Some Generalizations of Totient Function with Elementary Symmetric Sums

TL;DR

This paper extends totient functions using elementary symmetric polynomials, providing explicit product forms and exploring their properties, with applications to polynomial zero counting and linear congruences.

Contribution

It introduces new generalizations of totient functions based on elementary symmetric sums and links these to polynomial zero counts and linear congruence solutions.

Findings

Derived explicit product forms for generalized totient functions.

Established equivalence between totient generalizations, polynomial zero counts, and linear congruence solutions.

Provided a method for solving restricted linear congruences with gcd constraints.

Abstract

We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .