Arithmetical properties of double Möbius-Bernoulli numbers

Abdelmejid Bayad; Daeyeoul Kim; Yan Li

doi:10.1515/math-2019-0006

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Arithmetical properties of double Möbius-Bernoulli numbers

Abdelmejid Bayad , Daeyeoul Kim und Yan Li

Veröffentlicht/Copyright: 17. Februar 2019

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Abstract

Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers M_k(n), double Möbius-Bernoulli numbers M_k(n,n′), and Möbius-Bernoulli polynomials M_k(n)(x). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials M_k(n)(x), can be interpreted as critical values of the following Dirichlet type L-function

LHM(s;n,x):=∑d|n∑m=0∞μ(d)(md+x)s(for Re(s)>1),

which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.

Keywords: Möbius-Bernoulli numbers; Barnes-Bernoulli numbers; Dedekind sums

MSC 2010: 11A05; 33E99

1 Introduction

Curiously, Möbius-Bernoulli numbers and polynomials are closely related to Dedekind Sums, critical values of certain Dirichlet series, Barnes-Bernoulli numbers and of course also to the Bernoulli numbers. In this paper, we will clarify all relationships between these arithmetical objects.

This paper consists of three parts. The first part treats Möbius-Bernoulli numbers and polynomials( cf. Section 2 ). In the second part, we consider new Dirichlet type series and show that their critical values are related to the Möbius-Bernoulli numbers and polynomials( cf. Section 3). In the third part, we study double Möbius-Bernoulli numbers and connect them to Barnes-Bernoulli numbers and Dedekind sums( cf. Sections 4 and 5). The three parts are more or less independent.

2 Möbius-Bernoulli numbers and polynomials

We fix some notations, definitions and preliminaries used in this paper. As to us we specify the motivations of our work.

2.1 Identities on Möbius-Bernoulli numbers and polynomials

For n being a positive integer, we define the Möbius-Bernoulli polynomials by the generating function

∑k=0∞Mk(n)(x)tkk!=∑d|nμ(d)textedt−1,|t|<2πn, (2.1)

where as usual the Möbius function μ is given by

μ(m)=1ifm=1,(−1)kifm=p1…pk,p1,…,pk are distinct primes,0otherwise.

The Möbius-Bernoulli numbers M_k(n) are given by M_k(n) := M_k(n)(0). We recall the Bernoulli polynomial B_k (x) is defined by the series:

textet−1=∑k=0∞Bk(x)tkk!,|t|<2π. (2.2)

Note that B_k (x) are monic polynomials with rational coefficients and B_k := B_k (0) is the kth Bernoulli number. From the equations (2.1), (2.2) and Möbius inversion formula, we obtain

Theorem 1

Let n and k be nonnegative integers. We have

Mk(n)(x)=∑d|nμ(d)dk−1Bk(x/d), (2.3)

Bk(nx)=∑d|n(n/d)k−1Mk(d)(dx), (2.4)

at x = 0, we get

Mk(n)=Bk∏p|n1−pk−1 (2.5)

and Gauss type formula

∑d|ndk∏p|d1−1/pk=nk. (2.6)

Let n^* = p₁ ⋯ p_r be the square free part of n. To get the equation (2.5), we use the relation (2.3) at x = 0. Since μ(d) = 0 if d is not square-free, we have

Mk(n)=Bk∑d|n∗μ(d)dk−1=Bk∑{i1,⋯,is}⊂{1,2,⋯,r}μ(pi1⋯pis)(pi1⋯pis)k−1=Bk∑{i1,⋯,is}⊂{1,2,⋯,r}(−1)s(pi1⋯pis)k−1=Bk∏p prime |n(1−pk−1).

Among others, in this paper we will give an interpretation of the formula (2.5) in terms of critical values of Dirichlet L-series, and its generalization to double Möbius-Bernoulli numbers.

2.2 Generalized Bernoulli numbers

By Dirichlet character modulo a positive integer n, we mean as usual ℂ-valued function χ on ℤ such that χ(m) = 0 if m is not coprime to n, and χ induces a character on (ℤ/nℤ)^×. For a such χ, the generalized Bernoulli numbers

Bk,χ∈Q(χ(1),χ(2),⋯,χ(n−1))

are given through the generating function

∑a=1nχ(a)teatent−1=∑k=0∞Bk,χtkk!. (2.7)

From the equations (2.2) and (2.7) we have

Bk,χ=nk−1∑a=1n−1χ(a)Bk(a/n). (2.8)

The numbers B_k,χ are very intriguing and still mysterious. If k ≥ 1, then B_k,χ = 0 if χ(−1) = (−1)^k−1 (unless n = k = 1). In the simplest case where F is an imaginary quadratic field of discriminant −D, and χ is th unique quadratic character of conductor D such that χ(−1) = −1, that is, χ(a)=(a−D), and then we have the classical formula

B1,χ=−2hFwF,

where h_F is the class number of F and w_F is the number of roots of unity in F.

