏n=

( – qn)( – qn)

,

our aim here is first to prove the following lemma.

Lemma . Let q = eπ iτ , τ ∈H. Then we have(a)

℘

(τ

)= –

π

( + S + S).

(b)

℘

(τ +

)= –

π

( – S + S).

(c)

℘

(

)=π

( + S).

Proof(a) From (), we have

℘

(τ

)= –

π

((q;q)∞

(q)∞(q;q)∞+

q(q;q)∞(q;q)∞

)

= –π

( +

∞∑N=

qN( + (–)N

) ∑ω|N

ω odd

ω + ∑Nodd

σ (N)qN)



= –π

( +

∑N odd

qN∑ω|N

ω odd

ω + ∑

N even

qN∑ω|N

ω odd

ω + ∑N odd

σ (N)qN)

= –π

( +

∞∑N=

qN∑ω|N

ω odd

ω

)

= –π

( +

∞∑N=

σ,(N ; )qN)

= –π

( + S + S). ()

(b) From (), we have

℘

(τ +

)

= –π

((q;q)∞

(q)∞(q;q)∞–

q(q;q)∞(q;q)∞

)

= –π

( +

∞∑N=

qN( + (–)N

) ∑ω|N

ω odd

ω – ∑N odd

σ (N)qN)

= –π

( +

∑N odd

qN∑ω|N

ω odd

ω + ∑

N even

qN∑ω|N

ω odd

ω – ∑N odd

σ (N)qN)

= –π

( +

∞∑N=

(–)NqN∑ω|N

ω odd

ω

)

= –π

( +

∞∑N=

(–)Nσ,(N ; )qN)

= –π

( – S + S). ()

(c) From (), we have

℘

(

)=π

((q;q)∞

(q)∞(q;q)∞–

q(q;q)∞(q;q)∞

)

=π

( +

∞∑N=

qN( + (–)N

) ∑ω|N

ω odd

ω – ∑N odd

σ (N)qN)

=π

( +

∑N odd

qN∑ω|N

ω odd

ω + ∑

N even

qN∑ω|N

ω odd

ω – ∑N odd

σ (N)qN)

=π

( +

∑N even

qN∑ω|N

ω odd

ω

)



=π

( +

∑N even

σ,(N ; )qN)

=π

( + S). ()

This proves the lemma. �

Using the fact that e, e and e are the roots of the equation ℘(z) – g(τ )℘(z) – g(τ ) =, indeed we can express g(τ ) and g(τ ) in terms of S and S. More precisely, we have thefollowing lemma.

Lemma . We have

g(τ ) =π

[( + S) + S

]

and

g(τ ) =π

[( + S) – S ( + S)

].

Proof Note that g(τ ) = –[ee + ee + ee], and hence

g(τ ) = –[℘

(τ

)℘

(

)+℘

(

)℘

(τ +

)+℘

(τ +

)℘

(τ

)]

= –[–π

( + S + S)( + S) –

π

( + S)( – S + S)

+π

( – S + S)( + S + S)

]

=π

[( + S) + S

].

Also note that g(τ ) = eee, and hence

g(τ ) = ℘

(τ

)℘

(

)℘

(τ +

)

=π

( + S + S)( + S)( – S + S)

=π

[( + S) – S ( + S)

].

This completes the proof of the lemma. �

In the next theorem, we give q-series expressions for E(τ ) and E(τ ) with the coeffi-cients involving restricted divisor functions σ,(N ; ) and its convolution sums. Precisely,we prove the following theorem.



Theorem . We have

E(τ ) = · (π )

g(τ )

= + q +∞∑

M=

[σ,(M; ) +

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

]qM

and

E(τ ) = · (π )

g(τ )

= – q – ,q +∞∑

M=

[σ,(M; ) + ,

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ ,M–∑k=

k+l+m=M

σ,(k; )σ,(l; )σ,(m; )

– M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

– ,M–∑k=

k+l+m–=M

σ,(k – ; )σ,(l – ; )σ,(m; )

]qM.

Proof From Lemma ., we observe that

E(τ ) = · (π )

g(τ )

= · (π )

· π

[( + S) + S

]= + S + S + S

= + ∞∑M=

σ,(M; )qM

+ ∞∑

M=k+l=M

M–∑k=

σ,(k; )σ,(l; )qM

+ ∞∑M=

k+l–=M

M∑k=

σ,(k – ; )σ,(l – ; )qM

= + σ,(; )q + ∞∑

M=

σ,(M; )qM



+ ∞∑

M=

M–∑k=

k+l=M

σ,(k; )σ,(l; )qM

+ σ,(; )σ,(; )q + ∞∑

M=k+l–=M

M∑k=

σ,(k – ; )σ,(l – ; )qM

= + q +∞∑

M=

[σ,(M; ) +

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

]qM

and

E(τ ) = · (π )

g(τ )

