Experimenting with Sums of Primitive Roots of Unity

The primitive roots of unity are parametrized by the fractions where and via the complex function .
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 .
This Demonstration computes two sums over these fractions and computes a subsequent sign function that mimics the Möbius function for .
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 and suggests parametrization of the summation of the Möbius function by asymmetry in the distribution of the Farey sequence about .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
[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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+