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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...

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