Residue Classes Having Tardy Totients

The Distribution of Totients

This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φ-function. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) = m (the “multiplicity” of m), and Vk(x), the number of m ≤ x with A(m) = k. The first of the main results of the paper is a determination of the t...

A Survey of Single Machine Scheduling to Minimize Weighted Number of Tardy Jobs

This paper presents a review of single machine scheduling to minimize the weighted number of tardy jobs. The problem involves processing n jobs on a single machine, each having processing time pj and due date dj . The aim is to schedule the jobs to meet their due date. A job is tardy if the completion time of job j is Cj > dj and on-time otherwise. This paper assesses works done to minimize the...

On the Distribution of Perfect Totients

In this paper, we study the sum of iterates of the Euler function.

A Note on Square Totients

Extremal Orders of Certain Functions Associated with Regular Integers (mod n)

Let V (n) denote the number of positive regular integers (mod n) that are ≤ n, and let Vk(n) be a multidimensional generalization of the arithmetic function V (n). We find the Dirichlet series of Vk(n) and give the extremal orders of some totients involving arithmetic functions which generalize the sum-of-divisors function and the Dedekind function. We also give the extremal orders of other tot...