John Knopfmacher

Fourier analysis of arithmetical functions

SummarySome contributions are made towards a Fourier-type theory of arithmetical functions, which, without appealing to classical notions of periodicity, in many ways parallels the Bohr and Besicovitch theories of ordinary almost periodic functions.

Counting polynomials with a given number of zeros in a finite field

Let denote a polynomial ring in an indeterminate X over a Unite field Exact formulae are derived for (i) the number of polynomials of degree n in with a specified number k of zeros in and (ii) the average number of zeros and corresponding variance for a polynomial of degree n in The main emphasis is on the case when multiplicity of zeros is counted.

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.

The exact length of the Euclidean algorithm in [ X ]

A study is made of the length L ( h, k ) of the Euclidean algorithm for determining the g.c.d. of two polynomials h , k in [ X ], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.

Generalised Euler constants

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form It has been discovered independently by many authors that, in terms of this notation, the coefficient

Counting irreducible factors of polynomials over a finite field

Abstract Let F q [X] denote a polynomial ring in an indeterminate X over a finite field F q . Exact formulae are derived for (i) the number of polynomials of degree n in F q [X] with a specified number of irreducible factors of a fixed degree r in F q [X] and (ii) the average number of such irreducible factors and corresponding variance for a polynomial of degree n in F q [X] . The main emphasis is on the case when multiplicity of factors is counted. These results are then applied to derive the mean and variance for the total number of irreducible factors of polynomials of degree n in F q [X] .

Arithmetical Semigroups Related to Trees and Polyhedra

With emphasis on some natural asymptotic enumeration questions, a study is made of various arithmetical semigroups associated with isomorphism classes of finite graphs, trees and polyhedra. A suitable “abstract prime number theorem” is derived, particularly as an aid to solving the counting questions stated.

Inverse polynomial expansions of Laurent series II

An algorithm is considered, and shown to lead to various unusual and unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also partially characterized.

Arithmetical semigroups I: direct factors

There are two main theorems due to P. Erdös et al. on direct factors of IN. They are concerned with the asymptotic density and the distribution of primes. The concept of a direct factor is carried over to arithmetical semigroups as they were introduced by J. Knopfmacher and the two corresponding main theorems are stated and proved.

Note on Finite Topological Spaces

In a recent paper [4], H. Sharp, Jr., has discussed the problem of finding formulae for the following naturally defined integers: the numbers t ( n ), tc ( n ), t 0 ( n ), tc 0 ( n ), and ts ( n ) of all homeomorphism classes of finite topological spaces with n elements, which are respectively (i) arbitrary, (ii) connected, (iii) T 0 , (iv) connected and T 0 , (v) symmetric. Here, a finite topological space X is called symmetric provided the following relation ≦ is symmetric: x ≦ y if and only if x ∈ U v , the intersection of all open sets containing y .