# Experimenting with Sums of Primitive Roots of Unity

Experimenting with Sums of Primitive Roots of Unity

The primitive roots of unity are parametrized by the fractions where and via the complex function .

th

n

k/n

1≤k≤n

gcd(k,n)=1

exp(2πik/n)

Their sum is , where is the Möbius function. Conjugates are symmetric in the real axis, so is parametrized by only the fractions where and via the real function .

μ(n)

μ

μ(n)

k/n

1≤k<n/2

gcd(k,n)=1

2cos(2πk/n)

This Demonstration computes two sums over these fractions and computes a subsequent sign function that mimics the Möbius function for .

δ

n<~10^6

The set of fractions for all denominators up to are the elements of the Farey sequence of order that are less than .

δ

n

n

1

2

Observing the symmetries of and suggests parametrization of the summation of the Möbius function by asymmetry in the distribution of the Farey sequence about .

cos(2πδ)

1-4δ

1

4