Recently, for such numbers B_k,χ many others interesting explicit formulae are obtained. In particular, when χ is a quadratic character see [1], for a nontrivial primitive Dirichlet character χ we refer to [2, 3].

2.3 Critical values of certain Dirichlet L-series

The Dirichlet L-series:

L(s,χ)=∑m≥1χ(m)m−s( Re(s)>1)

is the L-series attached to any character χ.

The main interest of the numbers B_k,χ is that they give the value at non-positive integers of Dirichlet L-series. In fact, there is a well-known formula, proved by Hecke in [4]

L(1−k,χ)=−Bk,χk(k≥1). (2.9)

In this paper, we are going to study Möbius-Bernoulli and double Möbius-Bernoulli numbers: M_k(n), M_k(n, n′). We establish their relationship to Bernoulli, Apostol-Bernoulli and Barnes-Bernoulli numbers, and Dedekind sums. In part of this paper, the Möbius-Bernoulli, can be interpreted as critical values of the following Dirichlet type L-functions.

Lemma 2

For k, n being positive integers, and χ_n the Dirichlet principal character modulo n, then we have

Mk(n)/k=−L(1−k,χn). (2.10)

Proof of Lemma 2

We can get this result from relations (2.9) and Theorem 1. For Dirichlet character, the L-series has the Euler product

L(s,χ)=∏p∤n1−χ(p)p−s−1.

While χ = χ_n, we have

L(s,χn)=ζ(s)∏p|n1−p−s,

where ζ(s) is the Riemann zeta function. At s = 1 − k we obtain

Bk,χn=Bk∏p|n1−pk−1. (2.11)

The relation (2.5) and (2.9) completes the proof. □

From the equalities (2.5), (2.10) and Kummer’s congruence [5, Theorem 5, p.239] for Bernoulli numbers B_k we obtain the following Kummer’s type congruence formula.

Theorem 3

(Kummer’s type Congruences). Let p be a prime number, p − 1Ȥ k and k′ ≡ k (mod(p − 1)p^N) with N being a nonnegative integer. Then we have

Mk(n)k≡Mk′(n)k′(modpN+1) if p|n,1−pk−1Mk(n)k≡1−pk′−1Mk′(n)k′(modpN+1) otherwise.

Remark 4

One can use again the equalities (2.5), (2.10) and Von Satudt Clausen theorem [5, Theorem 3, p.233] to obtain von Staudt Clausen type result for M_k(n).

3 Dirichlet type L-series and Möbius-Bernoulli polynomials

3.1 Möbius L-functions of Hurwitz type

For n being a positive integer and x > 0, let

LHM(s;n,x):=∑d|n∑m=0∞μ(d)(md+x)s(for Re(s)>1). (3.1)

We call L_HM(s; n, x) Hurwitz-Möbius L-functions. Let

f(t;n,x):=∑k=0∞Mk(n)(x)tkk!=∑d|nμ(d)textedt−1

be the generating function of Möbius-Bernoulli polynomials, M_k (n)(x) (k ≥ 0). Consider

Γ(s)LHM(s;n,x)=∫0∞e−ttsdtt∑d|n∑m=0∞μ(d)(md+x)s=∑d|n∑m=0∞∫0∞μ(d)(md+x)se−ttsdtt(for Re(s)>1).

Substituting t by (md + x)t in the last equality, we get

Γ(s)LHM(s;n,x)=∑d|n∑m=0∞∫0∞μ(d)e−(md+x)ttsdtt=∫0∞∑d|nμ(d)∑m=0∞e−(md+x)ttsdtt=∫0∞∑d|nμ(d)e−xt1−e−dttsdtt=∫0∞∑d|nμ(d)−te−xte−dt−1ts−1dtt(for Re(s)>1).

The second equality is by Lebesgue’s dominated convergence theorem. Therefore, we have

LHM(s;n,x)=1Γ(s)∫0∞∑d|nμ(d)e−xt1−e−dttsdtt=1Γ(s)∫0∞f(−t;n,x)ts−1dtt(for Re(s)>1). (3.2)

This shows that the Hurwitz-Möbius L-function L_HM(s; n, x) is almost the Mellin transform of f(t; n, x), the generating function of Möbius-Bernoulli polynomials, M_k (n)(x) (k ≥ 0).

