WOLFRAM|DEMONSTRATIONS PROJECT

Experimenting with Sums of Primitive Roots of Unity

δ is any fraction k/n with numerator k < n/2 coprime to denominator n
3
The primitive
th
n
roots of unity are parametrized by the fractions
k/n
where
1≤k≤n
and
gcd(k,n)=1
via the complex function
exp(2πik/n)
.
Their sum is
μ(n)
, where
μ
is the Möbius function. Conjugates are symmetric in the real axis, so
μ(n)
is parametrized by only the fractions
k/n
where
1≤k<n/2
and
gcd(k,n)=1
via the real function
2cos(2πk/n)
.
This Demonstration computes two sums over these fractions
δ
and computes a subsequent sign function that mimics the Möbius function for
n<~10^6
.
The set of fractions
δ
for all denominators up to
n
are the elements of the Farey sequence of order
n
that are less than
1
2
.
Observing the symmetries of
cos(2πδ)
and
1-4δ
suggests parametrization of the summation of the Möbius function by asymmetry in the distribution of the Farey sequence about
1
4
.