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Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho, Palle E. T. Jorgensen (2015)

Special Matrices

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...

On ideals free of large prime factors

Eira J. Scourfield (2004)

Journal de Théorie des Nombres de Bordeaux

In 1989, E. Saias established an asymptotic formula for Ψ ( x , y ) = n x : p n p y with a very good error term, valid for exp ( log log x ) ( 5 / 3 ) + ϵ y x , x x 0 ( ϵ ) , ϵ > 0 . We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous function Ψ K ( x , y ) with the same error term and valid in the same region. Our main objective is to compare the formulae for Ψ ( x , y ) and Ψ K ( x , y ) , and in particular to compare the second term in the two expansions.

On the asymptotic behavior of some counting functions

Maciej Radziejewski, Wolfgang A. Schmid (2005)

Colloquium Mathematicae

The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class group satisfies...

On the asymptotic behavior of some counting functions, II

Wolfgang A. Schmid (2005)

Colloquium Mathematicae

The investigation of the counting function of the set of integral elements, in an algebraic number field, with factorizations of at most k different lengths gives rise to a combinatorial constant depending only on the class group of the number field and the integer k. In this paper the value of these constants, in case the class group is an elementary p-group, is estimated, and determined under additional conditions. In particular, it is proved that for elementary 2-groups these constants are equivalent...

Relative block semigroups and their arithmetical applications

Franz Halter-Koch (1992)

Commentationes Mathematicae Universitatis Carolinae

We introduce relative block semigroups as an appropriate tool for the study of certain phenomena of non-unique factorizations in residue classes. Thereby the main interest lies in rings of integers of algebraic number fields, where certain asymptotic results are obtained.

The distribution of powers of integers in algebraic number fields

Werner Georg Nowak, Johannes Schoißengeier (2004)

Journal de Théorie des Nombres de Bordeaux

For an arbitrary (not totally real) number field K of degree 3 , we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ ( τ ( γ p ) ) X for each embedding τ of K into the complex field. ( X a large real parameter, p 2 a fixed integer, and μ ( z ) = max ( | Re ( z ) | , | Im ( z ) | ) for any complex z .) This quantity is evaluated asymptotically in the form c p , K X n / p + R p , K ( X ) , with sharp estimates for the remainder R p , K ( X ) . The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation...

The Hooley-Huxley contour method for problems in number fields III : frobenian functions

Mark D. Coleman (2001)

Journal de théorie des nombres de Bordeaux

In this paper we study finite valued multiplicative functions defined on ideals of a number field and whose values on the prime ideals depend only on the Frobenius class of the primes in some Galois extension. In particular we give asymptotic results when the ideals are restricted to “small regions”. Special cases concern Ramanujan's tau function in small intervals and relative norms in “small regions” of elements from a full module of the Galois extension.

The size function h 0 for quadratic number fields

Paolo Francini (2001)

Journal de théorie des nombres de Bordeaux

We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function h 0 for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function k 0 ˜ , which is an analogue of exp h 0 defined on the class group, and we show it also assumes its...

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