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Experimenting with Sums of Primitive Roots of Unity

δ is any fraction k/n with numerator k < n/2 coprime to denominator n

The primitive

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

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

and computes a subsequent sign function that mimics the Möbius function for

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

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

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