On the average number of elements in a finite field with order or index in a prescribed residue class

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.Comment: 25 pages, 4 tables. A conjecture made in the previous version is now resolved. Tables are also improved, thanks to a C++ program written by Yves Gallo

The lengths of Hermitian Self-Dual Extended Duadic Codes

Duadic codes are a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. For every prime power q, we characterize the integers n such that over the finite field with q^2 elements there is a duadic code of length n having an Hermitian self-dual parity-check extension. We derive using analytic number theory asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.Comment: To appear in the Journal of Pure and Applied Algebra. 21 pages and 1 Table. Corollary 4.9 and Theorem 5.8 have been added. Some small changes have been mad

Nicolaas Govert de Bruijn, the enchanter of friable integers

N.G. de Bruijn carried out fundamental work on integers having only small prime factors and the Dickman-de Bruijn function that arises on computing the density of those integers. In this he used his earlier work on linear functionals and differential-difference equations. We review the relevant work and also some later improvements by others.Comment: 34 pages, 1 Figur

Counting carefree couples

A pair of natural numbers (a,b) such that a is both squarefree and coprime to b is called a carefree couple. A result conjectured by Manfred Schroeder (in his book `Number theory in science and communication') on carefree couples and a variant of it are established using standard arguments from elementary analytic number theory. Also a related conjecture of Schroeder on triples of integers that are pairwise coprime is proved.Comment: Updated version of 2005 update of 2000 version. Improved and expanded presentation. In estimate (2) now only a weaker error term than before is obtaine