**Volume 9** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some characteristics of primes within modular rings**

*Original research paper. Pages 49—58*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 3999 Kb) | Abstract

The squares of primes in the Modular Ring Z_{4} fall in even rows, *R*_{1}, in Class ̅1_{4} with *R*_{1} = 6*K*_{j} and *K*_{j} = ½ *n*(3*n* ± 1) (depending on the parity of *j*). The *n* values are found to equal the rows that the primes occupy when Z_{6} is set as a tabular array. The primes in Z_{4} equal a unique sum or difference of squares (*x*^{2} ± *y*^{2}) and via *n*, the (*x*, *y*) pairs can be identified within the Z_{6} structure. The *n* values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter n in the well-known function *P* = *h*2^{n} − 1 has been identified as the row in one or more of the Modular Rings Z_{10}, Z_{6} or Z_{4} that contains one or more primes.

**Properties of the restrictive factor**

*Original research paper. Pages 59—61*

Laurențiu Panaitopol

Full paper (PDF, 1087 Kb) | Abstract

In

Krassimir Atanassov has introduced the arithmetic function RF(n) defined by RF(1) = 1 and RF(

*n*) =

*q*_{1}^{α1−1}*q*_{2}^{α2−1}…

*q*_{k}^{αk−1} whenever

*n* =

*q*_{1}^{α1} *q*_{2}^{α2}…

*q*_{k}^{αk}, where

*q*_{1}, …,

*q*_{k} are pairwise different prime numbers and we have

*α*_{1}…

*α*_{k} ≥ 1. The function RF(

*n*) is called the restrictive factor. In the present paper we denote it simply by R(

*n*).

**A property of an arithmetic function**

*Original research paper. Pages 62—64*

Krassimir T. Atanassov

Full paper (PDF, 903 Kb) | Abstract

A digital arithmetical function described in

will be defined and new its properties will be described.

**Iterated Dirichlet series and the inverse of Ramanujan’s sum**

*Original research paper. Pages 65—72*

Kenneth R. Johnson

Full paper (PDF, 2246 Kb) | Abstract

The theory of Dirichlet series having number theoretic functions of a single variable as coefficients has a rich history. In this paper we present a parallel theory for iterated Dirichlet series with number theoretic functions of two variables as coefficients and find the Dirichlet product inverse of Ramanujan’s sum. The results presented here are easily accessible to an Advanced Calculus or undergraduate Number Theory course.

**Volume 9** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4