scholarly journals Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1710
Author(s):  
Nicuşor Minculete ◽  
Diana Savin
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Generalized Function ◽  
Algebraic Number Fields ◽  
Euler’S Function

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

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 A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

scholarly journals K-RATIONAL D-BRANE CRYSTALS

2012 ◽  
Vol 27 (22) ◽  
pp. 1250112
Author(s):  
ROLF SCHIMMRIGK
Keyword(s):  
String Theory ◽  
Algebraic Number ◽  
Special Class ◽  
Building Blocks ◽  
Number Fields ◽  
Central Point ◽  
Algebraic Number Fields ◽  
Special Values ◽  
Toroidal Compactifications ◽  
Base Points

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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