Prime number decomposition of the Fourier transform of a function of the greatest common divisor

TL;DR

This paper derives an explicit prime factorization formula for the Fourier transform of functions of the greatest common divisor, revealing new relations involving the Euler totient function.

Contribution

It provides a novel explicit prime decomposition of the Fourier transform of GCD-based functions and generalizes the known multiplicative property.

Findings

Explicit prime factorization formula for Fourier transform of GCD functions

Generalization to Fourier transforms of functions of GCD

New relations involving Euler totient function

Abstract

The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the prime factors of the argument, the explicit form of which is proven in this paper. Subsequently it is shown how the procedure can be generalized to the discrete Fourier transform of a function of the greatest common divisor. From this representation some interesting relations concerning the Euler totient function and generalizations thereof are established.