Using Proposition 10.2.2 of [6] and formula (2.2) of [7], we can get the following theorem.

Theorem 5

Notations as above. We have

LHM(s;n,x)=∑d|nμ(d)dsζ(s;xd),

where ζ(s; x) is the Hurwitz zeta function with parameter x > 0; L_HM(s; n, x) can be analytically continued to the whole complex plane, to a meromorphic function with a single pole at s = 1, simple with residue ϕ(n)n and

LHM(−k;n,x)=−Mk+1(n)(x)k+1,

where k ≥ 0 is an integer.

Proof

By definition (3.1), we have

LHM(s;n,x)=∑d|n∑m=0∞μ(d)(md+x)s=∑d|nμ(d)ds∑m=0∞1(m+x/d)s=∑d|nμ(d)dsζ(s;xd)( for Re(s)>1).

This completes the proof of the first statement.

Now by equation (3.2), we get

LHM(s;n,x)−1Γ(s)∫0∞∑d|nμ(d)de−tttsdtt=1Γ(s)∫0∞∑d|nμ(d)e−xt1−e−dt−∑d|nμ(d)de−tttsdtt.

On the other hand, we have

Let

f~(t;n,x)=∑d|nμ(d)e−xt1−e−dt−∑d|nμ(d)de−tt.

Then f͠(t; n, x) is C^∞ on [0, ∞) and tending to zero rapidly at infinity. We define

L(s,f~,x):=1Γ(s)∫0∞f~(t;n,x)tsdtt. (3.3)

By Proposition 10.2.2 of [6], we know that L(s, f͠, x) can be analytically continued to the whole complex plane, to a holomorphic function.

Obviously, the above computations show that

LHM(s;n,x)=L(s,f~,x)+1s−1∏p|n(1−1p). (3.4)

So L_HM(s; n, x) can be analytically continued to the whole complex plane, to a meromorphic function with a single pole at s = 1, simple with residue ϕ(n)n. This completes the proof of the second statement.

Also by Proposition 10.2.2 of [6], we have

L(−k,f~,x)=(−1)kdkf~dtk(0) (3.5)

for k ≥ 0 and k ∈ ℤ.

Obviously, f͠(t; n, x) is analytic around zero. Computing its Taylor expansion at 0, we get

Therefore,

L(−k,f~,x)=−1k+1Mk+1(n)(x)−∑d|nμ(d)d. (3.6)

From equations (3.4) and (3.6), we get

LHM(−k;n,x)=−1k+1∏p|n1−1p−1k+1Mk+1(n)(x)−∑d|nμ(d)d=−1k+1Mk+1(n)(x).

This completes the proof of the final statement. □

3.2 Modified Möbius L-functions

For n being a positive integer, let

g(t):=∑k=0∞M~k(n)tkk!=∑d|nμ(d)tedtent−1.

We call M͠_k (n) modified Möbius-Bernoulli numbers.

In the following, we will first show their relations with Bernoulli polynomials. For n = 1,

∑k=0∞M~k(n)tkk!=t+tet−1=t+∑k=0∞Bktkk!.

So M͠_k (1) is essentially Bernoulli number B_k.

Now we consider the general case. Then we have

∑k=0∞M~k(n)tkk!=∑d|nμ(d)tedtent−1.

Changing variable t to t/n, we get,

∑k=0∞M~k(n)1nktkk!=1n∑d|nμ(d)tedntet−1=1n∑d|nμ(d)∑k=0∞Bk(dn)tkk!.

Thus we have the following relation

M~k(n)=nk−1∑d|nμ(d)Bk(dn). (3.7)

For n being a positive integer, let

LM(s;n):=∑d|n∑m=0∞μ(d)(mn+d)s(for Re(s)>1). (3.8)

We call L_M(s; n) the modified Möbius L-functions. Note that if n = 1, then the modified Möbius L-function L_M(s; n) is just the usual Riemann zeta function ζ(s). Similarly as in the previous subsection, we can prove that

LM(s;n)=1Γ(s)∫0∞∑d|nμ(d)e−dt1−e−nttsdtt=1Γ(s)∫0∞g(−t)ts−1dtt(for Re(s)>1), (3.9)

which shows that the modified Möbius L-function L_M(s; n) is almost the Mellin transform of g(t), the generating function of modified Möbius-Bernoulli numbers, M͠_k (n) (k ≥ 0).

