§27.10 Periodic Number-Theoretic Functions

If $k$ is a fixed positive integer, then a number-theoretic function $f$ is periodic (mod $k$) if

Examples are the Dirichlet characters (mod $k$) and the greatest common divisor $\left(n,k\right)$ regarded as a function of $n$.

Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form

where $g(m)$ is also periodic (mod $k$), and is given by

An example is Ramanujan’s sum:

where ${\chi }_1$ is the principal character (mod $k$). This is the sum of the $n$th powers of the primitive $k$th roots of unity. It can also be expressed in terms of the Möbius function as a divisor sum:

More generally, if $f$ and $g$ are arbitrary, then the sum

is a periodic function of $n(modk)$ and has the finite Fourier-series expansion

where

Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. It is defined by the relation

In particular, $G(n,{\chi }_1)=c_k\left(n\right)$.

$G(n,\chi )$ is separable for some $n$ if

For any Dirichlet character $\chi (modk)$, $G(n,\chi )$ is separable for $n$ if $\left(n,k\right)=1$, and is separable for every $n$ if and only if $G(n,\chi )=0$ whenever $\left(n,k\right)>1$. For a primitive character $\chi (modk)$, $G(n,\chi )$ is separable for every $n$, and

Conversely, if $G(n,\chi )$ is separable for every $n$, then $\chi$ is primitive (mod $k$).

The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form