On the Fourier transform of the greatest common divisor

TL;DR

This paper investigates the discrete Fourier transform of the greatest common divisor, revealing its properties, connections to classical functions, and generalizing known identities involving the totient function and Ramanujan's sum.

Contribution

It introduces a new perspective on the Fourier transform of gcd, linking it to classical number-theoretic functions and generalizing several known identities.

Findings

The Fourier transform of gcd counts specific ordered pairs modulo m.

It expresses as a Dirichlet convolution involving Ramanujan's sum.

Generalizes identities for the totient function to the Fourier transform context.

Abstract

The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan's sum, and on the other hand it can be written as a generalised convolution product of the identity with the totient function. We show that this arithmetic function of two integers (a,m) counts the number of elements in the set of ordered pairs (i,j) such that i*j is equivalent to a modulo m. Furthermore we generalise a dozen known identities for the totient function, to identities which involve the discrete Fourier transform of the greatest common divisor, including its partial sums, and its Lambert series.