Changing variable t to t/n in (3.9), we get

LM(s;n)=∑d|nμ(d)ns1Γ(s)∫0∞e−dnt1−e−ttsdtt(for Re(s)>1). (3.10)

Similarly as Theorem 5, we can prove the following theorem

Theorem 6

Let n ≥ 1 be an integer. Then we have

LM(s;n)=n−s∑d|nμ(d)ζ(s;d/n).

We have L_M(s; 1) = ζ(s). For n ≥ 2, L_M(s; n) has holomorphic continuation to the whole s-complex plane and

LM(−k;n)=−M~k+1(n)k+1.

4 Double Möbius-Bernoulli and Barnes-Bernoulli numbers

In this section we introduce and give some properties of the double Möbius-Bernoulli numbers. We express these numbers in terms of Barnes-Bernoulli numbers. Using Barnes-Bernoulli numbers properties, explicit formulas will be given for double Möbius-Bernoulli in section 5.

Let a₁,a₂ be nonzero real numbers. The double Bernoulli-Barnes numbers B_k ((a₁, a₂ )) are defined through

z2(ea1z−1)(ea2z−1)=∑k=0∞Bk((a1,a2))zkk!,|t|<min2π|a1|,2π|a2|. (4.1)

We investigate, for n, n′ being positive integers, the double Möbius-Bernoulli numbers M_k (n, n′) given by

Mk(n,n′)=∑j=0kkjMj(n)Mk−j(n′). (4.2)

Theorem 7

Let n, n′ be positive integers and n^*, n′^* be their square free parts, respectively. We have the following results.

Mk(n,n′)=∑d|n,d′|n′μ(d)μ(d′)Bk((d,d′)). (4.3)

Mk(n)=Mk(n∗) (4.4)

Mk(n,n′)=Mk(n∗,n′∗). (4.5)

Proof of Theorem 10

Since μ(d) = 0 if d is not square-free, we obtain the relations (4.4) and (4.5). On the other hand, it is easy to show that

∑d|nμ(d)tedt−1∑d′|n′μ(d′)ted′t−1=∑d|n,d′|n′μ(d)μ(d′)t2(edt−1)(ed′t−1).

Expanding the Taylor series of both sides yields the following identity

∑k=0∞∑j=0kkjMj(n)Mk−j(n′)tkk!=∑k=0∞∑d|n,d′|n′μ(d)μ(d′)Bk((d,d′))tkk!.

Thus we obtain identity (4.3). □

5 Double Möbius-Bernoulli numbers and Dedekind sums

In this section, we give an effective method to compute the Möbius-Bernoulli numbers, based on the Apostol- Dedekind reciprocity law for the generalized Dedekind sums.

5.1 Generalized Dedekind Sums s_k(a, b)

Let a and b be positive integers. The Apostol–Dedekind sums s_k(a, b) are given by

sk(a,b)=∑r=0b−1rbBkarb. (5.1)

These sums are effectively computable through the Apostol- Dedekind reciprocity law [8] and its generalization in [9]. By use of the Apostol- Dedekind reciprocity we state the following results.

Theorem 8

Let a, b be positive integers, k positive integer, and d = gcd(a, b). Then we have

Bk((a,b))=k(ak−2sk−1(b,a)+bk−2sk−1(a,b))−kdk−1Bk−1−(k−1)d2(k−1)abBk. (5.2)

From the above theorem we get the following identities.

Corollary 9

Let p, p₁, p₂ be prime numbers, with p₁, p₂ different. We have

Bk((1,1))=−kBk−1−(k−1)Bk, (5.3)

Bk((p,1))=kpk−2sk−1(1,p)−kBk−1−(k−1)pBk, (5.4)

Bk((p,p))=−kpk−2Bk−1−(k−1)p2(k−2)Bk, (5.5)

Bk((p1,p2))=k(p2k−2sk−1(p1,p2)+p1k−2sk−1(p2,p1))−kBk−1−(k−1)p1p2Bk. (5.6)

The equality (5.3) is well-known since Euler [10, p.32]. However, the equalities (5.4), (5.5), (5.6) seem new identities.

Theorem 10

Let n, n′ be positive coprime integers and n^*, n′^* be their square free parts, respectively. We have

Mk(n,n′)=k∑d|n∗,d′|n′∗μ(d)μ(d′)dk−2sk−1(d′,d)+d′k−2sk−1(d,d′)−kBk−1δ1,n∗n′∗−(k−1)Bkφ(n∗n′∗)n∗n′∗, (5.7)k

where φ is the Euler’s function and δ_i,j is the Kronecker’s symbol.

