Experimenting with Sums of Primitive Roots of Unity

The roots of unity are the solutions of the equation . The solutions lie equally spaced around the unit circle.

A solution is called primitive if for some , which occurs when is coprime to [1]. The primitive roots of unity are given by for , so there are , where is the Euler totient function [2].

The Möbius function is defined to be if is the product of distinct primes, and zero otherwise, with [3].

Ramanujan's sum for gives for all positive integers [4].

The Farey sequence of order is the set of irreducible fractions between 0 and 1 with denominators less than or equal to , arranged in increasing order [5].

For the purpose of this Demonstration, we omit the fraction 0/1 and truncate the sequence to include only the fractions less than . The sequence is . The truncated Farey sequence is plotted as the black "barcode" and is scaled to the real interval where the fractions are those Farey fractions with denominator and the remaining bars are the Farey fractions with denominators less than .

Euler's formula (for as before) plots the primitive roots of unity on the unit circle in the complex plane. Upon summing, their symmetries allow cancellation of the imaginary parts and duplication of the real parts to give the shortened summation

, where stands here and in the following.

In the result, this sum is shown as , where δ denotes the fraction .

Plotting the points illustrates how this sum is a measure of their asymmetry about the point zero, seemingly miraculously always equal to or for any natural number .

The Demonstration then plots the points with a visual link to each point and computes the comparable sum .

Computations for up to show that the sign of matches the Möbius function for squarefree and the sign of (where is the floor function) matches the Möbius function for non-squarefree and is positive for squarefree .

The product of the sign functions gives an algorithm that returns the Möbius function value;

for .

Writing gives the Möbius function in terms of the totient function and the partial sum of the coprime numbers (to ) that it counts;

for .

Ranging through selected denominators shows the fractions that form the truncated Farey sequence .

Let be the Farey sequence of order ; a classical sum [6] of the primitive roots of unity gives

, which reduces to .

The analogous sum correlates to the classic discrepancy sum that measures distribution in the Farey sequence .

The sum measures asymmetry in the distribution of the Farey sequence about .

From the well-known uniform distribution modulo one of the Farey sequence, it follows that tends to zero as tends to infinity.

References

[1] Wikipedia. "Root of Unity." (May 8, 2013) en.wikipedia.org/wiki/Root_of_unity.

[2] Wikipedia. "Euler's Totient Function. (Jun 13, 2013) en.wikipedia.org/wiki/Euler%27s_totient_function.

[3] Wikipedia. "Möbius Function." (Jun 17, 2013) en.wikipedia.org/wiki/Mobius_function.

[4] Wikipedia. "Ramanujan's Sum." (Mar 19, 2013) en.wikipedia.org/wiki/Ramanujan%27s_sum.

[5] Wikipedia. "Farey Sequence." (Jun 9, 2013) en.wikipedia.org/wiki/Farey_sequence.

[6] Wikipedia. "Mertens Function." (May 3, 2013) en.wikipedia.org/wiki/Mertens_function.

[7] M. N. Huxley, "The Distribution of Farey Points, I," Acta Arithmetica, 18, 1971 pp. 281–287. matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18130.pdf.