Experimenting with Sums of Primitive Roots of Unity

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The primitive roots of unity are parametrized by the fractions
where
and
via the complex function
.
Contributed by: Jenda Vondra (July 2013)
The author is grateful to Ed Pegg Jr for assistance.
Open content licensed under CC BY-NC-SA
Snapshots
Details
Definitions
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
.
Demonstration
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
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.
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