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

.