= + S + ,S + ,S – S – ,SS

= + ∞∑M=

σ,(M; )qM + ,∞∑

M=k+l=M

M–∑k=

σ,(k; )σ,(l; )qM

+ ,∞∑

M=k+l+m=M

M–∑k=

σ,(k; )σ,(l; )σ,(m; )qM

– ∞∑M=

k+l–=M

M∑k=

σ,(k – ; )σ,(l – ; )qM

– ,∞∑

M=k+l+m–=M

M–∑k=

σ,(k – ; )σ,(l – ; )σ,(m; )qM

= + σ,(; )q + σ,(; )q + ∞∑

M=

σ,(M; )qM

+ ,σ,(; )σ,(; )q + ,∞∑

M=

M–∑k=

k+l=M

σ,(k; )σ,(l; )qM

+ ,∞∑

M=

M–∑k=

k+l+m=M

σ,(k; )σ,(l; )σ,(m; )qM

– σ,(; )σ,(; )q – [σ,(; )σ,(; ) + σ,(; )σ,(; )

]–

∞∑M=

M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )qM

– ,σ,(; )σ,(; )σ,(; )q



– ,∞∑

M=

M–∑k=

k+l+m–=M

σ,(k – ; )σ,(l – ; )σ,(m; )qM

= + q + q + ,q – q – ,q – ,q

+∞∑

M=

[σ,(M; ) + ,

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ ,M–∑k=

k+l+m=M

σ,(k; )σ,(l; )σ,(m; )

– M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

– ,M–∑k=

k+l+m–=M

σ,(k – ; )σ,(l – ; )σ,(m; )

]qM.

Now replacing q into q in the above expressions for E(τ ) and E(τ ), we obtain

E(τ ) = + q +∞∑

M=

[σ,(M; ) +

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

]qM ()

and

E(τ ) = – q – ,q +∞∑

M=

[σ,(M; ) + ,

M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ ,M–∑k=

k+l+m=M

σ,(k; )σ,(l; )σ,(m; )

– M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

– ,M–∑k=

k+l+m–=M

σ,(k – ; )σ,(l – ; )σ,(m; )

]qM. ()

This completes the proof of the theorem. �

It should be noted that E(τ ) and E(τ ) themselves have coefficients σ(M) and σ(M) intheir q-series expansions. However, the aim of the next theorem is to express σ(M) andσ(M) in terms of convolution sums involving restricted divisor functions σ,(N ; ).



Theorem . (a) If M ≥ is an integer, then

σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M∑k=

σ,(k – ; )σ,(M – k + ; ).

In particular, if M ≥ is odd, then

M–∑

k=

σ,(k; )σ,(M – k; ) =

[σ(M) – σ,(M; )

].

(b) If M ≥ is an integer, then

–σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M–∑k,l=

σ,(k; )σ,(l; )σ,(M – k – l; )

– M∑k=

σ,(k – ; )σ,(M – k + ; )

– M–∑k,l=

σ,(k – ; )σ,(l – ; )σ,(M – k – l + ; ).

In particular, if M ≥ is odd, then

–σ(M) + σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M–∑k,l=

σ,(k; )σ,(l; )σ,(M – k – l; )

– M–∑k,l=

σ,(k – ; )σ,(l – ; )σ,(M – k – l + ; ).

Proof From [, p.], we know that

E(τ ) = + ∑M≥

σ(M)qM ()

and

E(τ ) = – ∑M≥

σ(M)qM. ()



So comparing () and (), we find that

σ(M) = σ,(M; ) + M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

= σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M∑k=

σ,(k – ; )σ,(M – k + ; ), ()

whereM ≥ .On the other hand, Liouville [] proved

σ(M) =M∑k=

σ (k – )σ (M – k + ) ()

for oddM. From () and (), we note that we reprove a result in [, p.], namely,

σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; ) + σ(M)

and so,

σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

= σ (M) +

M–∑

k=

σ,(k; )σ,(M – k; ) ()

for oddM. By the same way, comparing () and (), we deduce

–σ(M) = σ,(M; ) + M–∑k=

k+l=M

σ,(k; )σ,(l; )

+ M–∑k=

k+l+m=M

σ,(k; )σ,(l; )σ,(m; )

– M∑k=

k+l–=M

σ,(k – ; )σ,(l – ; )

– M–∑k=

k+l+m–=M

σ,(k – ; )σ,(l – ; )σ,(m; )



= σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M–∑k,l=

σ,(k; )σ,(l; )σ,(M – k – l; )

– M∑k=

σ,(k – ; )σ,(M – k + ; )

– M–∑k,l=

σ,(k – ; )σ,(l – ; )σ,(M – k – l + ; ), ()

whereM ≥ . Note that for oddM ≥ , we get

–σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; )

+ M–∑k,l=

σ,(k; )σ,(l; )σ,(M – k – l; ) – σ(M)

– M–∑k,l=

σ,(k – ; )σ,(l – ; )σ,(M – k – l + ; ), ()

where we used Equation (). This completes the proof of the theorem. �

4 Proof of the theorems

Proof of Theorem . Glaisher proved that (see [, p.])

