Bernoulli numbers and certain convolution sums with divisor functions

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R E S E A R C H

Open Access

Bernoulli numbers and certain convolution

sums with divisor functions

Daeyeoul Kim

1

_{, Aeran Kim}

2

_{and Nazli Yildiz Ikikardes}

3*

*_{Correspondence:}

[email protected]

3_{Department of Elementary}

Mathematics Education, Necatibey Faculty of Education, Balikesir University, Balikesir, 10100, Turkey Full list of author information is available at the end of the article

Abstract

In this paper, we investigate the convolution sums

(a+b+c)x=n

a,

ax+by=n

ab,

ax+by+cz=n

abc,

ax+by+cz+du=n abcd,

wherea,b,c,d,x,y,z,u,n∈N. Many new equalities and inequalities involving convolution sums, Bernoulli numbers and divisor functions have also been given.

MSC: 11A05; 33E99

Keywords: inequality of Diophantine equations; Bernoulli numbers; convolution sums

1 Introduction

Throughout this paper,N,Z, andCwill denote the sets of positive integers, rational in-tegers, and complex numbers, respectively. The Bernoulli polynomialsBk(x), which are

usually deﬁned by the exponential generating function

text

et_{– }=

∞

k=

Bk(x)

tk

k!,

play an important role in different areas of mathematics, including number theory and the theory of finite differences. The Bernoulli polynomials satisfy the following well-known identity:

N

j=

jk=Bk+(N+ ) –Bk+()

k+  , k≥. ()

It is well known thatBk=Bk() are rational numbers. It can be shown thatBk+=  for

k≥, and is alternatively positive and negative for evenk. TheBk are called Bernoulli

numbers.

Forn,k∈Nwiths∈N∪ {}, we deﬁne

σs(n) =

d|n

ds, Fk(n) =

⎧ ⎨ ⎩

, ifk|n,

, ifkn.

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The exact evaluation of the basic convolution sum

n–

m=

σ(m)σ(n–m)

ﬁrst appeared in a letter from Besge to Liouville in . Ramanujan’s work has been ex-tended by many authors,e.g., see []. For example, the following identity

n–

m=

σ(m)σ(n–m) =

 

σ(n) + ( – n)σ(n)

()

is due to the works of Huardet al.[]. In [], Ramanujan also found nine identities, includ-ing (), of the form

n

m=

σr(m)σs(n–m) =Aσr+s+(n) +Bnσr+s–(n),

whereAandBare certain rational numbers. We refer to [] for a similar work. Lahiri [] obtained the most general result by evaluating the sum

m+···+mr=n

ma

 · · ·marrσb(m)· · ·σbr(mr) (r≥),

where the sum is over all positive integersm, . . . ,mr satisfyingm+· · ·+mr=n,ai∈

N∪ {}, andbi∈N.

The convolution identities have many beautiful applications in modern number theory, in particular in modular forms, since they appear in the coeﬃcients of the Fourier expan-sions of classical Eisenstein series. For example, a very well-known work of Serre onp-adic modular forms (see []). For some of the history of the subject, and for a selection of these articles, we mention [, ] and [], and especially [] and []. We also refer to [] and [].

In this paper, we shall investigate the convolution sums

(a+b+c)x=n

a,

ax+by=n

ab,

ax+by+cz=n

abc,

ax+by+cz+du=n

abcd.

In fact, we will prove the following results.

Theorem . Let n be a positive integer.Then we have

(a+b+c)x=n

a=  σ(n) –

 σ(n) +

 σ(n) >



B(n– ) ()

with n≥.

Remark . Letαbe a ﬁxed integer withα≥, and let

Pyr_α(x) =  

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be theαth order pyramid number. In fact, in (), ifn=pis a prime number, then we obtain

(a+b+c)x=p

a=Pyr_(p– ). ()

This result is similar to [, ()].

Theorem . Let M be an odd positive integer.Let R,r∈N∪ {}with R≥r.Then we have

A(R,r) :=

ax+by=R_M

ax=r_m

modd

ab= R–r–r+– σ(M) >

R–r– 

r+– B(q+ ) ()

with M= q+ .

Theorem . Let mbe an odd positive integer.Let r,r∈Nand r∈N∪ {}with r>

r>r.Then we have

A(r,r,r) :=

ax+by+cz=r_m_

ax+by=rm

ax=rm

modd

modd

abc= r–r–r–_r+_{– }_r+_{– }_σ

(m). ()

Theorem . Let m be an odd positive integer.Let r,r,r∈Nand r∈N∪ {}with

r>r>r>r.Then we have

ax+by+cz+du=rm

ax+by+cz=rm

ax+by=rm

ax=r_m_

modd

modd

modd

abcd=  ·

–r–r–r–_r+_{– }_r+_{– }_r+_{– }

×r_σ

(m) + (–)r–rr+r–b(m)

when∞_n_=b(n)qn=q∞_n_=( –qn)( –qn).

