scholarly journals Arithmetical properties of double Möbius-Bernoulli numbers

2019 ◽  
Vol 17 (1) ◽  
pp. 32-42
Author(s):  
Abdelmejid Bayad ◽  
Daeyeoul Kim ◽  
Yan Li
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Critical Values ◽  
Möbius Function ◽  
Dedekind Sums ◽  
Mobius Function ◽  
Dirichlet Type ◽  
Positive Integers

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 $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

scholarly journals An Analogue of a Result of Jacobsthal

1962 ◽  
Vol 13 (2) ◽  
pp. 139-142 ◽  
Author(s):  
Eckford Cohen
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Image Position ◽  
Positive Integers

Jacobsthal (4)has proved that the n×n matrixis invertible with the inverse,Here μ(x) denotes the Möbius function for positive integral x and is assumed to be 0 for other values; [x] has its usual meaning as the number of positive integers ≦x.

scholarly journals Sums of Products Involving Power Sums of φ(n) Integers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jitender Singh
Keyword(s):  
Closed Form ◽  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Rational Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Power Sums ◽  
Power Sum ◽  
Mobius Function ◽  
Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d,  n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

A Congruence for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (05) ◽  
pp. 571-574 ◽  
Author(s):  
M.V. Subbarao
Keyword(s):  
Fixed Points ◽  
Möbius Function ◽  
Arithmetic Functions ◽  
Left Member ◽  
Mobius Function ◽  
Image Position ◽  
Positive Integers

There is considerable literature concerning the century old result that for arbitrary positive integers a and m, 1.1 where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Does the Möbius Function Determine Multiplicative Arithmetic?

1995 ◽  
Vol 102 (4) ◽  
pp. 354-356
Author(s):  
D. Flath ◽  
A. Zulauf
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Multiplicative Arithmetic

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals From explicit estimates for primes to explicit estimates for the Möbius function

2013 ◽  
Vol 157 (4) ◽  
pp. 365-379 ◽  
Author(s):  
Olivier Ramaré
Keyword(s):  
Möbius Function ◽  
Mobius Function
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