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27 Functions of Number TheoryMultiplicative Number Theory

§27.10 Periodic Number-Theoretic Functions

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

27.10.1 f(n+k)=f(n),

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

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

27.10.2 f(n)=m=1kg(m)e2πimn/k,

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

27.10.3 g(m)=1kn=1kf(n)e-2πimn/k.

An example is Ramanujan’s sum:

27.10.4 ck(n)=m=1kχ1(m)e2πimn/k,

where χ1 is the principal character (mod k). This is the sum of the nth powers of the primitive kth roots of unity. It can also be expressed in terms of the Möbius function as a divisor sum:

27.10.5 ck(n)=d|(n,k)dμ(kd).

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

27.10.6 sk(n)=d|(n,k)f(d)g(kd)

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

27.10.7 sk(n)=m=1kak(m)e2πimn/k,


27.10.8 ak(m)=d|(m,k)g(d)f(kd)dk.

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

27.10.9 G(n,χ)=m=1kχ(m)e2πimn/k.

In particular, G(n,χ1)=ck(n).

G(n,χ) is separable for some n if

27.10.10 G(n,χ)=χ¯(n)G(1,χ).

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

27.10.11 |G(1,χ)|2=k.

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

The finite Fourier expansion of a primitive Dirichlet character χ(modk) has the form

27.10.12 χ(n)=G(1,χ)km=1kχ¯(m)e-2πimn/k.