Theorem . Let M be an odd positive integer.Let r,R∈N∪ {}.Then we have

R≤r<log(RMm )

ax+by=R_M

ax=r_m

modd

ab

=  

·R+– R++ σ(M) –

·R+M– R+– σ(M) .

Corollary . For R>r,we have the following lower bound of A(R,r)and the upper bound

of A(r,r,r),

A(R,r) >

 σ

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and

A(r,r,r) <

 σ

r–_m



.

2 Bernoulli number derived from Diophantine equations₍_a₊_b₊_c₎_x₌_na Lemma . Let n∈N.Let f :Z→Cbe an odd function.Then

(a,b,c,x)∈N

(a+b+c)x=n

f(a+b) +f(b–c)=

e|n e–

k=

(e–k– )f(e–k).

Proof We can write the equality as

(a+b+c)x=n

f(a+b) +f(b–c)

=

k≥

f(k)

(a+b+c)x=n a+b=k

 +

(a+b+c)x=n b–c=k

 –

(a+b+c)x=n b–c=–k



=

k≥

f(k)

(a+b+c)x=n a+b=k



=

e|n

(e– )f(e– ) + (e– )f(e– ) +· · ·+e– (e– )f()

=

e|n e–

k=

(e–k– )f(e–k).

This completes the proof of the lemma.

Proof of Theorem. Letf(x) =x. Then Lemma . becomes

(a+b+c)x=n

{a+ b–c}=

σ(n) –σ(n) + 

σ(n) ()

and

(a+b+c)x=n

a=  σ(n) –

 σ(n) +



σ(n). ()

Using (), we note that

p–

j=

j= 

B(p– ) –B =

B(p– )



sinceB= . It is easily checked that

p(p– )(p– )

 >

(p– )(p– )(p– )

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We can write that

(a+b+c)x=n

a> 

B(n– )

withn≥. This completes the proof of the theorem.

We list the ﬁrst ten values of₍_a₊_b₊_c₎_x₌_nain Table .

Remark . Let

f(x) :=

(a+b+c)t=x

a

and

g(x) := 

x(x– )(x– ) =Pyr(x– ).

Ifxis a prime integer, by () and (), thenf(x) =g(x).

The ﬁrst nine values off(x) andg(x) are given in Figure . In Figure , we plot the graphs for the values of the sumsf(x) andg(x) in Remark . whenx= , , , , , , , , .

Table 1 The ﬁrst ten values of₍_a₊_b₊_c₎_x₌_na

n 1 2 3 4 5 6 7 8 9 10

(a+b+c)x=na 0 0 1 4 10 21 35 60 85 130

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3 Two lemmas

where the Kronecker delta symbol is deﬁned by

δi,j=

Proof We note that

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Therefore, () becomes

This completes the proof of the lemma.

Example .

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Proof It is similar to Lemma ..

4 A study of_ax₊_by₌_nab

Proof of Theorem. We observe that

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Therefore, () becomes

ax+by=R_M

ax=r_m

modd

ab=r+– 

m<R–r_M m

σ(m)σ

R–rM–m

=r+– 

m<R–r_M σ(m)σ

R–rM–m

– 

m<R–r–_M

σ(m)σ

R–rM– m

+ 

m<R–r–_M

σ(m)σ

R–rM– m

=r+– ·R–r–σ(M), ()

where we refer to (),

m<n/

σ(m)σ(n– m) =

 

σ(n) + ( – n)σ(n) + σ(n/)

+ ( – n)σ(n/)

in [, (.)] and

m<n/

σ(m)σ(n– m) =

 

σ(n) + ( – n)σ(n) + σ(n/)

+ σ(n/) + ( – n)σ(n/)

in [, Theorem ]. Thus, we obtain

A(R,r) = R–r–

r+– σ(M)

> R–r–r+– σ(M) –σ(M)

> R–r–r+– 

q(q+ )(q+ ) 

≥R–r–r+– 

B(q+ ) –B



()

withM= q+ . This completes the proof of this theorem.

Theorem . Let M be an odd positive integer.Let R∈Nand r∈N∪ {}with R>r.Then we have

(a)

ax+by=R_M

ax=r_m

modd

xeven

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(b)

ax+by=R_M

ax=r_m

modd

xeven

yeven

ab= R–r–r– σ(M),

(c)

ax+by=R_M

ax=r_m

modd

xeven

yodd

ab=

ax+by=R_M

ax=r_m

modd

xodd

yeven

ab= R–r–r– σ(M),

(d)

ax+by=R_M

ax=r_m

modd

xodd

yodd

ab= R–r–σ(M).

