scholarly journals On a generalization of Hamburger’s theorem

1985 ◽  
Vol 98 ◽  
pp. 67-76 ◽  
Author(s):  
Akinori Yoshimoto
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Summation Formula ◽  
Number Fields ◽  
Gaussian Field ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Poisson’S Summation Formula ◽  
The Relationship

The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grössencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grössencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grössencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

scholarly journals Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1222
Author(s):  
Nicuşor Minculete ◽  
Diana Savin
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Point Of View ◽  
Algebraic Number Fields ◽  
Algebraic Point ◽  
Ring Of Integers ◽  
Euler’S Function

In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.

scholarly journals Units in families of totally complex algebraic number fields

2004 ◽  
Vol 2004 (45) ◽  
pp. 2383-2400
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Algebraic Number ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Number Fields ◽  
Imaginary Quadratic Field ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Multidimensional Continued Fraction ◽  
Fundamental Units

Multidimensional continued fraction algorithms associated withGLn(ℤk), whereℤkis the ring of integers of an imaginary quadratic fieldK, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.

scholarly journals On partial metric preserving functions and their characterization

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2315-2327
Author(s):  
Juan-José Miñana ◽  
Oscar Valero
Keyword(s):  
Functional Equation ◽  
Partial Metric ◽  
Generalized Metric ◽  
Metric Functions ◽  
Partial Metrics ◽  
The Relationship

In 1981, J. Bors?k and J. Dob?s characterized those functions that allow to transform a metric into another one in such a way that the topology of the metric to be transformed is preserved. Later on, in 1994, S.G. Matthews introduced a new generalized metric notion known as partial metric. In this paper, motivated in part by the applications of partial metrics, we characterize partial metric-preserving functions, i.e., those functions that help to transform a partial metric into another one. In particular we prove that partial metric-preserving functions are exactly those that are strictly monotone and concave. Moreover, we prove that the partial metric-preserving functions preserving the topology of the transformed partial metric are exactly those that are continuous. Furthermore, we give a characterization of those partial-metric preserving functions which preserve completeness and contractivity. Concretely, we prove that completeness is preserved by those partial metric-preserving functions that are non-bounded, and contractivity is kept by those partial metric-functions that satisfy a distinguished functional equation involving contractive constants. The relationship between metric-preserving and partial metric-preserving functions is also discussed. Finally, appropriate examples are introduced in order to illustrate the exposed theory.

scholarly journals A note on the characterization of CM-fields

1978 ◽  
Vol 26 (1) ◽  
pp. 26-30 ◽  
Author(s):  
P. E. Blanksby ◽  
J. H. Loxton
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Subject Classification ◽  
Algebraic Number Fields

AbstractThis note deals with some properties of algebraic number fields generated by numbers having all their conjugates on a circle. In particular, it is shown that an algebraic number field is a CM-field if and only if it is generated over the rationals by an element (not equal to ±1) whose conjugate all lie on the unit circle.Subject classification (Amer. Math. Soc. (MOS) 1970): 12 A 15, 12 A 40, 14 K 22.

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