Asymptotic free probability for arithmetic functions and factorization of Dirichlet series

We derive two new generalizations of the Busche–Ramanujan identities involving the multiple Dirichlet convolution of arithmetic functions of several variables. The proofs use formal multiple Dirichlet series and properties of symmetric polynomials of several variables.

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Some important Dirichlet series and arithmetic functions

The Prime Number Theorem ◽

10.1017/cbo9781139164986.003 ◽

2003 ◽

pp. 56-97

Keyword(s):

Dirichlet Series ◽

Arithmetic Functions

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Zeta-regularization of arithmetic sequences

EPJ Web of Conferences ◽

10.1051/epjconf/202024401008 ◽

2020 ◽

Vol 244 ◽

pp. 01008

Author(s):

Jean-Paul Allouche

Keyword(s):

Zeta Function ◽

Analytic Continuation ◽

Dirichlet Series ◽

Infinite Product ◽

The Other ◽

Arithmetic Functions ◽

Finite Product ◽

Arithmetic Sequences ◽

Classical Arithmetic

Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity.

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On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

Open Mathematics ◽

10.2478/s11533-012-0143-2 ◽

2013 ◽

Vol 11 (3) ◽

Cited By ~ 1

Author(s):

Manfred Kühleitner ◽

Werner Nowak

Keyword(s):

Dirichlet Series ◽

Lower Bounds ◽

Remainder Term ◽

Arithmetic Functions ◽

Euler Product ◽

Arbitrary Integer

AbstractThe paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

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Dirichlet product and the multiple Dirichlet series over function fields

Glasnik Matematicki ◽

10.3336/gm.55.2.06 ◽

2020 ◽

Vol 55 (2) ◽

pp. 253-265

Author(s):

Yoshinori Hamahata ◽

Keyword(s):

Dirichlet Series ◽

Function Fields ◽

Arithmetic Functions ◽

Multiple Dirichlet Series ◽

Dirichlet Product

We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.

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