Proof

(a) First, we note that

m<R–r_M m

σ(m)σ

R–rM–m= R–r–σ(M), ()

by (). Therefore,

ax+by=R_M

ax=r_m

modd

xeven

ab=

ax+by=R_M ax=r_m

modd

ab

=

ax+by=R_M

ax=r–_m

modd

ab

=

m<R–r_M m

a|r–_m a

b|r_(R–r_M_–_m₎

b

=r– r+– 

m<R–r_M m

σ(m)σ

R–rM–m

= R–r–r– r+– σ(M),

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(b) We observe that

so () becomes

rr– 

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Corollary . Let M be an odd positive integer.Let R∈Nand r∈N∪ {}with R>r.Then

Proof From (), we deduce that

n–

Proof By Theorem ., we have

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Similarly, the other terms of () are

Proof of Theorem. The proof starts as follows:

Then the second term of () is

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= 

Proof of Theorem. From Theorem ., we observe that

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Thus, we refer to

(a) The proof is similar to Theorem .. Let us consider that

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Then Thus, () becomes

(b) We sketch the proof as follows:

Proof of Corollary. Firstly, from (), we note that

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t–t_{= –(t}_{– )}_{+  and  < t}_–_t_≤

with  <t≤

t

. Putt= (  )r then

 <

r+_{– } r ≤



. ()

Thirdly, we considerf(t) =t( –t)_{with  <}_t_{< . Then, we easily check that  <}_f_(t)_≤ 

so

 <

r+_{– } r ≤



. ()

Consider

r  –

r_{– }  –  =

 – ()r–

 <  ()

withr> . From (), we deduce that

r–_<_σ



r– and

r–_<  σ

r–_. _()

From (), () and (), we compute thatA(r,r,r) <_σ(r–m). 5 A study of_ax₊_by₊_cz₊_du₌_nabcd

Corollary . Let m be an odd positive integer.Let r,r,r∈Nand r∈N∪ {}with

r>r>r>r.If r,r,r≡– (mod),then we have

r+_σ

(m)≡(–)r–r+·r+rb(m) (mod·r+r+r+).

Proof From Theorem ., we have

ax+by+cz+du=rm

ax+by+cz=r_m_

ax+by=rm

ax=rm

modd

modd

modd

abcd=  ·

–r–r–r–_r+_{– }_r+_{– }_r+_{– }

×r_σ

(m) + (–)r–rr+r–b(m) . ()

Sincer,r,r≡– (mod) by the assumption, therefore, r+– ≡ (mod). So from

(), we have

–r–r–r–_r_σ

(m) + (–)r–rr+r–b(m) ≡ (mod). ()

By multiplying () by r+r+r+_{, we obtain the proof.}

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6 Another convolution sums

Theorem . Let M∈NwithM.Let R∈Nand r∈N∪ {}with R≥r.Then we have

A∗_(R,r) :=

ax+by=R_M

ax=r_m m

ab=  ·

R–r–_r+_{– }

σ(M) ()

and if R>r,then

A∗_(R,r) >  ·σ

R–r–M.

Proof It is similar to Theorem .. So we obtain that

ax+by=R_M

ax=r_m m

ab= 

r+– 

m<R–r_M m

σ(m)σ

R–rM–m

= 

r+– 

m<R–r_M σ(m)σ

R–rM–m

–

m<R–r_M |m

σ(m)σ

R–rM–m

= 

r+_{– }

m<R–r_M σ(m)σ

R–r_M_–_m

–

m<R–r–_M

σ(m)σ

R–rM– m.

Then we refer to

m<n

σ(m)σ(n– m) =

 

σ(n) + ( – n)σ(n) + σ

n 

,

ifn≡ (mod) in [, Theorem ]. Therefore, we get (). By (), we note that

A∗_(R,r) =  

R–r–_r+_{– }

σ(M)

= ·

 

(R–r)_r+_{– }

σ(M). ()

It is well known that

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Table 2 Values ofb(n) (1≤n≤12)

n b(n) n b(n)

1 1 7 1,016

2 –8 8 –512

3 12 9 –2,043

4 64 10 1,680

5 –210 11 1,092

6 –96 12 768

withr≥. Combine () and (),

A∗_(R,r) > ·

(R–r)_σ (M)

> ··

((R–r)_{– )}

 –  σ(M)

=  ·σ

R–r–M

with (,M) = . This completes the proof of this theorem.

Appendix

The ﬁrst twelve values ofb(n) forn∈Nare given in Table .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon, 305-811, South Korea.}2_Department

of Mathematics, Institute of Pure and Applied Mathematics, Chonbuk National University, Chonbuk, Chonju, 561-756, South Korea. 3_{Department of Elementary Mathematics Education, Necatibey Faculty of Education, Balikesir University,}

Balikesir, 10100, Turkey.

Acknowledgements

The ﬁrst author was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean government (B21303).

Received: 20 June 2013 Accepted: 23 August 2013 Published: 23 September 2013

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doi:10.1186/1687-1847-2013-277