We combine Corollary 9 with Theorem 10, and we obtain the following explicit formulas.

Corollary 11

Let p₁, p₂, p be distinct primes and α ≥ 1, β ≥ 1 are integers. We have

Mk(p1α,p2β)=−(k−1)1−1/p11−1/p2Bk+kp2k−2sk−1(p1,p2)−sk−1(1,p2)+kp1k−2sk−1(p2,p1)−sk−1(1,p1),Mk(pα,pβ)=k1−pk−2Bk−1−(k−1)1+p2(k−2)−2/pBk−2kpk−2sk−1(1,p).

5.2 Proofs of the Theorem 8 and Theorem 10

Proof of Theorem 8

Combining (2.2) with (4.1), we get the relation between Bernoulli-Barnes numbers and Bernoulli numbers:

Bk((a1,a2))=∑m=0kkma1m−1a2k−m−1BmBk−m. (5.8)

By use of the equation (5.8), Apostol results in [8, Theorem 1], [11, Theorem 2] and Takacs generalization [9, Theorem 1], with x = y = 0, we achieve the proof of the Theorem 8. □

Proof of Theorem 10

Take n, n′, a, b be positive integers such that:

(a,b)=(n,n′)=1.

Then we have the equalities

Bk((a,b))=k(ak−2sk−1(b,a)+bk−2sk−1(a,b))−kBk−1−(k−1)abBk, (5.9)

Mk(n,n′)=Mk(n∗,n′∗)=∑d|n∗,d′|n′∗μ(d)μ(d′)Bk((d,d′)), (5.10)

φ(n)n=∑d|nμ(d)d−1, (5.11)

δ1,n=∑d|nμ(d). (5.12)

From these equalities we complete the proof of the Theorem 10. □

We conclude this paper by the following remark.

Remark 12

The generalized Dedekind sums s_k−1(a, b) are very easy to evaluate for small a and b. For example, using (5.1), we get values of s_k−1(a, b) in Table 1 with 1 ≤ a, b ≤ 5 and (a, b) = 1.

Table 1

Examples for s_k−1(a, b) with 1 ≤ a, b ≤ 5 and (a, b) = 1

(a, b)	s_k−1(a, b)
(a, 1)	0
(1,2),(3,2),(5,2)	12Bk−1(12)
(1,3),(4,3)	13Bk−1(13)+23Bk−1(23)
(1,4),(5,4)	14Bk−1(14)+12Bk−1(12)+34Bk−1(34)
(1,5)	15Bk−1(15)+25Bk−1(25)+35Bk−1(35)+45Bk−1(45)
(2,3),(5,3)	13Bk−1(23)+23Bk−1(13)
(2,5)	15Bk−1(25)+25Bk−1(45)+35Bk−1(15)+45Bk−1(35)
(3,4)	14Bk−1(34)+12Bk−1(12)+34Bk−1(14)
(3,5)	15Bk−1(35)+25Bk−1(15)+35Bk−1(45)+45Bk−1(25)
(4,5)	15Bk−1(45)+25Bk−1(35)+35Bk−1(25)+45Bk−1(15)

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Received: 2018-05-09

Accepted: 2019-01-09

Published Online: 2019-02-17

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https://doi.org/10.1515/math-2019-0006

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Möbius-Bernoulli numbers; Barnes-Bernoulli numbers; Dedekind sums

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Arithmetical properties of double Möbius-Bernoulli numbers

Artikel

Abstract

1 Introduction

2 Möbius-Bernoulli numbers and polynomials

2.1 Identities on Möbius-Bernoulli numbers and polynomials

Theorem 1

2.2 Generalized Bernoulli numbers

2.3 Critical values of certain Dirichlet L-series

Lemma 2

Proof of Lemma 2

Theorem 3

Remark 4

3 Dirichlet type L-series and Möbius-Bernoulli polynomials

3.1 Möbius L-functions of Hurwitz type

Theorem 5

Proof

3.2 Modified Möbius L-functions

Theorem 6

4 Double Möbius-Bernoulli and Barnes-Bernoulli numbers

Theorem 7

Proof of Theorem 10

5 Double Möbius-Bernoulli numbers and Dedekind sums

5.1 Generalized Dedekind Sums sk(a, b)

Theorem 8

Corollary 9

Theorem 10

Corollary 11

5.2 Proofs of the Theorem 8 and Theorem 10

Proof of Theorem 8

Proof of Theorem 10

Remark 12

References

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5.1 Generalized Dedekind Sums s_k(a, b)