Arithmetical properties of double Möbius-Bernoulli numbers

Abstract Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers Mk(n), double Möbius-Bernoulli numbers Mk(n,n′), and Möbius-Bernoulli polynomials Mk(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 Mk(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), $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.


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 rst 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.

Möbius-Bernoulli numbers and polynomials
We x some notations, de nitions and preliminaries used in this paper. As to us we specify the motivations of our work.

. Identities on Möbius-Bernoulli numbers and polynomials
For n being a positive integer, we de ne the Möbius-Bernoulli polynomials by the generating function ∞ k= M k (n)(x) t k k! = d|n µ(d) te xt e dt − , |t| < π n , (2.1) where as usual the Möbius function µ is given by The Möbius-Bernoulli numbers M k (n) are given by M k (n) := M k (n)( ). We recall the Bernoulli polynomial B k (x) is de ned by the series: and Gauss type formula Let n * = p · · · pr be the square free part of n. To get the equation (2.5), we use the relation (2.3) at x = . Since 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.

. 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) = if m is not coprime to n, and χ induces a character on Z/nZ × . For a such χ, the generalized Bernoulli numbers B k,χ ∈ Q(χ( ), χ( ), · · · , χ(n − )) are given through the generating function 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]. .

Critical values of certain Dirichlet L-series
The Dirichlet L-series : 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] 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 χn the Dirichlet principal character modulo n, then we have
Proof of Lemma 2. We can get this result from relations (2.9) and Theorem 1. For Dirichlet character, the Lseries has the Euler product where ζ (s) is the Riemann zeta function. At s = − k we obtain The relation (2.5) and (2.9) completes the proof.
Theorem 3 (Kummer's type Congruences). Let p be a prime number, p − | ̸ k and k ≡ k (mod (p − )p N ) with N being a nonnegative integer. Then we have Substituting t by (md + x)t in the last equality, we get The second equality is by Lebesgue's dominated convergence theorem. Therefore, we have where k ≥ is an integer.
Proof. By de nition (3.1), we have This completes the proof of the rst statement. Now by equation (3.2), we get On the other hand, we have Then f (t; n, x) is C ∞ on [ , ∞) and tending to zero rapidly at in nity. We de ne for k ≥ and k ∈ Z.
Obviously, f (t; n, x) is analytic around zero. Computing its Taylor expansion at , we get Therefore, From equations (3.4) and (3.6), we get This completes the proof of the nal statement. .

Modi ed Möbius L-functions
For n being a positive integer, let We call M k (n) modi ed Möbius-Bernoulli numbers.
In the following, we will rst show their relations with Bernoulli polynomials. For n = , So M k ( ) is essentially Bernoulli number B k . Now we consider the general case. Then we have Changing variable t to t/n, we get, Thus we have the following relation

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 a , a be nonzero real numbers. The double Bernoulli-Barnes numbers B k ((a , a )) are de ned through Theorem 7. Let n, n be positive integers and n * , n * be their square free parts, respectively. We have the following results. d )). Expanding the Taylor series of both sides yields the following identity Thus we obtain identity (4.3).

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

. 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 These sums are e ectively 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
From the above theorem we get the following identities.
Theorem 10. Let n, n be positive coprime integers and n * , n * be their square free parts, respectively. We have − kB k− δ ,n * n * − (k − )B k φ(n * n * ) n * n * , 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. 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− (a, b) are very easy to evaluate for small a and b. For example, using (5.1), we get values of s k− (a, b) in Table 1 with ≤ a, b ≤ and (a, b) = .