σ ()σ (n – ) + σ ()σ (n – ) + · · · + σ (n – )σ () =[σ(n) – σ(n)

]. ()

It follows directly from () that

σ ()σ (M – ) + σ ()σ (M – ) + · · · + σ (M – )σ ()

=M∑k=

σ,(k – ; )σ,(M – k + ; )

and

σ(M) = σ,(M; ) + M–∑k=

σ,(k; )σ,(M – k; ) +[σ(M) – σ(M)

].

Thus, we have proved the identity

M–∑k=

σ,(k; )σ,(M – k; ) =

[σ(M) – σ(M) – σ,(M; )

]. ()



Thus, for any integer M ≥ , we have

σ(M) ≡ σ(M) + σ,(M; ) (mod ).

This proves (i).We note that we can write () as

∑k<M

σ,(k; )σ,(M – k; ) + σ,

(M;

)σ,

(M;

)

=

[σ(M) – σ(M) – σ,(M; )

]. ()

IfM is odd, then () is equivalent to

· ∑k<M

σ,(k; )σ,(M – k; ) = σ(M) – σ(M) – σ,(M; )

and hence,

σ(M) ≡ σ(M) + σ,(M; ) (mod ).

LetM = mpnpe pe · · ·perr . Therefore, if ordpM = n is odd for an odd prime p, then we note

that

σ,

(M;

)σ,

(M;

)= σ,

(m–pnpe p

e · · ·perr ;

)σ,

(m–pnpe p

e · · ·perr ;

)=

{σ,

(m–;

)}{σ

(pnpe p

e · · ·perr

)}=

{σ

(pn

)}{σ,

(m–;

)}{σ

(pe p

e · · ·perr

)}= (l)

{σ,

(m–;

)}{σ

(pe p

e · · ·perr

)},since

σ(pn

)= + p + p + · · · + pn ≡ (mod )

and so we can write σ(pn) = l for some l ∈N. Therefore, from (), we obtain

∑k<M

σ,(k; )σ,(M – k; ) + · l{σ,(m–;

)}{σ

(pe p

e · · ·perr

)}

=

[σ(M) – σ(M) – σ,(M; )

].

This means that

σ(M) ≡ σ(M) + σ,(M; ) (mod ).

This proves (ii). �



Remark . Define

P(N) := {p : p|N ,p is a positive prime}

and

Pmax(N) := maxp∈P(N)

p.

For example, P() = {, } and Pmax() = . For any prime p ≥ , we note that

Pmax

( p–∑k=

σ,(k; )σ,(p – k; )

)= p.

We observe that

p–∑k=

σ,(k; )σ,(p – k; ) =

(p +

)=(p + )p(p – )

and (p, ) = . Thus, |(p – ) and this implies that the largest prime factor of p – isstrictly < p.

Remark . From the fact

σ,(k; ) = σ(k) – σ

(k

),

and from known results, it is possible to establish the following identity (in an elementaryway, without using the Eisenstein series), namely

M–∑k=

σ,(k; )σ,(M – k; ) =

σ(M) +σ

(M

)–

σ(M) +

σ

(M

).

One can also use this identity to obtain

p–∑k=

σ,(k; )σ,(p – k; ) =

(p +

).

Proof of Theorem . Inserting () and () into (), we obtain

–σ(M) = σ,(M; ) + ·

[σ(M) – σ(M) – σ,(M; )

]

+ M–∑k,l=

σ,(k; )σ,(l; )σ,(M – k – l; ) – · [σ(M) – σ(M)

]

– M–∑k,l=

σ,(k – ; )σ,(l – ; )σ,(M – k – l + ; ).



Thus, we deduce the congruence relation

–σ(M) + σ,(M; ) ≡ σ(M) – σ(M) (mod ).

This completes the proof of Theorem .. �

Proof of Theorem . We expand S as

∞∑n,l,m=

σ,(n – ; )σ,(l – ; )σ,(m – ; )q(n+l+m–)–

=∞∑N=

{N–∑M=

M∑m=

σ,(N – – M; )

× σ,(M – (m – );

)σ,(m – ; )

}qN–.

Then we notice that

N–∑M=

M∑m=

σ,(N – – M; )σ,(M – (m – );

)σ,(m – ; )

=N–∑M=

σ,(N – – M; )

{ M∑m=

σ,(M – (m – );

)σ,(m – ; )

}

=N–∑M=

σ(N – – M) · (σ(M) – σ(M)

)()

by [, Lemma .(b)] and σ,(odd; ) = σ(odd). Since σ(M) = σ(M) – σ(M ), theright-hand side of () is

=N–∑M=

σ(N – – M){σ(M) – σ

(M

)}

=N–∑M=

σ(N – – M)σ(M) –∑

M< N–

σ(N – – M)σ(M)

=

(σ(N – ) – σ(N – )

)–

{

,σ(N – ) –

σ(N – ) +

a(N – )

}

by [, Theorem ] and in [, Theorem .(iii)] which state that

∑m<N/

σ(m)σ(N – m)

=

(σ(N) – σ (N) + σ

(N

)+ ( – N)σ

(N

))()



and

∑m<N/

σ(m)σ(N – m) =

,σ(N) +

σ

(N

)+

σ

(N

)

+ – N

σ

(N

)–

σ(N) +

a(N) ()

respectively. This completes the proof. �

Proof of Corollary . We note that


(n – )σ,(n – ; )σ,(l – ; )σ,(m – ; )

=


((n + l +m – ) –

)σ,(n – ; )σ,(l – ; )σ,(m – ; )

=


(N – )σ,(n – ; )σ,(l – ; )σ,(m – ; ).

Therefore, from Theorem ., we obtain


nσ,(n – ; )σ,(l – ; )σ,(m – ; )

=N +


σ,(n – ; )σ,(l – ; )σ,(m – ; )

=N +

(σ(N – ) – a(N – )

).

Thus, the first assertion

(N + )a(N – ) ≡ (N + )σ(N – ) (mod )

follows.We note that = · , and thus whenever N + �≡ (mod ) and N + is odd, we

have (N + , ) = . Thus,

σ(N – ) ≡ a(N – ) (mod )

whenever N ≡ , (mod ). This proves the second assertion. �

Proof of Theorem . We note that

SS =∞∑

n,l,m=

σ,(n – ; )σ,(l; )σ,(m; )q(n+l+m)–

=∞∑N=

{N–∑M=

M–∑m=

σ,(N – – M; )σ,(M – m; )σ,(m; )

}qN–.



Thus, by σ,(m; ) = σ,(m; ) and [, ()], we obtain that

N–∑M=

M–∑m=

σ,(N – – M; )σ,(M – m; )σ,(m; )

=N–∑M=

σ,(N – – M; )

{M–∑m=

σ,(M –m; )σ,(m; )

}

=N–∑M=

σ(N – – M){

(σ(M) – σ(M) – σ,(M; )

)}. ()

We observe that

σ(M) – σ(M) – σ,(M; )

= σ(M) –{σ(M) – σ

(M

)}–

{σ(M) – σ

(M

)}

= σ(M) + σ

(M

)– σ(M) + σ

(M

).

Thus, the right-hand side of () is

=

[

N–∑M=

σ(N – – M)σ(M) + ∑

M< N–

σ(N – – M)σ(M)

– N–∑M=

σ(N – – M)σ(M) + ∑

M< N–

σ(N – – M)σ(M)

].

Now from (), () and (see [, (.)]), we have

∑m<N/

σ (m)σ (N – m)

=

(σ(N) + ( – N)σ(N) + σ

(N

)+ ( – N)σ

(N

))

and from (see [, Theorem ]), we have

∑m<N/

σ (m)σ (N – m)

=

(σ(N) + ( – N)σ(N) + σ

(N

)+ σ

(N

)+ ( – N)σ

(N

)).

Thus, the theorem follows. �

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally to the manuscript and typed, read and approved the final manuscript.



Author details1National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon, 305-811, South Korea.2Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Chonju,Chonbuk 561-756, South Korea. 3Current address: National Institute for Mathematical Sciences, Yuseong-daero 1689-gil,Yuseong-gu, Daejeon, 305-811, South Korea. 4Permanent address: School of Mathematics, Tata Institute of FundamentalResearch, Homi Bhabha Road, Mumbai, 400005, India.

AcknowledgementsDedicated to Professor Hari M Srivastava.The author, Ayyadurai Sankaranarayanan, wishes to thank the National Institute for Mathematical Sciences (NIMS),Daejeon, Republic of Korea, for its warm hospitality and generous support. This research was supported by the NationalInstitute for Mathematical Sciences (NIMS) grant funded by the Korean government (B21303).

Received: 17 January 2013 Accepted: 20 March 2013 Published: 2 April 2013

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doi:10.1186/1687-1847-2013-84Cite this article as: Kim et al.: Eisenstein series and their applications to some arithmetic identities and congruences.Advances in Difference Equations 2013 2013:84.


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